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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/CNAME b/CNAME new file mode 100644 index 00000000..7daf76d9 --- /dev/null +++ b/CNAME @@ -0,0 +1 @@ +hypercubing.xyz \ No newline at end of file diff --git a/MPU_puzzles/16-cell_FT.txt b/MPU_puzzles/16-cell_FT.txt new file mode 100644 index 00000000..df933be5 --- /dev/null +++ b/MPU_puzzles/16-cell_FT.txt @@ -0,0 +1,9 @@ +Puzzle 16-cell_FT +Dim 4 +NAxis 1 +Faces 1,1,1,1 +Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1 +Axis 1,1,1,1 +Twists 0,2,-1,-1/0,1,1,-2 1,-1,0,0/0,0,1,-1 +Cuts 0.6 -0.6 +FixedMask 2 \ No newline at end of file diff --git a/MPU_puzzles/1x1x1x1x2.txt b/MPU_puzzles/1x1x1x1x2.txt new file mode 100644 index 00000000..e7d65af5 --- /dev/null +++ b/MPU_puzzles/1x1x1x1x2.txt @@ -0,0 +1,11 @@ +Puzzle 1x1x1x1x2 +Dim 5 +NAxis 2 +Faces 0.5,0,0,0,0 0,0,0,0,1 +Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/1,0,0,1,0 1,0,0,0,0/0,0,0,0,1 +Axis 1,0,0,0,0 +Twists 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,0,0,0,1 +Cuts +Axis 0,0,0,0,1 +Twists 1,0,0,0,0/1,1,0,0,0 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/2x1x1x1x1.txt b/MPU_puzzles/2x1x1x1x1.txt new file mode 100644 index 00000000..322b51b7 --- /dev/null +++ b/MPU_puzzles/2x1x1x1x1.txt @@ -0,0 +1,23 @@ +Puzzle 1x1x1x1x2 +Dim 5 +NAxis 2 +Faces 0.5,0,0,0,0 0,0,0,0,1 +Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/1,0,0,1,0 1,0,0,0,0/0,0,0,0,1 +Axis 1,0,0,0,0 +Twists 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,0,0,0,1 +Cuts +Axis 0,0,0,0,1 +Twists 1,0,0,0,0/1,1,0,0,0 +Cuts 0 + +Puzzle 1x1x1x2x2 +Dim 5 +NAxis 2 +Faces 0.5,0,0,0,0 0,0,0,1,0 +Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1 +Axis 1,0,0,0,0 +Twists 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1 +Cuts +Axis 0,0,0,1,0 +Twists 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/0,0,0,0,1 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/2x2x1x1.txt b/MPU_puzzles/2x2x1x1.txt new file mode 100644 index 00000000..64766452 --- /dev/null +++ b/MPU_puzzles/2x2x1x1.txt @@ -0,0 +1,5 @@ +Puzzle 2x2x1x1 +Dim 0 +NAxis 2 +Faces 1,0,0,0 0,0,0,0.5 +Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,1,1 \ No newline at end of file diff --git a/MPU_puzzles/2x2x2x1.txt b/MPU_puzzles/2x2x2x1.txt new file mode 100644 index 00000000..44f134c3 --- /dev/null +++ b/MPU_puzzles/2x2x2x1.txt @@ -0,0 +1,11 @@ +Puzzle 2x2x2x1 +Dim 4 +NAxis 2 +Faces 1,0,0,0 0,0,0,0.5 +Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1 +Cuts 0 0 +Axis 0,0,0,1 +Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 +Cuts \ No newline at end of file diff --git a/MPU_puzzles/2x2x2x1x1.txt b/MPU_puzzles/2x2x2x1x1.txt new file mode 100644 index 00000000..959ef068 --- /dev/null +++ b/MPU_puzzles/2x2x2x1x1.txt @@ -0,0 +1,11 @@ +Puzzle 1x1x2x2x2 +Dim 5 +NAxis 2 +Faces 0.5,0,0,0,0 0,0,1,0,0 +Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0 0,0,1,0,0/0,0,1,0,1 +Axis 1,0,0,0,0 +Twists 0,1,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0 +Cuts +Axis 0,0,1,0,0 +Twists 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/2x2x2x2x1.txt b/MPU_puzzles/2x2x2x2x1.txt new file mode 100644 index 00000000..a69bdbc6 --- /dev/null +++ b/MPU_puzzles/2x2x2x2x1.txt @@ -0,0 +1,11 @@ +Puzzle 1x2x2x2x2 +Dim 5 +NAxis 2 +Faces 0.5,0,0,0,0 0,1,0,0,0 +Group 1,0,0,0,0/0,1,0,0,0 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,1,0,1,0 0,1,0,0,0/0,1,0,0,1 +Axis 1,0,0,0,0 +Twists 0,1,0,0,0/0,1,1,0,0 +Cuts +Axis 0,1,0,0,0 +Twists 1,0,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/2x2x2x2x2.txt b/MPU_puzzles/2x2x2x2x2.txt new file mode 100644 index 00000000..3b38bd91 --- /dev/null +++ b/MPU_puzzles/2x2x2x2x2.txt @@ -0,0 +1,8 @@ +Puzzle 2^5 +Dim 5 +NAxis 1 +Faces 1,0,0,0,0 +Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/1,0,0,1,0 1,0,0,0,0/1,0,0,0,1 +Axis 1,0,0,0,0 +Twists 0,1,0,0,0/0,1,1,0,0 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/2x2x2x3.txt b/MPU_puzzles/2x2x2x3.txt new file mode 100644 index 00000000..1c21b62c --- /dev/null +++ b/MPU_puzzles/2x2x2x3.txt @@ -0,0 +1,11 @@ +Puzzle 2x2x2x3 +Dim 4 +NAxis 2 +Faces 1,0,0,0 0,0,0,1.5 +Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1 +Cuts 0 +Axis 0,0,0,1 +Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 +Cuts 0.5 -0.5 \ No newline at end of file diff --git a/MPU_puzzles/2x2x3x3.txt b/MPU_puzzles/2x2x3x3.txt new file mode 100644 index 00000000..79fb1968 --- /dev/null +++ b/MPU_puzzles/2x2x3x3.txt @@ -0,0 +1,11 @@ +Puzzle 2x2x3x3 +Dim 4 +NAxis 2 +Faces 1,0,0,0 0,0,1.5,0 +Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1 +Axis 1,0,0,0 +Twists 0,0,1,0/0,0,1,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0,1,1 +Cuts 0 +Axis 0,0,1,0 +Twists 1,0,0,0/1,1,0,0 0,0,0,1/1,0,0,0 0,0,0,1/1,1,0,0 +Cuts 0.5 -0.5 \ No newline at end of file diff --git a/MPU_puzzles/2x2x3x4.txt b/MPU_puzzles/2x2x3x4.txt new file mode 100644 index 00000000..efed398d --- /dev/null +++ b/MPU_puzzles/2x2x3x4.txt @@ -0,0 +1,14 @@ +Puzzle 2x2x3x4 +Dim 4 +NAxis 3 +Faces 1,0,0,0 0,0,1.5,0 0,0,0,2 +Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1 +Cuts 0 +Axis 0,0,1,0 +Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0 +Cuts 0.5 -0.5 +Axis 0,0,0,1 +Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0 +Cuts 1 0 -1 \ No newline at end of file diff --git a/MPU_puzzles/2x3x3x3.txt b/MPU_puzzles/2x3x3x3.txt new file mode 100644 index 00000000..b2b660c4 --- /dev/null +++ b/MPU_puzzles/2x3x3x3.txt @@ -0,0 +1,11 @@ +Puzzle 2x3x3x3 +Dim 4 +NAxis 2 +Faces 1.5,0,0,0 0,0,0,1 +Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1 +Cuts -0.5 0.5 +Axis 0,0,0,1 +Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/2x3x4x5.txt b/MPU_puzzles/2x3x4x5.txt new file mode 100644 index 00000000..3dcda35b --- /dev/null +++ b/MPU_puzzles/2x3x4x5.txt @@ -0,0 +1,17 @@ +Puzzle 2x3x4x5 +Dim 4 +NAxis 4 +Faces 1,0,0,0 0,1.5,0,0 0,0,2,0 0,0,0,2.5 +Group 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1 +Cuts 0 +Axis 0,1,0,0 +Twists 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,0,1,0/0,0,0,1 +Cuts 0.5 -0.5 +Axis 0,0,1,0 +Twists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0 +Cuts 1 0 -1 +Axis 0,0,0,1 +Twists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0 +Cuts 1.5 0.5 -0.5 -1.5 \ No newline at end of file diff --git a/MPU_puzzles/3D.txt b/MPU_puzzles/3D.txt new file mode 100644 index 00000000..8f7b13fb --- /dev/null +++ b/MPU_puzzles/3D.txt @@ -0,0 +1,21 @@ +Block 3D + +Puzzle 3x3x3 +Dim 3 +NAxis 1 +Faces 1,0,0 +Group 1,0,0/1,1,0 1,0,0/1,0,1 +Axis 1,0,0 +Twists 0,1,0/0,1,1 +Cuts -0.33 0.33 + +Puzzle Skewb +Dim 3 +NAxis 1 +Faces 1,0,0 +Group 1,0,0/1,1,0 1,0,0/1,0,1 +Axis 1,1,1 +Twists 1,-1,0/1,0,-1 +Cuts 0 + +EndBlock \ No newline at end of file diff --git a/MPU_puzzles/3^4_Skewb.txt b/MPU_puzzles/3^4_Skewb.txt new file mode 100644 index 00000000..7e538bdc --- /dev/null +++ b/MPU_puzzles/3^4_Skewb.txt @@ -0,0 +1,9 @@ +Puzzle 3^4 Skewb +Dim 4 +NAxis 1 +Faces 1,0,0,0 +Simplified +Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1 +Axis 1,1,1,1 +Twists 0,2,-1,-1/0,1,1,-2 1,-1,0,0/0,0,1,-1 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/3x3x3x1.txt b/MPU_puzzles/3x3x3x1.txt new file mode 100644 index 00000000..2e1baa1d --- /dev/null +++ b/MPU_puzzles/3x3x3x1.txt @@ -0,0 +1,11 @@ +Puzzle 3x3x3x1 +Dim 4 +NAxis 2 +Faces 1.5,0,0,0 0,0,0,0.5 +Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1 +Cuts 0.5 -0.5 +Axis 0,0,0,1 +Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 +Cuts \ No newline at end of file diff --git a/MPU_puzzles/4x4x4x4.txt b/MPU_puzzles/4x4x4x4.txt new file mode 100644 index 00000000..c4ae3b9e --- /dev/null +++ b/MPU_puzzles/4x4x4x4.txt @@ -0,0 +1,8 @@ +Puzzle 4^4 +Dim 4 +NAxis 1 +Faces 1,0,0,0 +Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,1,1,0 0,1,-1,0/0,0,0,1 0,2,-1,-1/0,1,1,-2 +Cuts 0.5 0 -0.5 \ No newline at end of file diff --git a/MPU_puzzles/5-5_Duotegum.txt b/MPU_puzzles/5-5_Duotegum.txt new file mode 100644 index 00000000..14fa2004 --- /dev/null +++ b/MPU_puzzles/5-5_Duotegum.txt @@ -0,0 +1,8 @@ +Puzzle 5-5_Duotegum +Dim 4 +NAxis 1 +Faces -1,1,0,0 +Group 1,0,0,0/0.809016994,0,0.587785252,0 1,1,0,0/0,0,1,1 +Axis -1,1,0,0 +Twists 0,0,1,0/0,0,0,1 1,1,0,0/0,0,1,1 1,1,0,0/0,0,-1,1 +Cuts 0.65 \ No newline at end of file diff --git a/MPU_puzzles/Chamfered_Pentagonal_Duoprism.txt b/MPU_puzzles/Chamfered_Pentagonal_Duoprism.txt new file mode 100644 index 00000000..65ae601f --- /dev/null +++ b/MPU_puzzles/Chamfered_Pentagonal_Duoprism.txt @@ -0,0 +1,11 @@ +Puzzle Chamfered_Pentagonal_Duoprism +Dim 4 +NAxis 2 +Faces -1.41429,0,0,0 1.41429,0,0,0 -1,1,0,0 +Group 1,0,0,0/0.809016994,0,0.587785252,0 1,1,0,0/0,0,1,1 +Axis -1,0,0,0 +Twists 0,1,0,0/0,0.809016994,0,0.587785252 0,0,1,0/0,0,0,1 +Cuts 1.3 -1.23 +Axis -1,1,0,0 +Twists 0,0,1,0/0,0,0,1 1,1,0,0/0,0,1,1 1,1,0,0/0,0,-1,1 +Cuts 0.85 \ No newline at end of file diff --git a/MPU_puzzles/Champered_Tesseract.txt b/MPU_puzzles/Champered_Tesseract.txt new file mode 100644 index 00000000..a76c4bc6 --- /dev/null +++ b/MPU_puzzles/Champered_Tesseract.txt @@ -0,0 +1,11 @@ +Puzzle Chamfered_Tesseract +Dim 4 +NAxis 2 +Faces 1,0,0,0 0.70710678,0.70710678,0,0 +Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,1,0,1 0,0,1,0/0,0,1,1 +Cuts 0.85 -0.85 +Axis 1,1,0,0 +Twists 0,0,1,0/0,0,1,1 1,-1,0,0/0,0,1,0 1,-1,0,0/0,0,1,1 +Cuts 0.57 -0.57 \ No newline at end of file diff --git a/MPU_puzzles/Compy_Rainbow.txt b/MPU_puzzles/Compy_Rainbow.txt new file mode 100644 index 00000000..35e21661 --- /dev/null +++ b/MPU_puzzles/Compy_Rainbow.txt @@ -0,0 +1,9 @@ +Puzzle Compy_Rainbow +Dim 3 +NAxis 1 +Faces 1,0,0 0.6667,0.6667,0.6667 +Group 1,0,0/1,1,0 1,0,0/1,0,1 +Axis 1,1,1 +Twists 1,-1,0/1,0,-1 +Cuts -0.45 0.45 +FixedMask 2 \ No newline at end of file diff --git a/MPU_puzzles/Cuboctahedron.txt b/MPU_puzzles/Cuboctahedron.txt new file mode 100644 index 00000000..b815f0f6 --- /dev/null +++ b/MPU_puzzles/Cuboctahedron.txt @@ -0,0 +1,11 @@ +Puzzle Cuboctahedron +Dim 3 +NAxis 2 +Faces 1,0,0 0.667,0.667,0.667 +Group 1,0,0/1,1,0 1,0,0/1,0,1 +Axis 1,0,0 +Twists 0,1,0/0,1,1 +Cuts 0.5 -0.5 +Axis 1,1,1 +Twists 1,-1,0/1,0,-1 +Cuts 0.5 -0.5 \ No newline at end of file diff --git a/MPU_puzzles/Lunas_pxq_guide.txt b/MPU_puzzles/Lunas_pxq_guide.txt new file mode 100644 index 00000000..95134c2c --- /dev/null +++ b/MPU_puzzles/Lunas_pxq_guide.txt @@ -0,0 +1,11 @@ +Puzzle {p}x{q} +Dim 4 +NAxis 2 +Faces 1,0,0,0 0,0,1,0 +Group 1,0,0,0/1,tan(pi/p),0,0 0,0,1,0/0,0,1,tan(pi/q) +Axis 1,0,0,0 +Twists 0,1,0,0/0,0,0,1 0,0,1,0/0,0,1,tan(pi/q) +Cuts ... +Axis 0,0,1,0 +Twists 0,0,0,1/0,1,0,0 1,0,0,0/1,tan(pi/p),0,0 +Cuts ... \ No newline at end of file diff --git a/MPU_puzzles/Oct-Rhomb.txt b/MPU_puzzles/Oct-Rhomb.txt new file mode 100644 index 00000000..84eeff87 --- /dev/null +++ b/MPU_puzzles/Oct-Rhomb.txt @@ -0,0 +1,5 @@ +Puzzle Oct-Rhomb +Dim 4 +NAxis 0 +Faces 1,0,0,0 0,0,2.98787794289171,4.51904939276421 0,4.23606797749979,4.23606797749979,4.23606797749979 0,0,0,4.314039705040488 +Group 1,0,0,0 0,1,0,1/0,0,-1,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0.809017,0.209735,-0.549093 \ No newline at end of file diff --git a/MPU_puzzles/Octahedral_Prism.txt b/MPU_puzzles/Octahedral_Prism.txt new file mode 100644 index 00000000..2b8997d4 --- /dev/null +++ b/MPU_puzzles/Octahedral_Prism.txt @@ -0,0 +1,11 @@ +Puzzle Octahedral_Prism +Dim 4 +NAxis 2 +Faces 1,0,0,0 0,1,1,1 +Group 1,0,0,0/0,1,0,0 0,1,0,0/0,1,1,0 0,1,0,0/0,1,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,1,1,0 0,1,1,0/0,0,1,1 0,1,0,0/0,0,1,1 +Cuts 0.5 -0.5 +Axis 0,1,1,1 +Twists 0,1,-1,0/0,1,0,-1 1,0,0,0/0,1,-1,0 +Cuts 0.5 -0.5 \ No newline at end of file diff --git a/MPU_puzzles/Simplex_Prism.txt b/MPU_puzzles/Simplex_Prism.txt new file mode 100644 index 00000000..6c95946e --- /dev/null +++ b/MPU_puzzles/Simplex_Prism.txt @@ -0,0 +1,11 @@ +Puzzle Simplex_Prism +Dim 5 +NAxis 2 +Faces 0,0,0,0,1 1,0,0,0,0 +Group 1,0,0,0,0 0,0,1,1,0/0,0,1,-1,0 0,2,-1,-1,0/0,1,1,-2,0 0,2,-2,0,0/0,1,1,-1,-2.236068 +Axis 0,0,0,0,1 +Twists 1,0,0,0,0/0,1,-1,0,0 0,0,1,1,0/0,0,1,-1,0 0,2,-1,-1,0/0,1,1,-2,0 +Cuts 0 +Axis 1,0,0,0,0 +Twists 0,2,-1,-1,0/0,1,1,-2,0 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/Snub24cell.txt b/MPU_puzzles/Snub24cell.txt new file mode 100644 index 00000000..4971d41d --- /dev/null +++ b/MPU_puzzles/Snub24cell.txt @@ -0,0 +1,14 @@ +Puzzle Snub24cell +Dim 4 +NAxis 2 +Faces 1,0,0,0 0.809017,0.809017,0,0 0.9045085,0.6545085,0.25,0 +#Faces 1,0,0,0 0.809017,0.809017,0,0 0.9045085,0.6545085,0.25,0 0.9045085,0.6545085,-0.25,0 +Group 0,2,-1,-1/0,1,1,-2 0,1,1,2/0,2,-1,1 2,-2,-2,0/1,-1,-1,3 +Axis 1,0,0,0 +Twists 0,2,-1,-1/0,1,1,-2 0,0,1,0/0,0,0,1 0,1,1,2/0,2,-1,1 +Cuts 0.9 -0.9 +FixedMask 2 +Axis 0.809017,0.809017,0,0 +Twists 2,-2,-2,0/1,-1,-1,3 1,-1,-1,-3/2,-2,-2,0 0,0,0,1/0,0,1,0 +Cuts 0.95 -0.95 +FixedMask 2 \ No newline at end of file diff --git a/MPU_puzzles/Square_Antiprism_Prism.txt b/MPU_puzzles/Square_Antiprism_Prism.txt new file mode 100644 index 00000000..58ad073f --- /dev/null +++ b/MPU_puzzles/Square_Antiprism_Prism.txt @@ -0,0 +1,14 @@ +Puzzle Square_Antiprism_Prism +Dim 4 +NAxis 3 +Faces -0.5,0,0,0 0,0,0.42044820,0 0,0.56903559,0.14014940,0 +Group 0,0,1,0/0,0.38268343,0,0.92387953 0,0,0,1/0,1,0,1 1,0,0,0/0,0,0,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,1,0,1 0,0,1,0/0,0.38268343,0,0.92387953 0,0,1,0/0,-0.38268343,0,0.92387953 +Cuts 0.1666 -0.1666 +Axis 0,0,1,0 +Twists 0,1,0,0/0,1,0,1 1,0,0,0/0,1,0,0 1,0,0,0/0,1,0,1 +Cuts 0.1235 -0.1235 +Axis 0,0.56903559,0.14014940,0 +Twists 1,0,0,0/0,0,0,1 +Cuts 0.621 \ No newline at end of file diff --git a/MPU_puzzles/Triangular-antitegmatic_Icosachoron.txt b/MPU_puzzles/Triangular-antitegmatic_Icosachoron.txt new file mode 100644 index 00000000..46d10e17 --- /dev/null +++ b/MPU_puzzles/Triangular-antitegmatic_Icosachoron.txt @@ -0,0 +1,9 @@ +Puzzle Triangular-antitegmatic_Icosachoron +Dim 4 +NAxis 1 +Faces 1,0,0,0 +Group 1,0,0,0/0.5,0.866025404,0,0 0,0.577350269,0.816496581,0/0,0,0.612372436,0.790569415 +#1,0,0,0/0.5,sqrt(3)/2,0,0 0,1/sqrt(3),sqrt(2/3),0/0,0,1/sqrt(6),sqrt(5/6) +Axis 1,0,0,0 +Twists 0,0.577350269,0.816496581,0/0,0,0.612372436,0.790569415 0,0.790569,-0.559017,0.25/0,0.57735,1.22474,0.912871 +Cuts 0.75 -0.75 \ No newline at end of file diff --git a/MPU_puzzles/{3,3}x{4}.txt b/MPU_puzzles/{3,3}x{4}.txt new file mode 100644 index 00000000..5e57ac8b --- /dev/null +++ b/MPU_puzzles/{3,3}x{4}.txt @@ -0,0 +1,11 @@ +Puzzle {3,3}x{4} +Dim 5 +NAxis 2 +Faces 1,1,1,0,0 0,0,0,1.73205081,0 +Group 1,1,0,0,0/1,0,-1,0,0 1,1,0,0,0/0,1,-1,0,0 0,0,0,1,0/0,0,0,1,1 +Axis 1,1,1,0,0 +Twists 0,0,0,1,0/0,0,0,1,1 1,-1,0,0,0/1,0,-1,0,0 1,-1,0,0,0/0,0,0,1,0 +Cuts -0.33333 +Axis 0,0,0,1,0 +Twists 1,1,0,0,0/1,0,-1,0,0 0,0,0,0,1/1,-1,0,0,0 +Cuts 0 \ No newline at end of file diff --git a/MPU_puzzles/{4}x{4}3.txt b/MPU_puzzles/{4}x{4}3.txt new file mode 100644 index 00000000..ff4275bf --- /dev/null +++ b/MPU_puzzles/{4}x{4}3.txt @@ -0,0 +1,11 @@ +Puzzle {4}x{4} 3 +Dim 4 +NAxis 2 +Faces 1,0,0,0 0,0,1,0 +Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1 +Axis 1,0,0,0 +Twists 0,1,0,0/0,0,1,0 0,0,1,0/0,0,1,1 +Cuts 0.5 -0.5 +Axis 0,0,1,0 +Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1 +Cuts 0.5 -0.5 \ No newline at end of file diff --git a/assets/animcube/AnimCube2.js b/assets/animcube/AnimCube2.js new file mode 100644 index 00000000..7879c1d7 --- /dev/null +++ b/assets/animcube/AnimCube2.js @@ -0,0 +1 @@ +"use strict";function AnimCube2(params){var cubeDim=2,config=[],bgColor,hlColor,textColor,buttonBgColor,sliderColor,sliderBgColor,buttonBorderColor,cubeColor,colors=[],cube=[],scube=[],initialCube=[],initialSCube=[],faceNormals=[[0,-1,0],[0,1,0],[0,0,-1],[0,0,1],[-1,0,0],[1,0,0]],cornerCoords=[[-1,-1,-1],[1,-1,-1],[1,-1,1],[-1,-1,1],[-1,1,-1],[1,1,-1],[1,1,1],[-1,1,1]],faceCorners=[[0,1,2,3],[4,7,6,5],[0,4,5,1],[2,6,7,3],[0,3,7,4],[1,5,6,2]],oppositeCorners=[[0,3,2,1],[0,3,2,1],[3,2,1,0],[3,2,1,0],[0,3,2,1],[0,3,2,1]],adjacentFaces=[[2,5,3,4],[4,3,5,2],[4,1,5,0],[5,1,4,0],[0,3,1,2],[2,1,3,0]],twistedLayer,twistedMode,faceTwistDirs=[1,1,-1,-1,-1,-1],eye=[0,0,-1],eyeX=[1,0,0],eyeY=[],initialEye=[],initialEyeX=[],initialEyeY=[],currentAngle,originalAngle,speed,doubleSpeed,natural=!0,toTwist,interrupted,restarted,mirrored,editable,repeatable,clickProgress,twisting,spinning,animating,dragging,demo,persp,scale,align,hint,faceShift,hintHoriz,hintVert,hintBorder,moveCounter,move=[],demoMove=[],initialMove=[],initialReversedMove=[],curMove,movePos,moveDir,moveOne,moveAnimated,metric,infoText=[],curInfoText,buttonBar,buttonHeight,drawButtons=!0,pushed,buttonPressed=-1,progressHeight=6,textHeight=12,moveText,moveTextSpace,outlined=!0,snap=!1,yzAlt,superCube=!1,scrambleToggle=!1,scramble=0,randMoveCount=0,scw=0,borderWidth=0,rotateAllowed=1,posFaceTransform=[3,2,0,5,1,4],posFaceletTransform=[[2,0,3,1],[1,3,0,2],[0,1,2,3],[0,1,2,3],[2,0,3,1],[0,1,2,3]];function onModuleLoad(){var e=getParameter("config");if(null==e)init();else{var t=location.pathname,r=t.substring(t.lastIndexOf("/")+1);loadConfigFile(0==r.length?t+e:t.replace(r,e))}}function loadConfigFile(e){var t=new XMLHttpRequest;t.onreadystatechange=function(){4==t.readyState&&(200==t.status?parseConfigFile(t.responseText):console.log("Error loading config file: "+e),init())},t.open("GET",e,!0),t.send()}function parseConfigFile(e){for(var t=e.split("\n"),r=0;r="0"&&e.charAt(t)<="9"&&(borderWidth=10*borderWidth+parseInt(e[t]));borderWidth>=0&&borderWidth<=20&&setBorderWidth(borderWidth/100)}if(superCube)for(t=0;t<6;t++)for(r=0;r<4;r++)scube[t][r]=0;var a="lluu";if(null!=(e=getParameter("colorscheme"))&&6==e.length)for(t=0;t<6;t++){var i=23;for(r=0;r<23;r++)if(e[t].toLowerCase()=="0123456789wyorgbldmcpnk".charAt(r)){i=r;break}for(r=0;r<4;r++)cube[t][r]=i}if("1"==(e=getParameter("scramble"))?scramble=1:"2"==e&&(scramble=2),2==scramble)for(t=0;t<6;t++)for(r=0;r<4;r++)initialCube[t][r]=cube[t][r],initialSCube[t][r]=scube[t][r];if(0==scramble){if(null!=(e=getParameter("pos"))&&24==e.length){a="uuuuff","gray"==bgColor&&(bgColor="white");for(t=0;t<6;t++){var n=posFaceTransform[t];for(r=0;r<4;r++){var s=posFaceletTransform[t][r];cube[n][s]=23;for(var l=0;l<14;l++)if(e.charAt(9*t+r)=="DFECABdfecabgh".charAt(l)){cube[n][s]=l+4;break}}}}if(null!=(e=getParameter("facelets"))&&24==e.length)for(t=0;t<6;t++)for(r=0;r<4;r++){cube[t][r]=23;for(l=0;l<23;l++)if(e[4*t+r].toLowerCase()=="0123456789wyorgbldmcpnk".charAt(l)){cube[t][r]=l;break}}if(null!=(e=getParameter("superfacelets"))&&24==e.length)for(t=0;t<6;t++)for(r=0;r<4;r++)for(l=0;l<4;l++)if(e[4*t+r].toLowerCase()=="0123".charAt(l)){scube[t][r]=l;break}}if(moveText=0,yzAlt=!1,null!=(e=getParameter("sign"))&&"1"==e&&(moveText=5,yzAlt=!0),null!=(e=getParameter("wca"))&&"1"==e&&(moveText=5,yzAlt=!0),null!=(e=getParameter("yz"))&&("0"==e?yzAlt=!1:"1"==e&&(yzAlt=!0)),null!=(e=getParameter("randmoves"))){var c=0;for(t=0;t="0"&&e.charAt(t)<="9"&&(c=10*c+parseInt(e[t]));c>0&&(randMoveCount=c)}("random"==(e=getParameter("move"))||scramble>0)&&(e=randMoves(2,randMoveCount)),move=null==e?[]:getMove(e,!0),movePos=0,curInfoText=-1,0==scramble&&(null!=(e=getParameter("initmove"))&&("random"==e&&(e=randMoves(2,randMoveCount)),initialMove="#"==e?move:getMove(e,!1)),null!=(e=getParameter("initrevmove"))&&("random"==e&&(e=randMoves(2,randMoveCount)),initialReversedMove="#"==e?move:getMove(e,!1)),null!=(e=getParameter("demo"))&&("random"==e&&(e=randMoves(2,randMoveCount)),(demoMove="#"==e?move:getMove(e,!0)).length>0&&demoMove[0].length>0&&(demo=!0))),e=getParameter("position"),vNorm(vMul(eyeY,eye,eyeX)),null==e&&(e=a);var d=Math.PI/12;for(t=0;t="0"&&e.charAt(t)<="9"&&(speed=10*speed+parseInt(e[t]));if(null!=(e=getParameter("doublespeed")))for(t=0;t="0"&&e.charAt(t)<="9"&&(doubleSpeed=10*doubleSpeed+parseInt(e[t]));if(0==speed&&(speed=10),0==doubleSpeed&&(doubleSpeed=3*speed/2),persp=0,null==(e=getParameter("perspective")))persp=2;else for(t=0;t="0"&&e.charAt(t)<="9"&&(persp=10*persp+parseInt(e[t]));var g,h=0;if(null!=(e=getParameter("scale")))for(t=0;t="0"&&e.charAt(t)<="9"&&(h=10*h+parseInt(e[t]));if(scale=1/(1+h/10),hint=!1,null!=(e=getParameter("hint"))){hint=!0,faceShift=0;for(t=0;t="0"&&e.charAt(t)<="9"&&(faceShift=10*faceShift+parseInt(e[t]));faceShift<1?hint=!1:faceShift/=10}(hintHoriz=3.7,null!=(e=getParameter("hinthoriz")))&&((g=parseFloat(e))>0&&(hintHoriz=g));(hintVert=3.7,null!=(e=getParameter("hintvert")))&&((g=parseFloat(e))>0&&(hintVert=g));(hintBorder=0,null!=(e=getParameter("hintborder"))&&"1"==e&&(hintBorder=1),buttonHeight=13,null!=(e=getParameter("buttonheight")))&&((g=parseInt(e))>=9&g<=25&&(buttonHeight=g));(progressHeight=0==move.length?0:6,buttonBar=1,"0"==(e=getParameter("buttonbar"))?(buttonBar=0,buttonHeight=0,progressHeight=0):"1"==e?buttonBar=1:"2"!=e&&0!=move.length||(buttonBar=2,progressHeight=0),e=getParameter("edit"),editable="0"!=e,e=getParameter("repeat"),repeatable="0"!=e,e=getParameter("clickprogress"),clickProgress="0"!=e,"0"==(e=getParameter("movetext"))?moveText=0:"1"==e?moveText=1:("5"==e||"6"==e)&&(moveText=5),moveTextSpace=1,"0"==(e=getParameter("movetextspace"))&&(moveTextSpace=0),null!=(e=getParameter("textsize")))&&((g=parseInt(e))>=5&g<=40&&(textHeight=g));(e=getParameter("fonttype"),outlined=null==e||"1"==e,metric=0,null!=(e=getParameter("metric"))&&("1"==e?metric=1:"2"==e?metric=2:"3"==e&&(metric=3)),align=1,null!=(e=getParameter("align")))&&((g=parseInt(e))>=0&g<=3&&(align=g),g>=3&g<=99&&(align=g/100));null!=(e=getParameter("snap"))&&"1"==e&&(snap=!0);for(t=0;t<6;t++)for(r=0;r<4;r++)initialCube[t][r]=cube[t][r],initialSCube[t][r]=scube[t][r];initialMove.length>0&&doMove(cube,initialMove[0],0,initialMove[0].length,!1),initialReversedMove.length>0&&doMove(cube,initialReversedMove[0],0,initialReversedMove[0].length,!0),2==scramble&&doMove(cube,move[0],0,move[0].length,!0);for(t=0;t<3;t++)initialEye[t]=eye[t],initialEyeX[t]=eyeX[t],initialEyeY[t]=eyeY[t];colorAverage(bgColor)<128?(textColor="white",hlColor=brighter(bgColor)):(textColor="black",hlColor=darker(bgColor)),buttonBorderColor=colorAverage(buttonBgColor)<128?"white":"black",sliderColor=textColor,null!=(e=getParameter("slidercolor"))&&6==e.length&&validateColor(e)&&(sliderColor="#"+e),sliderBgColor=darker(bgColor),null!=(e=getParameter("sliderbgcolor"))&&6==e.length&&validateColor(e)&&(sliderBgColor="#"+e),null!=(e=getParameter("troughcolor"))&&6==e.length&&validateColor(e)&&(sliderBgColor="#"+e),cubeColor="black",null!=(e=getParameter("cubecolor"))&&6==e.length&&validateColor(e)&&(cubeColor="#"+e),moveCounter=1,"0"==(e=getParameter("counter"))&&(moveCounter=0),curInfoText=move.length>0&&move[0][0]>=1e3?0:-1,init2(),demo&&startAnimation(-1)}function getParameter(e){var t=searchParams[e];return void 0===t?config[e]:t}function setBorderWidth(e){border[0][0]=border[0][1]=border[1][1]=border[3][0]=e,border[1][0]=border[2][0]=border[2][1]=border[3][1]=1-e}var moveModes=[0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,2,2,2,2,2,2],moveCodes=[0,1,2,3,4,5,1,2,4,5,2,0,5,2,0,0,1,2,3,4,5];function getMove(e,t){if(t){for(var r=e.indexOf("{");-1!=r;)r=e.indexOf("{",r+1);if(null==infoText)curInfoText=0,infoText=[];else{for(var o=[],a=0;a=e.length)return"";for(var o="",a=t;a=e.length)return"";if(e[t]>=1e3)return"";if(-1==e[t])return".";var r=turnSymbol[moveText-1][Math.floor(e[t]/4)%6][Math.floor(e[t]/24)];return"~"==r.charAt(0)?r.substring(1)+modifierStrings[(e[t]+2)%4]:r+modifierStrings[e[t]%4]}var metricChar=["","q","h","s"];function realMoveLength(e){for(var t=0,r=0;r=1e3;)r++;if(o==t)break;r=1e3)return 0;var t=e%4,r=Math.floor(e/4)%6,o=1;switch(metric){case 1:1!=t&&3!=t||(o*=2);case 2:1!=r&&4!=r&&5!=r||(o*=2);case 3:3==r&&(o=0),3!=metric||4!=r&&5!=r||(o*=2)}return o}function initInfoText(e){curInfoText=e.length>0&&e[0]>=1e3?e[0]-1e3:-1}function doMove(e,t,r,o,a){for(var i=a?r+o:r;;){if(a){if(i<=r)break;i--}if(t[i]>=1e3)curInfoText=a?-1:t[i]-1e3;else if(t[i]>=0){var n=t[i]%4+1,s=Math.floor(t[i]/4)%6;4==n&&(n=2),a&&(n=4-n),twistLayers(e,Math.floor(t[i]/24),n,s)}if(!a&&++i>=r+o)break}}var jobNumber=0,nowServing=0;function startAnimation(e){if(stopAnimation(),(demo||0!=move.length&&0!=move[curMove].length)&&(!demo||0!=demoMove.length&&0!=demoMove[0].length)){switch(moveDir=1,moveOne=!1,moveAnimated=!0,e){case 0:break;case 1:moveDir=-1;break;case 2:moveOne=!0;break;case 3:moveDir=-1,moveOne=!0;break;case 4:moveAnimated=!1}run(jobNumber++,moveDir)}}function stopAnimation(){1==animating&&(restarted=!0)}function clear(){movePos=0,natural=!0,mirrored=!1;for(var e=0;e<6;e++)for(var t=0;t<4;t++)cube[e][t]=initialCube[e][t],scube[e][t]=initialSCube[e][t];initialMove.length>0&&void 0!==initialMove[curMove]&&doMove(cube,initialMove[curMove],0,initialMove[curMove].length,!1),initialReversedMove.length>0&&void 0!==initialReversedMove[curMove]&&doMove(cube,initialReversedMove[curMove],0,initialReversedMove[curMove].length,!0),move.length>0&&initInfoText(move[curMove]),scramble>0&&(move=getMove(randMoves(2,randMoveCount),!1)),2==scramble&&doMove(cube,move[0],0,move[0].length,!0);for(e=0;e<3;e++)eye[e]=initialEye[e],eyeX[e]=initialEyeX[e],eyeY[e]=initialEyeY[e];setTimeout(paint,0)}var cubeBlocks=[[[0,2],[0,2]],[[0,2],[0,2]],[[0,2],[0,2]],[[0,2],[0,2]],[[0,2],[0,2]],[[0,2],[0,2]]],topBlocks=[],botBlocks=[],topBlockTable=[[[0,0],[0,0]],[[0,2],[0,2]],[[0,2],[0,1]],[[0,1],[0,2]],[[0,2],[1,2]],[[1,2],[0,2]]],topBlockFaceDim=[[1,0,3,3,2,3],[0,1,5,5,4,5],[2,3,1,0,3,2],[4,5,0,1,5,4],[3,2,2,4,1,0],[5,4,4,2,0,1]],botBlockFaceDim=[[0,1,5,5,4,5],[1,0,3,3,2,3],[4,5,0,1,5,4],[2,3,1,0,3,2],[5,4,4,2,0,1],[3,2,2,4,1,0]];function splitCube(e){for(var t=0;t<6;t++)topBlocks[t]=topBlockTable[topBlockFaceDim[e][t]],botBlocks[t]=topBlockTable[botBlockFaceDim[e][t]];natural=!1}function twistLayers(e,t,r,o){5==o?twistLayer(e,1^t,4-r):2!=o&&3!=o&&4!=o||twistLayer(e,1^t,r),twistLayer(e,t,4-r)}var cycleOrder=[0,1,3,2],cycleFactors=[1,2,-1,-2],cycleOffsets=[0,1,3,2],cycleLayerSides=[[3,3,3,0],[2,1,1,1],[3,3,0,0],[2,1,1,2],[3,2,0,0],[2,2,0,1]],twistBuffer=[];function twistLayer(e,t,r){twistLayer2(e,t,r),1==superCube&&r>0&&r<4&&(twistLayer2(scube,t,r),twistSuperLayer(t,r))}function twistLayer2(e,t,r){for(var o=0;o<4;o++)twistBuffer[(o+r)%4]=e[t][cycleOrder[o]];for(o=0;o<4;o++)e[t][cycleOrder[o]]=twistBuffer[o];var a=2*r;for(o=0;o<4;o++)for(var i=adjacentFaces[t][o],n=cycleLayerSides[t][o],s=cycleFactors[n],l=cycleOffsets[n],c=0;c<2;c++,a++)twistBuffer[a%8]=e[i][c*s+l];for(o=0,a=0;o<4;o++)for(i=adjacentFaces[t][o],n=cycleLayerSides[t][o],s=cycleFactors[n],l=cycleOffsets[n],c=0;c<2;c++,a++)e[i][c*s+l]=twistBuffer[a]}var superTwistArr=[[[0,1,0],[0,2,1],[0,2,4],[0,1,5]],[[2,1,0],[1,2,1],[1,2,4],[2,1,5]],[[2,1,3],[0,1,1],[0,1,2],[0,2,0]],[[0,1,3],[1,2,0],[2,1,2],[2,1,1]]],width,height,lastX,lastY,lastDragX,lastDragY,dragAreas;function twistSuperLayer(e,t,r){for(var o=0;o<4;o++)scube[e][o]=(scube[e][o]+4-t)%4;2==e&&superTwist2(0,4-t),3==e&&superTwist2(1,t),4==e&&superTwist(2,t),5==e&&superTwist(3,t)}function superTwist(e,t){superTwist1(superTwistArr[e][0]),superTwist1(superTwistArr[e][t])}function superTwist1(e){for(var t=e[0],r=0;r<2;t+=e[1],r++)scube[e[2]][t]=(scube[e[2]][t]+2)%4}function superTwist2(e,t){for(var r=0;r<4;r++)for(var o=superTwistArr[e][r],a=o[0],i=0;i<2;a+=o[1],i++)scube[o[2]][a]=(scube[o[2]][a]+t)%4}var dragCornersX=[],dragCornersY=[],dragDirsX=[],dragDirsY=[],dragBlocks=[[[0,0],[2,0],[2,1],[0,1]],[[2,0],[2,2],[1,2],[1,0]],[[2,2],[0,2],[0,1],[2,1]],[[0,2],[0,0],[1,0],[1,2]]],areaDirs=[[1,0],[0,1],[-1,0],[0,-1]],twistDirs=[[1,1,1,1],[1,1,1,1],[1,-1,1,-1],[1,-1,1,-1],[-1,1,-1,1],[1,-1,1,-1]],dragLayers=[],dragModes=[],dragX,dragY,rotCos=[[[1,0,0],[0,0,0],[0,0,1]],[[1,0,0],[0,1,0],[0,0,0]],[[0,0,0],[0,1,0],[0,0,1]]],rotSin=[[[0,0,1],[0,0,0],[-1,0,0]],[[0,1,0],[-1,0,0],[0,0,0]],[[0,0,0],[0,0,1],[0,-1,0]]],rotVec=[[[0,0,0],[0,1,0],[0,0,0]],[[0,0,0],[0,0,0],[0,0,1]],[[1,0,0],[0,0,0],[0,0,0]]],rotSign=[1,-1,1,-1,1,-1],tempEye=[],tempEyeX=[],tempEyeY=[],tempEye2=[],tempEyeX2=[],tempEyeY2=[],perspEye=[],perspEyeI=[],perspNormal=[],eyeArray=[],eyeArrayX=[],eyeArrayY=[],eyeOrder=[[1,0],[1,0],[1,1],[1,1],[1,1],[1,2]],blockArray=[],blockMode=[[0,2],[2,1],[2,2],[2,2],[2,2],[2,2]],drawOrder=[[0,1],[1,0],[0,1]];function paint(){graphics.save(),graphics.fillStyle=bgColor;var e=1!=buttonBar||0!=progressHeight&&!demo&&0!=move[curMove].length?height:height-dpr;if(setClip(graphics,0,0,width,e),graphics.fillRect(0,0,width,e),dragAreas=0,natural)fixBlock(eye,eyeX,eyeY,cubeBlocks,3);else{for(var t=Math.cos(originalAngle+currentAngle),r=Math.sin(originalAngle+currentAngle)*rotSign[twistedLayer],o=0;o<3;o++){tempEye[o]=0,tempEyeX[o]=0;for(var a=0;a<3;a++){var i=Math.floor(twistedLayer/2);tempEye[o]+=eye[a]*(rotVec[i][o][a]+rotCos[i][o][a]*t+rotSin[i][o][a]*r),tempEyeX[o]+=eyeX[a]*(rotVec[i][o][a]+rotCos[i][o][a]*t+rotSin[i][o][a]*r)}}vMul(tempEyeY,tempEye,tempEyeX);var n=Math.cos(originalAngle-currentAngle),s=Math.sin(originalAngle-currentAngle)*rotSign[twistedLayer];for(o=0;o<3;o++){tempEye2[o]=0,tempEyeX2[o]=0;for(a=0;a<3;a++){i=Math.floor(twistedLayer/2);tempEye2[o]+=eye[a]*(rotVec[i][o][a]+rotCos[i][o][a]*n+rotSin[i][o][a]*s),tempEyeX2[o]+=eyeX[a]*(rotVec[i][o][a]+rotCos[i][o][a]*n+rotSin[i][o][a]*s)}}vMul(tempEyeY2,tempEye2,tempEyeX2),eyeArray[0]=eye,eyeArrayX[0]=eyeX,eyeArrayY[0]=eyeY,eyeArray[1]=tempEye,eyeArrayX[1]=tempEyeX,eyeArrayY[1]=tempEyeY,eyeArray[2]=tempEye2,eyeArrayX[2]=tempEyeX2,eyeArrayY[2]=tempEyeY2,blockArray[0]=topBlocks,blockArray[1]=botBlocks,vSub(vScale(vCopy(perspEye,eye),5+persp),vScale(vCopy(perspNormal,faceNormals[twistedLayer]),1/3)),vSub(vScale(vCopy(perspEyeI,eye),5+persp),vScale(vCopy(perspNormal,faceNormals[1^twistedLayer]),1/3));var l,c=vProd(perspEye,faceNormals[twistedLayer]),d=vProd(perspEyeI,faceNormals[1^twistedLayer]);l=c<0&&d>0?0:c>0&&d<0?1:2;for(o=0;o<2;o++){a=drawOrder[l][o];var u=eyeOrder[twistedMode][a];fixBlock(eyeArray[u],eyeArrayX[u],eyeArrayY[u],blockArray[a],blockMode[twistedMode][a])}}if(pushed||animating||(buttonPressed=-1),!(scramble>0&&2==buttonBar)){if(!demo&&move.length>0){if(move[curMove].length>0){if(progressHeight>0){graphics.lineWidth=lineWidth,graphics.strokeStyle=buttonBorderColor;var g=(width-2)*realMovePos(move[curMove],movePos)/realMoveLength(move[curMove]);graphics.fillStyle=sliderBgColor,graphics.fillRect(dph,height-progressHeight-dph,width-dpr,progressHeight),graphics.fillStyle=sliderColor,graphics.fillRect(dph,height-progressHeight-dph,g,progressHeight),graphics.beginPath(),graphics.rect(dph,height-progressHeight-dph,width-dpr,progressHeight),graphics.stroke()}graphics.font="bold "+textHeight+"px helvetica";var h=moveLength(move[curMove],movePos)+"/"+moveLength(move[curMove],-1)+metricChar[metric],v=graphics.measureText(h).width,f=width-v-2,m=height-progressHeight-Math.floor(4*dpr);moveText>0?(moveCounter&&drawString(graphics,h,outlined?f-dpr:f,m-textHeight),drawMoveTextFunc(graphics,m)):moveCounter&&drawString(graphics,h,outlined?f-dpr:f,m)}if(move.length>1){graphics.font="bold "+textHeight+"px helvetica";h=curMove+1+"/"+move.length,v=graphics.measureText(h).width,f=width-v-2*buttonHeight-Math.floor(5*dpr);drawString(graphics,h,f,adjTextHeight()),drawButton(graphics,7,width-2*buttonHeight,0),drawButton(graphics,8,width-buttonHeight,0)}}curInfoText>=0&&(graphics.font="bold "+textHeight+"px helvetica",drawString(graphics,infoText[curInfoText],outlined?dpr:0,adjTextHeight()))}graphics.restore(),drawButtons&&0!=buttonBar&&drawButtonsFunc(graphics)}function adjTextHeight(){return utextHeight<10?Math.floor(10*dpr):utextHeight<12?Math.floor(12*dpr):utextHeight<14?Math.floor(14*dpr):textHeight}var fillX=[],fillY=[],coordsX=[],coordsY=[],cooX=[[],[],[],[],[],[]],cooY=[[],[],[],[],[],[]],border=[[.1,.1],[.9,.1],[.9,.9],[.1,.9]],factors=[[0,0],[0,1],[1,1],[1,0]],tempNormal=[];function fixBlock(e,t,r,o,a){for(var i=0;i<8;i++){var n=(c=width0&&g>0)for(var h=0,v=o[i][1][0];h-(1-scale)){u=o[i][0][1]-o[i][0][0],g=o[i][1][1]-o[i][1][0];if(u>0&&g>0)for(h=0,v=o[i][1][0];h0&&g>0){for(l=2==u?0==o[i][1][0]?0:2:0==o[i][0][0]?3:1,p=0;p<4;p++)getCorners(i,p,dragCornersX[dragAreas],dragCornersY[dragAreas],dragBlocks[l][p][0],dragBlocks[l][p][1],!1);dragDirsX[dragAreas]=(b*areaDirs[l][0]+y*areaDirs[l][1])*twistDirs[i][l],dragDirsY[dragAreas]=(w*areaDirs[l][0]+M*areaDirs[l][1])*twistDirs[i][l],dragLayers[dragAreas]=twistedLayer,dragModes[dragAreas]=0,dragAreas++}}}function getCorners(e,t,r,o,a,i,n){a/=2,i/=2;var s=cooX[e][0]+(cooX[e][1]-cooX[e][0])*a,l=cooY[e][0]+(cooY[e][1]-cooY[e][0])*a,c=cooX[e][3]+(cooX[e][2]-cooX[e][3])*a,d=cooY[e][3]+(cooY[e][2]-cooY[e][3])*a;r[t]=Math.floor(.5+s+(c-s)*i),o[t]=Math.floor(.5+l+(d-l)*i),n&&(r[t]=width-r[t])}var textOffsetInit=[1,1,-1,-1,-1,1,1,-1,-1,0,1,0,0,1,0,-1],textOffset=[];function drawString(e,t,r,o){if(outlined){e.fillStyle="black";for(var a=0;awidth&&(c=Math.min(1,width/2-n-s/2),c=Math.max(c,width-n-s-l-2)),s>0&&(e.fillStyle=hlColor,e.lineWidth=2,e.strokeStyle="black",e.beginPath(),utextHeight<=14?e.fillRect(c+n-1,height-progressHeight-textHeight-Math.floor(4*dpr),s+2,textHeight+Math.floor(3*dpr)):e.fillRect(c+n-1,height-progressHeight-textHeight-Math.floor(2*dpr),s+2,textHeight+Math.floor(dpr))),n>0&&drawString(e,o,c,t),s>0&&drawString(e,a,c+n,t),l>0&&drawString(e,i,c+n+s,t)}function selectButton(e,t){if(0==buttonBar)return-1;if(move.length>1&&e>=width-2*buttonHeight&&e=0&&t1&&e>=width-buttonHeight&&e=0&&t=0&&e=height-buttonHeight&&t=r&&e=height&&t0&&6==buttonPressed?darker(buttonBgColor):buttonBgColor,e.fillRect(dph,height-buttonHeight,buttonHeight,buttonHeight),e.lineWidth=lineWidth,e.strokeStyle=buttonBorderColor,e.beginPath(),e.rect(dph,height-buttonHeight-dph,buttonHeight,buttonHeight),e.stroke(),void drawButton(e,0,buttonHeight/2,height-(buttonHeight+1)/2-t);if(1!=buttonBar);else{for(var r=0,o=0;o<7;o++){var a=Math.floor((width-r)/(7-o));e.fillStyle=buttonPressed==o?darker(buttonBgColor):buttonBgColor,e.fillRect(r,height,a,buttonHeight),e.lineWidth=lineWidth,e.strokeStyle=buttonBorderColor,e.beginPath(),0==o?e.rect(r+dph,height-dph,a-dpr,buttonHeight):e.rect(r-dph,height-dph,a,buttonHeight),e.stroke(),e.strokeStyle="black",drawButton(e,o,r+a/2,height+buttonHeight/2-t),r+=a}drawButtons=!1}}var ds=[];function drawButton(e,t,r,o){switch(r=Math.floor(r),o=Math.floor(o),t){case 0:drawRect(e,r-ds[4],o-ds[3],ds[3],ds[6]+1),drawArrow(e,r+ds[4],o,-1);break;case 1:drawRect(e,r+ds[1],o-ds[3],ds[3],ds[6]+1),drawArrow(e,r-ds[1],o,-1);break;case 2:drawArrow(e,r+ds[1],o,-1);break;case 3:animating?drawRect(e,r-ds[4],o-ds[3],ds[7],ds[7]):(drawRect(e,r-ds[4],o-ds[2],ds[7],ds[5]),drawRect(e,r-ds[2],o-ds[4],ds[3],ds[9]));break;case 4:drawArrow(e,r-ds[2],o,1);break;case 5:drawRect(e,r-ds[4],o-ds[3],ds[3],ds[6]+1),drawArrow(e,r,o,1);break;case 6:drawRect(e,r+ds[1],o-ds[3],ds[3],ds[6]+1),drawArrow(e,r-ds[4],o,1);break;case 7:var a=7==buttonPressed?darker(buttonBgColor):buttonBgColor;drawRect2(e,r-2*dpr,o+dpr,buttonHeight,o+buttonHeight,a),drawArrow(e,r+2*dpr+buttonHeight/2-3*dpr,o+buttonHeight/2+dph,-1);break;case 8:a=8==buttonPressed?darker(buttonBgColor):buttonBgColor;drawRect2(e,r-2*dpr,o+dpr,buttonHeight,o+buttonHeight,a),drawArrow(e,r-dpr+buttonHeight/2-3*dpr,o+buttonHeight/2+dph,1)}}function drawArrow(e,t,r,o){var a=3*dpr,i=[],n=[];i[0]=t,i[1]=t+o,i[2]=t+4*dpr*o,i[3]=t+o,i[4]=t,n[0]=r-a,n[1]=r-a,n[2]=r,n[3]=r+a,n[4]=r+a,drawArrow2(e,i,n)}function drawArrow2(e,t,r){e.beginPath(),e.moveTo(t[0]+dph,r[0]+dph);for(var o=1;o<5;o++)e.lineTo(t[o]+dph,r[o]+dph);e.closePath(),e.fillStyle="white",e.strokeStyle="black",e.fill(),e.lineWidth=lineWidth,e.stroke()}function drawRect(e,t,r,o,a){e.lineWidth=lineWidth,e.beginPath(),e.rect(t+dph,r+dph,o-1,a-1),e.fillStyle="white",e.fill(),e.strokeStyle="black",e.stroke()}function drawRect2(e,t,r,o,a,i){e.lineWidth=lineWidth,e.beginPath(),e.rect(t+dph,r+dph,o-1,a-1),e.fillStyle=i,e.fill(),e.strokeStyle="black",e.stroke()}function drawPolygon(e,t,r,o){e.beginPath(),e.moveTo(t[0],r[0]),e.lineTo(t[1],r[1]),e.lineTo(t[2],r[2]),e.lineTo(t[3],r[3]),e.closePath(),e.strokeStyle=o,e.lineWidth=.7*dpr,e.stroke()}function fillPolygon(e,t,r,o){e.beginPath(),e.moveTo(t[0],r[0]),e.lineTo(t[1],r[1]),e.lineTo(t[2],r[2]),e.lineTo(t[3],r[3]),e.closePath(),e.fillStyle=o,e.fill()}var superRotate=[[0,1,2,3],[3,0,1,2],[2,3,0,1],[1,2,3,0]],offsetX,offsetY;function drawSuperArrow(e,t,r,o,a,i){var n=[],s=[];if(2!=scw||"#ffffff"!=i){for(var l=0;l<4;l++)n[l]=Math.floor(t[l]+.05*(t[superRotate[2][l]]-t[l])),s[l]=Math.floor(r[l]+.05*(r[superRotate[2][l]]-r[l]));0==o&&(a=(a+1)%4),4==o&&(a=(a+3)%4);var c=superRotate[a][0],d=superRotate[a][1],u=superRotate[a][2],g=superRotate[a][3],h=.26*(n[u]-n[d]),v=.26*(s[u]-s[d]),f=(n[c]-n[d])/2,m=s[d]+(s[c]-s[d])/2,p=s[u]+(s[g]-s[u])/2,b=(n[g]-n[u])/2,w=1^a;e.fillStyle=i,e.beginPath(),e.moveTo(n[c]+(n[g]-n[c])/2,s[c]+(s[g]-s[c])/2),e.lineTo(n[w]+f,m),e.lineTo(n[w]+h+f,m+v),e.lineTo(n[w]+h,s[w]+v),w=(w+1)%4,e.lineTo(n[w]-h,s[w]-v),e.lineTo(n[w]-h+b,p-v),e.lineTo(n[w]+b,p),e.closePath(),e.fill(),e.lineWidth=.6*dpr,e.strokeStyle="black",e.stroke()}}function colorToHex(e){return"white"==e?"#FFFFFF":"black"==e?"#000000":"#808080"}function colorAverage(e){return"#"!=e.substring(0,1)&&(e=colorToHex(e)),(299*parseInt(e.substring(1,3),16)+587*parseInt(e.substring(3,5),16)+114*parseInt(e.substring(5,7),16))/1e3}function brighter(e){"#"!=e.substring(0,1)&&(e=colorToHex(e));var t=parseInt(e.substring(1,3),16),r=parseInt(e.substring(3,5),16),o=parseInt(e.substring(5,7),16);return rgbToHex((t=Math.floor(1.3*t))>255?255:t,(r=Math.floor(1.3*r))>255?255:r,(o=Math.floor(1.3*o))>255?255:o)}function darker(e){"#"!=e.substring(0,1)&&(e=colorToHex(e));var t=parseInt(e.substring(1,3),16),r=parseInt(e.substring(3,5),16),o=parseInt(e.substring(5,7),16);return rgbToHex(t=Math.floor(.7*t),r=Math.floor(.7*r),o=Math.floor(.7*o))}function run(e,t){if(e>nowServing)setTimeout(run,0,e,t);else{if(!demo&&(0==move.length||0==move[curMove].length))return animating=!1,drawButtons=!0,void paint();if(!moveAnimated){for(var r=move[curMove];movePos=1e3)curInfoText=r[movePos]-1e3;else if(-1!=r[movePos]){var o=r[movePos]%4+1,a=Math.floor(r[movePos]/4)%6,i=Math.floor(r[movePos]/24);twistLayers(cube,i,4==o?2:o,a)}movePos++}return animating=!1,drawButtons=!0,paint(),void nowServing++}var n,s,l,c,d,u,g=t;interrupted=!1,requestAnimationFrame((function e(){if(v){if(v=!1,u=!1,repeatable)g>0?movePos>=r.length&&(movePos=0,initInfoText(r)):(curInfoText=-1,0==movePos&&(movePos=r.length));else if(g>0&&movePos>=r.length||g<0&&0==movePos)return restarted=!1,animating=!1,nowServing++,drawButtons=!0,void paint();animating=!0,drawButtons=!0}if(f&&(f=!1,g<0&&(b=!1,0==movePos?(b=!0,p=!0):movePos--),!b)){if(h=!1,-1==r[movePos]){if(paint(),!moveOne)for(n=Date.now();Date.now()-n<1e3;);}else r[movePos]>=1e3?curInfoText=g>0?r[movePos]-1e3:-1:h=!0;if(h){o=r[movePos]%4+1,a=Math.floor(r[movePos]/4)%6;var t=o<3;if(4==o&&(o=2),g<0&&(t=!t,o=4-o),i=Math.floor(r[movePos]/24),twisting=!1,natural=!0,spinning=!0,originalAngle=0,faceTwistDirs[i]>0&&(t=!t),moveAnimated){d=Math.PI/2,c=t?1:-1;var w=67*speed;2==o&&(d=Math.PI,w=67*doubleSpeed),twisting=!0,twistedLayer=i,twistedMode=a,splitCube(i),n=Date.now(),s=n,l=c*d/w,currentAngle=0}}else m=!0}b||(h&&(moveAnimated&¤tAngle*c0?(++movePos=1e3&&(curInfoText=r[movePos]-1e3,movePos++),movePos==r.length&&(demo?clearDemo(r):p=!0)):curInfoText=-1,(interrupted||restarted||u)&&(p=!0)));if(p)return p=!1,v=!0,jobNumber<=nowServing+1&&(animating=!1),drawButtons=!0,(0==buttonPressed||buttonPressed>6)&&clear(),paint(),demo&&(clear(),demo=!1),restarted=!1,nowServing++,void(1==movePos&&r[0]>=1e3&&movePos--);requestAnimationFrame(e)}));var h=!1,v=!0,f=!0,m=!1,p=!1,b=!1;r=demo?demoMove[0]:move[curMove]}}function clearDemo(e){movePos=0;for(var t=0;t<6;t++)for(var r=0;r<4;r++)cube[t][r]=initialCube[t][r],scube[t][r]=initialSCube[t][r];initialMove.length>0&&void 0!==initialMove[curMove]&&doMove(cube,initialMove[curMove],0,initialMove[curMove].length,!1),initialReversedMove.length>0&&void 0!==initialReversedMove[curMove]&&doMove(cube,initialReversedMove[curMove],0,initialReversedMove[curMove].length,!0),initInfoText(e)}document.addEventListener("touchstart",mousedown),document.addEventListener("touchmove",mousemove,{passive:!1}),document.addEventListener("touchend",mouseup),document.addEventListener("mousedown",mousedown),document.addEventListener("mousemove",mousemove),document.addEventListener("mouseup",mouseup),document.addEventListener("contextmenu",contextmenu);var mouseIsDown=!1,showContextMenu=!0,divs=document.getElementsByTagName("div"),wrapDiv=divs.length>0&&"wrap"==divs[0].className;function touchfunc(e){wrapDiv?divs[0].style.overflow=e:document.body.style.overflow=e}function mouseup(e){if(mouseIsDown&&void 0!==e.touches&&(e.preventDefault(),touchfunc("auto")),mouseIsDown?setTimeout((function(){showContextMenu=!0}),100):showContextMenu=!0,mouseIsDown=!1,dragging=!1,pushed)pushed=!1,drawButtons=!0,paint();else if(twisting&&!spinning){twisting=!1,originalAngle+=currentAngle,currentAngle=0;for(var t=originalAngle;t<0;)t+=32*Math.PI;var r=Math.floor(8*t/Math.PI)%16;(snap||r%4==0||r%4==3)&&(r=Math.floor((r+2)/4),faceTwistDirs[twistedLayer]>0&&(r=(4-r)%4),originalAngle=0,natural=!0,twistLayers(cube,twistedLayer,r,twistedMode)),paint()}}function mousedown(e){var t=canvas.getBoundingClientRect(),r=Math.floor(t.left),o=Math.floor(t.top);if(void 0===e.touches)var a=e.clientX,i=e.clientY;else a=e.touches[0].clientX,i=e.touches[0].clientY;ar+width/dpr||io+(height+buttonHeight)/dpr||(e.preventDefault(),mouseIsDown=!0,showContextMenu=!1,void 0!==e.touches&&touchfunc("hidden"),offsetX=r,offsetY=o,lastDragX=lastX=getX(e),lastDragY=lastY=getY(e),toTwist=!1,(buttonPressed=selectButton(lastX,lastY))>=0?button():progressHeight>0&&move.length>0&&move[curMove].length>0&&lastY>height-progressHeight&&lastY<=height?clickProgress&&(stopAnimation(),progress(jobNumber++)):(mirrored&&(lastDragX=lastX=width-lastX),void 0===e.touches?!editable||animating||0!=e.button||e.shiftKey||(toTwist=!0):editable&&!animating&&(toTwist=!0)))}function button(){pushed=!0,3==buttonPressed?animating?stopAnimation():mirrored=!mirrored:0==buttonPressed?scramble>0&&2==buttonBar?1==scrambleToggle?(scrambleToggle=!1,stopAnimation(),clear()):(scrambleToggle=!0,startAnimation(buttonAction[buttonPressed=6])):(stopAnimation(),clear()):7==buttonPressed||8==buttonPressed?(stopAnimation(),setTimeout(clear,0),curMove=7==buttonPressed?curMove>0?curMove-1:move.length-1:curMovenowServing)setTimeout(progress,0,e);else{var t=realMoveLength(move[curMove]),r=Math.floor(((lastX-1)*t*2/(width-2)+1)/2);(r=Math.max(0,Math.min(t,r)))>0&&(r=arrayMovePos(move[curMove],r)),r>movePos&&doMove(cube,move[curMove],movePos,r-movePos,!1),r0&&(r=arrayMovePos(move[curMove],r)),r>movePos&&doMove(cube,move[curMove],movePos,r-movePos,!1),r0&&v<1&&f>0&&f<1){if(i*i+n*n<144)return;if(dragX=dragDirsX[s],dragY=dragDirsY[s],Math.abs(dragX*i+dragY*n)/Math.sqrt((dragX*dragX+dragY*dragY)*(i*i+n*n))>.75){twisting=!0,twistedLayer=dragLayers[s],twistedMode=dragModes[s];break}}}toTwist=!1,lastX=lastDragX,lastY=lastDragY}i=(o-lastX)/dpr,n=(a-lastY)/dpr,!twisting||animating?rotateAllowed&&(vNorm(vAdd(eye,vScale(vCopy(eyeD,eyeX),-.016*i))),vNorm(vMul(eyeX,eyeY,eye)),vNorm(vAdd(eye,vScale(vCopy(eyeD,eyeY),.016*n))),vNorm(vMul(eyeY,eye,eyeX)),lastX=o,lastY=a):(natural&&splitCube(twistedLayer),currentAngle=.03*(dragX*i+dragY*n)/Math.sqrt(dragX*dragX+dragY*dragY)),paint()}}function getX(e){return void 0===e.touches?(e.clientX-offsetX)*dpr:(e.touches[0].clientX-offsetX)*dpr}function getY(e){return void 0===e.touches?(e.clientY-offsetY)*dpr:(e.touches[0].clientY-offsetY)*dpr}function contextmenu(e){showContextMenu||e.preventDefault()}function resize(){clearTimeout(timer),timer=setTimeout((function(){scaleCanvas(),drawButtons=!0,paint()}),20)}function init2(){graphics=canvas.getContext("2d"),ubuttonHeight=buttonHeight,uprogressHeight=progressHeight,utextHeight=textHeight,scaleCanvas(),parNode.appendChild(canvas),paint()}function scaleCanvas(){height=parNode.clientHeight-1,width=parNode.clientWidth-1,setCanvasCssSize(),dpr=devicePixelRatio,dph=dpr/2,height=Math.floor(height*dpr),width=Math.floor(width*dpr),setCanvasSize(),lineWidth=dpr,buttonHeight=Math.floor(ubuttonHeight*dpr),progressHeight=Math.floor(uprogressHeight*dpr),textHeight=Math.floor(utextHeight*dpr);for(var e=1;e<10;e++)ds[e]=e*dpr;for(e=0;e4?3:2)),5==e&&1==o&&(o=0),d+=""+a[s]+i[o]+n[r]+" ")}return d}window.addEventListener("resize",resize);var searchParams=[];function parseSearchParams(){for(var e=params.split("&"),t=0;t="0"&&e.charAt(r)<="9"&&(borderWidth=10*borderWidth+parseInt(e[r]));borderWidth>=0&&borderWidth<=20&&setBorderWidth(borderWidth/100)}if(superCube)for(r=0;r<6;r++)for(t=0;t<9;t++)scube[r][t]=0;var a="lluu";if(null!=(e=getParameter("colorscheme"))&&6==e.length)for(r=0;r<6;r++){var i=23;for(t=0;t<23;t++)if(e[r].toLowerCase()=="0123456789wyorgbldmcpnk".charAt(t)){i=t;break}for(t=0;t<9;t++)cube[r][t]=i}if("1"==(e=getParameter("scramble"))?scramble=1:"2"==e&&(scramble=2),0==scramble){if(null!=(e=getParameter("pos"))&&54==e.length){a="uuuuff","gray"==bgColor&&(bgColor="white");for(r=0;r<6;r++){var n=posFaceTransform[r];for(t=0;t<9;t++){var s=posFaceletTransform[r][t];cube[n][s]=23;for(var l=0;l<14;l++)if(e.charAt(9*r+t)=="DFECABdfecabgh".charAt(l)){cube[n][s]=l+4;break}}}}if(null!=(e=getParameter("facelets"))&&54==e.length)for(r=0;r<6;r++)for(t=0;t<9;t++){cube[r][t]=23;for(l=0;l<23;l++)if(e[9*r+t].toLowerCase()=="0123456789wyorgbldmcpnk".charAt(l)){cube[r][t]=l;break}}if(null!=(e=getParameter("superfacelets"))&&54==e.length)for(r=0;r<6;r++)for(t=0;t<9;t++)for(l=0;l<4;l++)if(e[9*r+t].toLowerCase()=="0123".charAt(l)){scube[r][t]=l;break}}if(moveText=0,yzAlt=!1,signNotation=!1,null!=(e=getParameter("sign"))&&"1"==e&&(signNotation=!0,moveText=5,yzAlt=!0),wcaNotation=!1,null!=(e=getParameter("wca"))&&"1"==e&&(wcaNotation=!0,moveText=6,yzAlt=!0),null!=(e=getParameter("yz"))&&("0"==e?yzAlt=!1:"1"==e&&(yzAlt=!0)),null!=(e=getParameter("randmoves"))){var c=0;for(r=0;r="0"&&e.charAt(r)<="9"&&(c=10*c+parseInt(e[r]));c>0&&(randMoveCount=c)}("random"==(e=getParameter("move"))||scramble>0)&&(e=randMoves(3,randMoveCount)),move=null==e?[]:getMove(e,!0),movePos=0,curInfoText=-1,0==scramble&&(null!=(e=getParameter("initmove"))&&("random"==e&&(e=randMoves(3,randMoveCount)),initialMove="#"==e?move:getMove(e,!1)),null!=(e=getParameter("initrevmove"))&&("random"==e&&(e=randMoves(3,randMoveCount)),initialReversedMove="#"==e?move:getMove(e,!1)),null!=(e=getParameter("demo"))&&("random"==e&&(e=randMoves(3,randMoveCount)),(demoMove="#"==e?move:getMove(e,!0)).length>0&&demoMove[0].length>0&&(demo=!0))),e=getParameter("position"),vNorm(vMul(eyeY,eye,eyeX)),null==e&&(e=a);var d=Math.PI/12;for(r=0;r="0"&&e.charAt(r)<="9"&&(speed=10*speed+parseInt(e[r]));if(null!=(e=getParameter("doublespeed")))for(r=0;r="0"&&e.charAt(r)<="9"&&(doubleSpeed=10*doubleSpeed+parseInt(e[r]));if(0==speed&&(speed=10),0==doubleSpeed&&(doubleSpeed=3*speed/2),persp=0,null==(e=getParameter("perspective")))persp=2;else for(r=0;r="0"&&e.charAt(r)<="9"&&(persp=10*persp+parseInt(e[r]));var g,h=0;if(null!=(e=getParameter("scale")))for(r=0;r="0"&&e.charAt(r)<="9"&&(h=10*h+parseInt(e[r]));if(scale=1/(1+h/10),hint=!1,null!=(e=getParameter("hint"))){hint=!0,faceShift=0;for(r=0;r="0"&&e.charAt(r)<="9"&&(faceShift=10*faceShift+parseInt(e[r]));faceShift<1?hint=!1:faceShift/=10}(hintHoriz=3.7,null!=(e=getParameter("hinthoriz")))&&((g=parseFloat(e))>0&&(hintHoriz=g));(hintVert=3.7,null!=(e=getParameter("hintvert")))&&((g=parseFloat(e))>0&&(hintVert=g));(hintBorder=0,null!=(e=getParameter("hintborder"))&&"1"==e&&(hintBorder=1),buttonHeight=13,null!=(e=getParameter("buttonheight")))&&((g=parseInt(e))>=9&g<=25&&(buttonHeight=g));(progressHeight=0==move.length?0:6,buttonBar=1,"0"==(e=getParameter("buttonbar"))?(buttonBar=0,buttonHeight=0,progressHeight=0):"1"==e?buttonBar=1:"2"!=e&&0!=move.length||(buttonBar=2,progressHeight=0),e=getParameter("edit"),editable="0"!=e,e=getParameter("repeat"),repeatable="0"!=e,e=getParameter("clickprogress"),clickProgress="0"!=e,(e=getParameter("movetext"))>=0&&e<=6&&(moveText=parseInt(e)),moveTextSpace=1,"0"==(e=getParameter("movetextspace"))&&(moveTextSpace=0),null!=(e=getParameter("textsize")))&&((g=parseInt(e))>=5&g<=40&&(textHeight=g));(e=getParameter("fonttype"),outlined=null==e||"1"==e,metric=0,null!=(e=getParameter("metric"))&&("1"==e?metric=1:"2"==e?metric=2:"3"==e&&(metric=3)),align=1,null!=(e=getParameter("align")))&&((g=parseInt(e))>=0&g<=3&&(align=g),g>=3&g<=99&&(align=g/100));null!=(e=getParameter("snap"))&&"1"==e&&(snap=!0);for(r=0;r<6;r++)for(t=0;t<9;t++)initialCube[r][t]=cube[r][t],initialSCube[r][t]=scube[r][t];initialMove.length>0&&doMove(cube,initialMove[0],0,initialMove[0].length,!1),initialReversedMove.length>0&&doMove(cube,initialReversedMove[0],0,initialReversedMove[0].length,!0),2==scramble&&doMove(cube,move[0],0,move[0].length,!0);for(r=0;r<3;r++)initialEye[r]=eye[r],initialEyeX[r]=eyeX[r],initialEyeY[r]=eyeY[r];colorAverage(bgColor)<128?(textColor="white",hlColor=brighter(bgColor)):(textColor="black",hlColor=darker(bgColor)),buttonBorderColor=colorAverage(buttonBgColor)<128?"white":"black",sliderColor=textColor,null!=(e=getParameter("slidercolor"))&&6==e.length&&validateColor(e)&&(sliderColor="#"+e),sliderBgColor=darker(bgColor),null!=(e=getParameter("sliderbgcolor"))&&6==e.length&&validateColor(e)&&(sliderBgColor="#"+e),null!=(e=getParameter("troughcolor"))&&6==e.length&&validateColor(e)&&(sliderBgColor="#"+e),cubeColor="black",null!=(e=getParameter("cubecolor"))&&6==e.length&&validateColor(e)&&(cubeColor="#"+e),moveCounter=1,"0"==(e=getParameter("counter"))&&(moveCounter=0),curInfoText=move.length>0&&move[0][0]>=1e3?0:-1,init2(),demo&&startAnimation(-1)}function getParameter(e){var r=searchParams[e];return void 0===r?config[e]:r}function setBorderWidth(e){border[0][0]=border[0][1]=border[1][1]=border[3][0]=e,border[1][0]=border[2][0]=border[2][1]=border[3][1]=1-e}var moveModes=[0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,2,2,2,2,2,2],moveCodes=[0,1,2,3,4,5,1,2,4,5,2,0,5,2,0,0,1,2,3,4,5];function getMove(e,r){if(r){for(var t=e.indexOf("{");-1!=t;)t=e.indexOf("{",t+1);if(null==infoText)curInfoText=0,infoText=[];else{for(var o=[],a=0;a=e.length)return"";for(var o="",a=r;a=e.length)return"";if(e[r]>=1e3)return"";if(-1==e[r])return".";var t=turnSymbol[moveText-1][Math.floor(e[r]/4)%6][Math.floor(e[r]/24)];return"~"==t.charAt(0)?t.substring(1)+modifierStrings[(e[r]+2)%4]:t+modifierStrings[e[r]%4]}var metricChar=["","q","h","s"];function realMoveLength(e){for(var r=0,t=0;t=1e3;)t++;if(o==r)break;t=1e3)return 0;var r=e%4,t=Math.floor(e/4)%6,o=1;switch(metric){case 1:1!=r&&3!=r||(o*=2);case 2:1!=t&&4!=t&&5!=t||(o*=2);case 3:3==t&&(o=0),3!=metric||4!=t&&5!=t||(o*=2)}return o}function initInfoText(e){curInfoText=e.length>0&&e[0]>=1e3?e[0]-1e3:-1}function doMove(e,r,t,o,a){for(var i=a?t+o:t;;){if(a){if(i<=t)break;i--}if(r[i]>=1e3)curInfoText=a?-1:r[i]-1e3;else if(r[i]>=0){var n=r[i]%4+1,s=Math.floor(r[i]/4)%6;4==n&&(n=2),a&&(n=4-n),twistLayers(e,Math.floor(r[i]/24),n,s)}if(!a&&++i>=t+o)break}}var jobNumber=0,nowServing=0;function startAnimation(e){if(stopAnimation(),(demo||0!=move.length&&0!=move[curMove].length)&&(!demo||0!=demoMove.length&&0!=demoMove[0].length)){switch(moveDir=1,moveOne=!1,moveAnimated=!0,e){case 0:break;case 1:moveDir=-1;break;case 2:moveOne=!0;break;case 3:moveDir=-1,moveOne=!0;break;case 4:moveAnimated=!1}run(jobNumber++,moveDir)}}function stopAnimation(){1==animating&&(restarted=!0)}function clear(){movePos=0,natural=!0,mirrored=!1;for(var e=0;e<6;e++)for(var r=0;r<9;r++)cube[e][r]=initialCube[e][r],scube[e][r]=initialSCube[e][r];initialMove.length>0&&void 0!==initialMove[curMove]&&doMove(cube,initialMove[curMove],0,initialMove[curMove].length,!1),initialReversedMove.length>0&&void 0!==initialReversedMove[curMove]&&doMove(cube,initialReversedMove[curMove],0,initialReversedMove[curMove].length,!0),move.length>0&&initInfoText(move[curMove]),scramble>0&&(move=getMove(randMoves(3,randMoveCount),!1)),2==scramble&&doMove(cube,move[0],0,move[0].length,!0);for(e=0;e<3;e++)eye[e]=initialEye[e],eyeX[e]=initialEyeX[e],eyeY[e]=initialEyeY[e];setTimeout(paint,0)}var cubeBlocks=[[[0,3],[0,3]],[[0,3],[0,3]],[[0,3],[0,3]],[[0,3],[0,3]],[[0,3],[0,3]],[[0,3],[0,3]]],topBlocks=[],midBlocks=[],botBlocks=[],topBlockTable=[[[0,0],[0,0]],[[0,3],[0,3]],[[0,3],[0,1]],[[0,1],[0,3]],[[0,3],[2,3]],[[2,3],[0,3]]],midBlockTable=[[[0,0],[0,0]],[[0,3],[1,2]],[[1,2],[0,3]]],topBlockFaceDim=[[1,0,3,3,2,3],[0,1,5,5,4,5],[2,3,1,0,3,2],[4,5,0,1,5,4],[3,2,2,4,1,0],[5,4,4,2,0,1]],midBlockFaceDim=[[0,0,2,2,1,2],[0,0,2,2,1,2],[1,2,0,0,2,1],[1,2,0,0,2,1],[2,1,1,1,0,0],[2,1,1,1,0,0]],botBlockFaceDim=[[0,1,5,5,4,5],[1,0,3,3,2,3],[4,5,0,1,5,4],[2,3,1,0,3,2],[5,4,4,2,0,1],[3,2,2,4,1,0]];function splitCube(e){for(var r=0;r<6;r++)topBlocks[r]=topBlockTable[topBlockFaceDim[e][r]],botBlocks[r]=topBlockTable[botBlockFaceDim[e][r]],midBlocks[r]=midBlockTable[midBlockFaceDim[e][r]];natural=!1}function twistLayers(e,r,t,o){switch(o){case 3:twistLayer(e,1^r,t,!1);case 2:twistLayer(e,r,4-t,!1);case 1:twistLayer(e,r,4-t,!0);break;case 5:twistLayer(e,1^r,4-t,!1),twistLayer(e,r,4-t,!1);break;case 4:twistLayer(e,1^r,t,!1);default:twistLayer(e,r,4-t,!1)}}var cycleOrder=[0,1,2,5,8,7,6,3],cycleFactors=[1,3,-1,-3,1,3,-1,-3],cycleOffsets=[0,2,8,6,3,1,5,7],cycleLayerSides=[[3,3,3,0],[2,1,1,1],[3,3,0,0],[2,1,1,2],[3,2,0,0],[2,2,0,1]],cycleCenters=[[7,7,7,4],[6,5,5,5],[7,7,4,4],[6,5,5,6],[7,6,4,4],[6,6,4,5]],twistBuffer=[];function twistLayer(e,r,t,o){twistLayer2(e,r,t,o),1==superCube&&t>0&&t<4&&(twistLayer2(scube,r,t,o),twistSuperLayer(r,t,o))}function twistLayer2(e,r,t,o){if(!o){for(var a=0;a<8;a++)twistBuffer[(a+2*t)%8]=e[r][cycleOrder[a]];for(a=0;a<8;a++)e[r][cycleOrder[a]]=twistBuffer[a]}var i=3*t;for(a=0;a<4;a++)for(var n=adjacentFaces[r][a],s=o?cycleCenters[r][a]:cycleLayerSides[r][a],l=cycleFactors[s],c=cycleOffsets[s],d=0;d<3;d++,i++)twistBuffer[i%12]=e[n][d*l+c];for(a=0,i=0;a<4;a++)for(n=adjacentFaces[r][a],s=o?cycleCenters[r][a]:cycleLayerSides[r][a],l=cycleFactors[s],c=cycleOffsets[s],d=0;d<3;d++,i++)e[n][d*l+c]=twistBuffer[i]}var superTwistArr=[[[0,1,0],[0,3,1],[0,3,4],[0,1,5]],[[6,1,0],[2,3,1],[2,3,4],[6,1,5]],[[3,1,0],[1,3,1],[1,3,4],[3,1,5]],[[6,1,3],[0,1,1],[0,1,2],[0,3,0]],[[0,1,3],[2,3,0],[6,1,2],[6,1,1]],[[3,1,3],[3,1,1],[3,1,2],[1,3,0]],[[3,1,3],[1,3,0],[3,1,2],[3,1,1]]],width,height,lastX,lastY,lastDragX,lastDragY,dragAreas;function twistSuperLayer(e,r,t){if(0==t){for(var o=0;o<9;o++)scube[e][o]=(scube[e][o]+4-r)%4;2==e&&superTwist2(0,4-r),3==e&&superTwist2(1,r),4==e&&superTwist(3,r),5==e&&superTwist(4,r)}1==t&&(2==e&&superTwist2(2,4-r),3==e&&superTwist2(2,r),4==e&&superTwist(5,r),5==e&&superTwist(6,r))}function superTwist(e,r){superTwist1(superTwistArr[e][0]),superTwist1(superTwistArr[e][r])}function superTwist1(e){for(var r=e[0],t=0;t<3;r+=e[1],t++)scube[e[2]][r]=(scube[e[2]][r]+2)%4}function superTwist2(e,r){for(var t=0;t<4;t++)for(var o=superTwistArr[e][t],a=o[0],i=0;i<3;a+=o[1],i++)scube[o[2]][a]=(scube[o[2]][a]+r)%4}var dragCornersX=[],dragCornersY=[],dragDirsX=[],dragDirsY=[],dragBlocks=[[[0,0],[3,0],[3,1],[0,1]],[[3,0],[3,3],[2,3],[2,0]],[[3,3],[0,3],[0,2],[3,2]],[[0,3],[0,0],[1,0],[1,3]],[[0,1],[3,1],[3,2],[0,2]],[[2,0],[2,3],[1,3],[1,0]]],areaDirs=[[1,0],[0,1],[-1,0],[0,-1],[1,0],[0,1]],twistDirs=[[1,1,1,1,1,-1],[1,1,1,1,1,-1],[1,-1,1,-1,1,1],[1,-1,1,-1,-1,1],[-1,1,-1,1,-1,-1],[1,-1,1,-1,1,1]],dragLayers=[],dragModes=[],dragX,dragY,rotCos=[[[1,0,0],[0,0,0],[0,0,1]],[[1,0,0],[0,1,0],[0,0,0]],[[0,0,0],[0,1,0],[0,0,1]]],rotSin=[[[0,0,1],[0,0,0],[-1,0,0]],[[0,1,0],[-1,0,0],[0,0,0]],[[0,0,0],[0,0,1],[0,-1,0]]],rotVec=[[[0,0,0],[0,1,0],[0,0,0]],[[0,0,0],[0,0,0],[0,0,1]],[[1,0,0],[0,0,0],[0,0,0]]],rotSign=[1,-1,1,-1,1,-1],tempEye=[],tempEyeX=[],tempEyeY=[],tempEye2=[],tempEyeX2=[],tempEyeY2=[],perspEye=[],perspEyeI=[],perspNormal=[],eyeArray=[],eyeArrayX=[],eyeArrayY=[],eyeOrder=[[1,0,0],[0,1,0],[1,1,0],[1,1,1],[1,0,1],[1,0,2]],blockArray=[],blockMode=[[0,2,2],[2,1,2],[2,2,2],[2,2,2],[2,2,2],[2,2,2]],drawOrder=[[0,1,2],[2,1,0],[0,2,1]];function paint(){graphics.save(),graphics.fillStyle=bgColor;var e=1!=buttonBar||0!=progressHeight&&!demo&&0!=move[curMove].length?height:height-dpr;if(setClip(graphics,0,0,width,e),graphics.fillRect(0,0,width,e),dragAreas=0,natural)fixBlock(eye,eyeX,eyeY,cubeBlocks,3);else{for(var r=Math.cos(originalAngle+currentAngle),t=Math.sin(originalAngle+currentAngle)*rotSign[twistedLayer],o=0;o<3;o++){tempEye[o]=0,tempEyeX[o]=0;for(var a=0;a<3;a++){var i=Math.floor(twistedLayer/2);tempEye[o]+=eye[a]*(rotVec[i][o][a]+rotCos[i][o][a]*r+rotSin[i][o][a]*t),tempEyeX[o]+=eyeX[a]*(rotVec[i][o][a]+rotCos[i][o][a]*r+rotSin[i][o][a]*t)}}vMul(tempEyeY,tempEye,tempEyeX);var n=Math.cos(originalAngle-currentAngle),s=Math.sin(originalAngle-currentAngle)*rotSign[twistedLayer];for(o=0;o<3;o++){tempEye2[o]=0,tempEyeX2[o]=0;for(a=0;a<3;a++){i=Math.floor(twistedLayer/2);tempEye2[o]+=eye[a]*(rotVec[i][o][a]+rotCos[i][o][a]*n+rotSin[i][o][a]*s),tempEyeX2[o]+=eyeX[a]*(rotVec[i][o][a]+rotCos[i][o][a]*n+rotSin[i][o][a]*s)}}vMul(tempEyeY2,tempEye2,tempEyeX2),eyeArray[0]=eye,eyeArrayX[0]=eyeX,eyeArrayY[0]=eyeY,eyeArray[1]=tempEye,eyeArrayX[1]=tempEyeX,eyeArrayY[1]=tempEyeY,eyeArray[2]=tempEye2,eyeArrayX[2]=tempEyeX2,eyeArrayY[2]=tempEyeY2,blockArray[0]=topBlocks,blockArray[1]=midBlocks,blockArray[2]=botBlocks,vSub(vScale(vCopy(perspEye,eye),5+persp),vScale(vCopy(perspNormal,faceNormals[twistedLayer]),1/3)),vSub(vScale(vCopy(perspEyeI,eye),5+persp),vScale(vCopy(perspNormal,faceNormals[1^twistedLayer]),1/3));var l,c=vProd(perspEye,faceNormals[twistedLayer]),d=vProd(perspEyeI,faceNormals[1^twistedLayer]);l=c<0&&d>0?0:c>0&&d<0?1:2;for(o=0;o<3;o++){a=drawOrder[l][o];var u=eyeOrder[twistedMode][a];fixBlock(eyeArray[u],eyeArrayX[u],eyeArrayY[u],blockArray[a],blockMode[twistedMode][a])}}if(pushed||animating||(buttonPressed=-1),!(scramble>0&&2==buttonBar)){if(!demo&&move.length>0){if(move[curMove].length>0){if(progressHeight>0){graphics.lineWidth=lineWidth,graphics.strokeStyle=buttonBorderColor;var g=(width-2)*realMovePos(move[curMove],movePos)/realMoveLength(move[curMove]);graphics.fillStyle=sliderBgColor,graphics.fillRect(dph,height-progressHeight-dph,width-dpr,progressHeight),graphics.fillStyle=sliderColor,graphics.fillRect(dph,height-progressHeight-dph,g,progressHeight),graphics.beginPath(),graphics.rect(dph,height-progressHeight-dph,width-dpr,progressHeight),graphics.stroke()}graphics.font="bold "+textHeight+"px helvetica";var h=moveLength(move[curMove],movePos)+"/"+moveLength(move[curMove],-1)+metricChar[metric],v=graphics.measureText(h).width,f=width-v-2,m=height-progressHeight-Math.floor(4*dpr);moveText>0?(moveCounter&&drawString(graphics,h,outlined?f-dpr:f,m-textHeight),drawMoveTextFunc(graphics,m)):moveCounter&&drawString(graphics,h,outlined?f-dpr:f,m)}if(move.length>1){graphics.font="bold "+textHeight+"px helvetica";h=curMove+1+"/"+move.length,v=graphics.measureText(h).width,f=width-v-2*buttonHeight-Math.floor(5*dpr);drawString(graphics,h,f,adjTextHeight()),drawButton(graphics,7,width-2*buttonHeight,0),drawButton(graphics,8,width-buttonHeight,0)}}curInfoText>=0&&(graphics.font="bold "+textHeight+"px helvetica",drawString(graphics,infoText[curInfoText],outlined?dpr:0,adjTextHeight()))}graphics.restore(),drawButtons&&0!=buttonBar&&drawButtonsFunc(graphics)}function adjTextHeight(){return utextHeight<10?Math.floor(10*dpr):utextHeight<12?Math.floor(12*dpr):utextHeight<14?Math.floor(14*dpr):textHeight}var fillX=[],fillY=[],coordsX=[],coordsY=[],cooX=[[],[],[],[],[],[]],cooY=[[],[],[],[],[],[]],border=[[.1,.1],[.9,.1],[.9,.9],[.1,.9]],factors=[[0,0],[0,1],[1,1],[1,0]],tempNormal=[];function fixBlock(e,r,t,o,a){for(var i=0;i<8;i++){var n=(c=width0&&g>0)for(var h=0,v=o[i][1][0];h-(1-scale)){u=o[i][0][1]-o[i][0][0],g=o[i][1][1]-o[i][1][0];if(u>0&&g>0)for(h=0,v=o[i][1][0];h=4&&(dragLayers[dragAreas]&=-2),dragModes[dragAreas]=Math.floor(l/4),18==++dragAreas)break}else if(0==a){if(i!=twistedLayer&&u>0&&g>0){for(l=3==u?0==o[i][1][0]?0:2:0==o[i][0][0]?3:1,p=0;p<4;p++)getCorners(i,p,dragCornersX[dragAreas],dragCornersY[dragAreas],dragBlocks[l][p][0],dragBlocks[l][p][1],!1);dragDirsX[dragAreas]=(b*areaDirs[l][0]+y*areaDirs[l][1])*twistDirs[i][l],dragDirsY[dragAreas]=(w*areaDirs[l][0]+M*areaDirs[l][1])*twistDirs[i][l],dragLayers[dragAreas]=twistedLayer,dragModes[dragAreas]=0,dragAreas++}}else if(1==a&&i!=twistedLayer&&u>0&&g>0){for(l=3==u?4:5,p=0;p<4;p++)getCorners(i,p,dragCornersX[dragAreas],dragCornersY[dragAreas],dragBlocks[l][p][0],dragBlocks[l][p][1],!1);dragDirsX[dragAreas]=(b*areaDirs[l][0]+y*areaDirs[l][1])*twistDirs[i][l],dragDirsY[dragAreas]=(w*areaDirs[l][0]+M*areaDirs[l][1])*twistDirs[i][l],dragLayers[dragAreas]=twistedLayer,dragModes[dragAreas]=1,dragAreas++}}}function getCorners(e,r,t,o,a,i,n){a/=3,i/=3;var s=cooX[e][0]+(cooX[e][1]-cooX[e][0])*a,l=cooY[e][0]+(cooY[e][1]-cooY[e][0])*a,c=cooX[e][3]+(cooX[e][2]-cooX[e][3])*a,d=cooY[e][3]+(cooY[e][2]-cooY[e][3])*a;t[r]=Math.floor(.5+s+(c-s)*i),o[r]=Math.floor(.5+l+(d-l)*i),n&&(t[r]=width-t[r])}var textOffsetInit=[1,1,-1,-1,-1,1,1,-1,-1,0,1,0,0,1,0,-1],textOffset=[];function drawString(e,r,t,o){if(outlined){e.fillStyle="black";for(var a=0;awidth&&(c=Math.min(1,width/2-n-s/2),c=Math.max(c,width-n-s-l-2)),s>0&&(e.fillStyle=hlColor,e.lineWidth=2,e.strokeStyle="black",e.beginPath(),utextHeight<=14?e.fillRect(c+n-1,height-progressHeight-textHeight-Math.floor(4*dpr),s+2,textHeight+Math.floor(3*dpr)):e.fillRect(c+n-1,height-progressHeight-textHeight-Math.floor(2*dpr),s+2,textHeight+Math.floor(dpr))),n>0&&drawString(e,o,c,r),s>0&&drawString(e,a,c+n,r),l>0&&drawString(e,i,c+n+s,r)}function selectButton(e,r){if(0==buttonBar)return-1;if(move.length>1&&e>=width-2*buttonHeight&&e=0&&r1&&e>=width-buttonHeight&&e=0&&r=0&&e=height-buttonHeight&&r=t&&e=height&&r0&&6==buttonPressed?darker(buttonBgColor):buttonBgColor,e.fillRect(dph,height-buttonHeight,buttonHeight,buttonHeight),e.lineWidth=lineWidth,e.strokeStyle=buttonBorderColor,e.beginPath(),e.rect(dph,height-buttonHeight-dph,buttonHeight,buttonHeight),e.stroke(),void drawButton(e,0,buttonHeight/2,height-(buttonHeight+1)/2-r);if(1!=buttonBar);else{for(var t=0,o=0;o<7;o++){var a=Math.floor((width-t)/(7-o));e.fillStyle=buttonPressed==o?darker(buttonBgColor):buttonBgColor,e.fillRect(t,height,a,buttonHeight),e.lineWidth=lineWidth,e.strokeStyle=buttonBorderColor,e.beginPath(),0==o?e.rect(t+dph,height-dph,a-dpr,buttonHeight):e.rect(t-dph,height-dph,a,buttonHeight),e.stroke(),e.strokeStyle="black",drawButton(e,o,t+a/2,height+buttonHeight/2-r),t+=a}drawButtons=!1}}var ds=[];function drawButton(e,r,t,o){switch(t=Math.floor(t),o=Math.floor(o),r){case 0:drawRect(e,t-ds[4],o-ds[3],ds[3],ds[6]+1),drawArrow(e,t+ds[4],o,-1);break;case 1:drawRect(e,t+ds[1],o-ds[3],ds[3],ds[6]+1),drawArrow(e,t-ds[1],o,-1);break;case 2:drawArrow(e,t+ds[1],o,-1);break;case 3:animating?drawRect(e,t-ds[4],o-ds[3],ds[7],ds[7]):(drawRect(e,t-ds[4],o-ds[2],ds[7],ds[5]),drawRect(e,t-ds[2],o-ds[4],ds[3],ds[9]));break;case 4:drawArrow(e,t-ds[2],o,1);break;case 5:drawRect(e,t-ds[4],o-ds[3],ds[3],ds[6]+1),drawArrow(e,t,o,1);break;case 6:drawRect(e,t+ds[1],o-ds[3],ds[3],ds[6]+1),drawArrow(e,t-ds[4],o,1);break;case 7:var a=7==buttonPressed?darker(buttonBgColor):buttonBgColor;drawRect2(e,t-2*dpr,o+dpr,buttonHeight,o+buttonHeight,a),drawArrow(e,t+2*dpr+buttonHeight/2-3*dpr,o+buttonHeight/2+dph,-1);break;case 8:a=8==buttonPressed?darker(buttonBgColor):buttonBgColor;drawRect2(e,t-2*dpr,o+dpr,buttonHeight,o+buttonHeight,a),drawArrow(e,t-dpr+buttonHeight/2-3*dpr,o+buttonHeight/2+dph,1)}}function drawArrow(e,r,t,o){var a=3*dpr,i=[],n=[];i[0]=r,i[1]=r+o,i[2]=r+4*dpr*o,i[3]=r+o,i[4]=r,n[0]=t-a,n[1]=t-a,n[2]=t,n[3]=t+a,n[4]=t+a,drawArrow2(e,i,n)}function drawArrow2(e,r,t){e.beginPath(),e.moveTo(r[0]+dph,t[0]+dph);for(var o=1;o<5;o++)e.lineTo(r[o]+dph,t[o]+dph);e.closePath(),e.fillStyle="white",e.strokeStyle="black",e.fill(),e.lineWidth=lineWidth,e.stroke()}function drawRect(e,r,t,o,a){e.lineWidth=lineWidth,e.beginPath(),e.rect(r+dph,t+dph,o-1,a-1),e.fillStyle="white",e.fill(),e.strokeStyle="black",e.stroke()}function drawRect2(e,r,t,o,a,i){e.lineWidth=lineWidth,e.beginPath(),e.rect(r+dph,t+dph,o-1,a-1),e.fillStyle=i,e.fill(),e.strokeStyle="black",e.stroke()}function drawPolygon(e,r,t,o){e.beginPath(),e.moveTo(r[0],t[0]),e.lineTo(r[1],t[1]),e.lineTo(r[2],t[2]),e.lineTo(r[3],t[3]),e.closePath(),e.strokeStyle=o,e.lineWidth=.7*dpr,e.stroke()}function fillPolygon(e,r,t,o){e.beginPath(),e.moveTo(r[0],t[0]),e.lineTo(r[1],t[1]),e.lineTo(r[2],t[2]),e.lineTo(r[3],t[3]),e.closePath(),e.fillStyle=o,e.fill()}var superRotate=[[0,1,2,3],[3,0,1,2],[2,3,0,1],[1,2,3,0]],offsetX,offsetY;function drawSuperArrow(e,r,t,o,a,i){var n=[],s=[];if(2!=scw||"#ffffff"!=i){for(var l=0;l<4;l++)n[l]=Math.floor(r[l]+.05*(r[superRotate[2][l]]-r[l])),s[l]=Math.floor(t[l]+.05*(t[superRotate[2][l]]-t[l]));0==o&&(a=(a+1)%4),4==o&&(a=(a+3)%4);var c=superRotate[a][0],d=superRotate[a][1],u=superRotate[a][2],g=superRotate[a][3],h=.26*(n[u]-n[d]),v=.26*(s[u]-s[d]),f=(n[c]-n[d])/2,m=s[d]+(s[c]-s[d])/2,p=s[u]+(s[g]-s[u])/2,b=(n[g]-n[u])/2,w=1^a;e.fillStyle=i,e.beginPath(),e.moveTo(n[c]+(n[g]-n[c])/2,s[c]+(s[g]-s[c])/2),e.lineTo(n[w]+f,m),e.lineTo(n[w]+h+f,m+v),e.lineTo(n[w]+h,s[w]+v),w=(w+1)%4,e.lineTo(n[w]-h,s[w]-v),e.lineTo(n[w]-h+b,p-v),e.lineTo(n[w]+b,p),e.closePath(),e.fill(),e.lineWidth=.6*dpr,e.strokeStyle="black",e.stroke()}}function colorToHex(e){return"white"==e?"#FFFFFF":"black"==e?"#000000":"#808080"}function colorAverage(e){return"#"!=e.substring(0,1)&&(e=colorToHex(e)),(299*parseInt(e.substring(1,3),16)+587*parseInt(e.substring(3,5),16)+114*parseInt(e.substring(5,7),16))/1e3}function brighter(e){"#"!=e.substring(0,1)&&(e=colorToHex(e));var r=parseInt(e.substring(1,3),16),t=parseInt(e.substring(3,5),16),o=parseInt(e.substring(5,7),16);return rgbToHex((r=Math.floor(1.3*r))>255?255:r,(t=Math.floor(1.3*t))>255?255:t,(o=Math.floor(1.3*o))>255?255:o)}function darker(e){"#"!=e.substring(0,1)&&(e=colorToHex(e));var r=parseInt(e.substring(1,3),16),t=parseInt(e.substring(3,5),16),o=parseInt(e.substring(5,7),16);return rgbToHex(r=Math.floor(.7*r),t=Math.floor(.7*t),o=Math.floor(.7*o))}function run(e,r){if(e>nowServing)setTimeout(run,0,e,r);else{if(!demo&&(0==move.length||0==move[curMove].length))return animating=!1,drawButtons=!0,void paint();if(!moveAnimated){for(var t=move[curMove];movePos=1e3)curInfoText=t[movePos]-1e3;else if(-1!=t[movePos]){var o=t[movePos]%4+1,a=Math.floor(t[movePos]/4)%6,i=Math.floor(t[movePos]/24);twistLayers(cube,i,4==o?2:o,a)}movePos++}return animating=!1,drawButtons=!0,paint(),void nowServing++}var n,s,l,c,d,u,g=r;interrupted=!1,requestAnimationFrame((function e(){if(v){if(v=!1,u=!1,repeatable)g>0?movePos>=t.length&&(movePos=0,initInfoText(t)):(curInfoText=-1,0==movePos&&(movePos=t.length));else if(g>0&&movePos>=t.length||g<0&&0==movePos)return restarted=!1,animating=!1,nowServing++,drawButtons=!0,void paint();animating=!0,drawButtons=!0}if(f){Date.now();if(f=!1,g<0&&(b=!1,0==movePos?(b=!0,p=!0):movePos--),!b){if(h=!1,-1==t[movePos]){if(paint(),!moveOne)for(n=Date.now();Date.now()-n<1e3;);}else t[movePos]>=1e3?curInfoText=g>0?t[movePos]-1e3:-1:h=!0;if(h){o=t[movePos]%4+1,a=Math.floor(t[movePos]/4)%6;var r=o<3;if(4==o&&(o=2),g<0&&(r=!r,o=4-o),i=Math.floor(t[movePos]/24),twisting=!1,natural=!0,spinning=!0,originalAngle=0,faceTwistDirs[i]>0&&(r=!r),moveAnimated){d=Math.PI/2,c=r?1:-1;var w=67*speed;2==o&&(d=Math.PI,w=67*doubleSpeed),twisting=!0,twistedLayer=i,twistedMode=a,splitCube(i),n=Date.now(),s=n,l=c*d/w,currentAngle=0}}else m=!0}}b||(h&&(moveAnimated&¤tAngle*c0?(++movePos=1e3&&(curInfoText=t[movePos]-1e3,movePos++),movePos==t.length&&(demo?clearDemo(t):p=!0)):curInfoText=-1,(interrupted||restarted||u)&&(p=!0)));if(p)return p=!1,v=!0,jobNumber<=nowServing+1&&(animating=!1),drawButtons=!0,(0==buttonPressed||buttonPressed>6)&&clear(),paint(),demo&&(clear(),demo=!1),restarted=!1,nowServing++,void(1==movePos&&t[0]>=1e3&&movePos--);requestAnimationFrame(e)}));var h=!1,v=!0,f=!0,m=!1,p=!1,b=!1;t=demo?demoMove[0]:move[curMove]}}function clearDemo(e){movePos=0;for(var r=0;r<6;r++)for(var t=0;t<9;t++)cube[r][t]=initialCube[r][t],scube[r][t]=initialSCube[r][t];initialMove.length>0&&void 0!==initialMove[curMove]&&doMove(cube,initialMove[curMove],0,initialMove[curMove].length,!1),initialReversedMove.length>0&&void 0!==initialReversedMove[curMove]&&doMove(cube,initialReversedMove[curMove],0,initialReversedMove[curMove].length,!0),initInfoText(e)}document.addEventListener("touchstart",mousedown),document.addEventListener("touchmove",mousemove,{passive:!1}),document.addEventListener("touchend",mouseup),document.addEventListener("mousedown",mousedown),document.addEventListener("mousemove",mousemove),document.addEventListener("mouseup",mouseup),document.addEventListener("contextmenu",contextmenu);var mouseIsDown=!1,showContextMenu=!0,divs=document.getElementsByTagName("div"),wrapDiv=divs.length>0&&"wrap"==divs[0].className;function touchfunc(e){wrapDiv?divs[0].style.overflow=e:document.body.style.overflow=e}function mouseup(e){if(mouseIsDown&&void 0!==e.touches&&(e.preventDefault(),touchfunc("auto")),mouseIsDown?setTimeout((function(){showContextMenu=!0}),100):showContextMenu=!0,mouseIsDown=!1,dragging=!1,pushed)pushed=!1,drawButtons=!0,paint();else if(twisting&&!spinning){twisting=!1,originalAngle+=currentAngle,currentAngle=0;for(var r=originalAngle;r<0;)r+=32*Math.PI;var t=Math.floor(8*r/Math.PI)%16;(snap||t%4==0||t%4==3)&&(t=Math.floor((t+2)/4),faceTwistDirs[twistedLayer]>0&&(t=(4-t)%4),originalAngle=0,natural=!0,twistLayers(cube,twistedLayer,t,twistedMode)),paint()}}function mousedown(e){var r=canvas.getBoundingClientRect(),t=Math.floor(r.left),o=Math.floor(r.top);if(void 0===e.touches)var a=e.clientX,i=e.clientY;else a=e.touches[0].clientX,i=e.touches[0].clientY;at+width/dpr||io+(height+(1==buttonBar?buttonHeight:0))/dpr||(e.preventDefault(),mouseIsDown=!0,showContextMenu=!1,void 0!==e.touches&&touchfunc("hidden"),offsetX=t,offsetY=o,lastDragX=lastX=getX(e),lastDragY=lastY=getY(e),toTwist=!1,(buttonPressed=selectButton(lastX,lastY))>=0?button():progressHeight>0&&move.length>0&&move[curMove].length>0&&lastY>height-progressHeight&&lastY<=height?clickProgress&&(stopAnimation(),progress(jobNumber++)):(mirrored&&(lastDragX=lastX=width-lastX),void 0===e.touches?!editable||animating||0!=e.button||e.shiftKey||(toTwist=!0):editable&&!animating&&(toTwist=!0)))}function button(){pushed=!0,3==buttonPressed?animating?stopAnimation():mirrored=!mirrored:0==buttonPressed?scramble>0&&2==buttonBar?1==scrambleToggle?(scrambleToggle=!1,stopAnimation(),clear()):(scrambleToggle=!0,startAnimation(buttonAction[buttonPressed=6])):(stopAnimation(),clear()):7==buttonPressed||8==buttonPressed?(stopAnimation(),setTimeout(clear,0),curMove=7==buttonPressed?curMove>0?curMove-1:move.length-1:curMovenowServing)setTimeout(progress,0,e);else{var r=realMoveLength(move[curMove]),t=Math.floor(((lastX-1)*r*2/(width-2)+1)/2);(t=Math.max(0,Math.min(r,t)))>0&&(t=arrayMovePos(move[curMove],t)),t>movePos&&doMove(cube,move[curMove],movePos,t-movePos,!1),t0&&(t=arrayMovePos(move[curMove],t)),t>movePos&&doMove(cube,move[curMove],movePos,t-movePos,!1),t0&&v<1&&f>0&&f<1){if(i*i+n*n<144)return;if(dragX=dragDirsX[s],dragY=dragDirsY[s],Math.abs(dragX*i+dragY*n)/Math.sqrt((dragX*dragX+dragY*dragY)*(i*i+n*n))>.75){twisting=!0,twistedLayer=dragLayers[s],twistedMode=dragModes[s];break}}}toTwist=!1,lastX=lastDragX,lastY=lastDragY}i=(o-lastX)/dpr,n=(a-lastY)/dpr,!twisting||animating?rotateAllowed&&(vNorm(vAdd(eye,vScale(vCopy(eyeD,eyeX),-.016*i))),vNorm(vMul(eyeX,eyeY,eye)),vNorm(vAdd(eye,vScale(vCopy(eyeD,eyeY),.016*n))),vNorm(vMul(eyeY,eye,eyeX)),lastX=o,lastY=a):(natural&&splitCube(twistedLayer),currentAngle=.03*(dragX*i+dragY*n)/Math.sqrt(dragX*dragX+dragY*dragY)),paint()}}function getX(e){return void 0===e.touches?(e.clientX-offsetX)*dpr:(e.touches[0].clientX-offsetX)*dpr}function getY(e){return void 0===e.touches?(e.clientY-offsetY)*dpr:(e.touches[0].clientY-offsetY)*dpr}function contextmenu(e){showContextMenu||e.preventDefault()}function resize(){clearTimeout(timer),timer=setTimeout((function(){scaleCanvas(),drawButtons=!0,paint()}),20)}function init2(){graphics=canvas.getContext("2d"),ubuttonHeight=buttonHeight,uprogressHeight=progressHeight,utextHeight=textHeight,scaleCanvas(),parNode.appendChild(canvas),paint()}function scaleCanvas(){height=parNode.clientHeight-1,width=parNode.clientWidth-1,setCanvasCssSize(),dpr=devicePixelRatio,dph=dpr/2,height=Math.floor(height*dpr),width=Math.floor(width*dpr),setCanvasSize(),lineWidth=dpr,buttonHeight=Math.floor(ubuttonHeight*dpr),progressHeight=Math.floor(uprogressHeight*dpr),textHeight=Math.floor(utextHeight*dpr);for(var e=1;e<10;e++)ds[e]=e*dpr;for(e=0;e4?3:2)),5==e&&1==o&&(o=0),d+=""+a[s]+i[o]+n[t]+" ")}return d}window.addEventListener("resize",resize);var searchParams=[];function parseSearchParams(){for(var e=params.split("&"),r=0;r="0"&&e.charAt(r)<="9"&&(borderWidth=10*borderWidth+parseInt(e[r]));borderWidth>=0&&borderWidth<=20&&setBorderWidth(borderWidth/100)}if(superCube)for(r=0;r<6;r++)for(t=0;t<16;t++)scube[r][t]=0;var a="lluu";if(null!=(e=getParameter("colorscheme"))&&6==e.length)for(r=0;r<6;r++){var i=23;for(t=0;t<23;t++)if(e[r].toLowerCase()=="0123456789wyorgbldmcpnk".charAt(t)){i=t;break}for(t=0;t<16;t++)cube[r][t]=i}if("1"==(e=getParameter("scramble"))?scramble=1:"2"==e&&(scramble=2),2==scramble)for(r=0;r<6;r++)for(t=0;t<16;t++)initialCube[r][t]=cube[r][t],initialSCube[r][t]=scube[r][t];if(0==scramble){if(null!=(e=getParameter("pos"))&&96==e.length){a="uuuuff","gray"==bgColor&&(bgColor="white");for(r=0;r<6;r++){var n=posFaceTransform[r];for(t=0;t<16;t++){var s=posFaceletTransform[r][t];cube[n][s]=23;for(var l=0;l<14;l++)if(e.charAt(16*r+t)=="DFECABdfecabgh".charAt(l)){cube[n][s]=l+4;break}}}}if(null!=(e=getParameter("facelets"))&&96==e.length)for(r=0;r<6;r++)for(t=0;t<16;t++){cube[r][t]=23;for(l=0;l<23;l++)if(e[16*r+t].toLowerCase()=="0123456789wyorgbldmcpnk".charAt(l)){cube[r][t]=l;break}}if(null!=(e=getParameter("superfacelets"))&&96==e.length)for(r=0;r<6;r++)for(t=0;t<16;t++)for(l=0;l<4;l++)if(e[16*r+t].toLowerCase()=="0123".charAt(l)){scube[r][t]=l;break}}if(moveText=0,yzAlt=!1,signNotation=!1,null!=(e=getParameter("sign"))&&"1"==e&&(signNotation=!0,moveText=5,yzAlt=!0),wcaNotation=!1,null!=(e=getParameter("wca"))&&"1"==e&&(wcaNotation=!0,moveText=6,yzAlt=!0),null!=(e=getParameter("yz"))&&("0"==e?yzAlt=!1:"1"==e&&(yzAlt=!0)),null!=(e=getParameter("randmoves"))){var c=0;for(r=0;r="0"&&e.charAt(r)<="9"&&(c=10*c+parseInt(e[r]));c>0&&(randMoveCount=c)}("random"==(e=getParameter("move"))||scramble>0)&&(e=randMoves(4,randMoveCount)),move=null==e?[]:getMove(e,!0),movePos=0,curInfoText=-1,0==scramble&&(null!=(e=getParameter("initmove"))&&("random"==e&&(e=randMoves(4,randMoveCount)),initialMove="#"==e?move:getMove(e,!1)),null!=(e=getParameter("initrevmove"))&&("random"==e&&(e=randMoves(4,randMoveCount)),initialReversedMove="#"==e?move:getMove(e,!1)),null!=(e=getParameter("demo"))&&("random"==e&&(e=randMoves(4,randMoveCount)),(demoMove="#"==e?move:getMove(e,!0)).length>0&&demoMove[0].length>0&&(demo=!0))),e=getParameter("position"),vNorm(vMul(eyeY,eye,eyeX)),null==e&&(e=a);var d=Math.PI/12;for(r=0;r="0"&&e.charAt(r)<="9"&&(speed=10*speed+parseInt(e[r]));if(null!=(e=getParameter("doublespeed")))for(r=0;r="0"&&e.charAt(r)<="9"&&(doubleSpeed=10*doubleSpeed+parseInt(e[r]));if(0==speed&&(speed=10),0==doubleSpeed&&(doubleSpeed=3*speed/2),persp=0,null==(e=getParameter("perspective")))persp=2;else for(r=0;r="0"&&e.charAt(r)<="9"&&(persp=10*persp+parseInt(e[r]));var g,h=0;if(null!=(e=getParameter("scale")))for(r=0;r="0"&&e.charAt(r)<="9"&&(h=10*h+parseInt(e[r]));if(scale=1/(1+h/10),hint=!1,null!=(e=getParameter("hint"))){hint=!0,faceShift=0;for(r=0;r="0"&&e.charAt(r)<="9"&&(faceShift=10*faceShift+parseInt(e[r]));faceShift<1?hint=!1:faceShift/=10}(hintHoriz=3.7,null!=(e=getParameter("hinthoriz")))&&((g=parseFloat(e))>0&&(hintHoriz=g));(hintVert=3.7,null!=(e=getParameter("hintvert")))&&((g=parseFloat(e))>0&&(hintVert=g));(hintBorder=0,null!=(e=getParameter("hintborder"))&&"1"==e&&(hintBorder=1),buttonHeight=13,null!=(e=getParameter("buttonheight")))&&((g=parseInt(e))>=9&g<=25&&(buttonHeight=g));(progressHeight=0==move.length?0:6,buttonBar=1,"0"==(e=getParameter("buttonbar"))?(buttonBar=0,buttonHeight=0,progressHeight=0):"1"==e?buttonBar=1:"2"!=e&&0!=move.length||(buttonBar=2,progressHeight=0),e=getParameter("edit"),editable="0"!=e,e=getParameter("repeat"),repeatable="0"!=e,e=getParameter("clickprogress"),clickProgress="0"!=e,"0"==(e=getParameter("movetext"))?moveText=0:"1"==e?moveText=1:"5"==e?moveText=5:"6"==e&&(moveText=6),moveTextSpace=1,"0"==(e=getParameter("movetextspace"))&&(moveTextSpace=0),null!=(e=getParameter("textsize")))&&((g=parseInt(e))>=5&g<=40&&(textHeight=g));(e=getParameter("fonttype"),outlined=null==e||"1"==e,metric=0,null!=(e=getParameter("metric"))&&("1"==e?metric=1:"2"==e?metric=2:"3"==e&&(metric=3)),align=1,null!=(e=getParameter("align")))&&((g=parseInt(e))>=0&g<=3&&(align=g),g>=3&g<=99&&(align=g/100));null!=(e=getParameter("snap"))&&"1"==e&&(snap=!0);for(r=0;r<6;r++)for(t=0;t<16;t++)initialCube[r][t]=cube[r][t],initialSCube[r][t]=scube[r][t];initialMove.length>0&&doMove(cube,initialMove[0],0,initialMove[0].length,!1),initialReversedMove.length>0&&doMove(cube,initialReversedMove[0],0,initialReversedMove[0].length,!0),2==scramble&&doMove(cube,move[0],0,move[0].length,!0);for(r=0;r<3;r++)initialEye[r]=eye[r],initialEyeX[r]=eyeX[r],initialEyeY[r]=eyeY[r];colorAverage(bgColor)<128?(textColor="white",hlColor=brighter(bgColor)):(textColor="black",hlColor=darker(bgColor)),buttonBorderColor=colorAverage(buttonBgColor)<128?"white":"black",sliderColor=textColor,null!=(e=getParameter("slidercolor"))&&6==e.length&&validateColor(e)&&(sliderColor="#"+e),sliderBgColor=darker(bgColor),null!=(e=getParameter("sliderbgcolor"))&&6==e.length&&validateColor(e)&&(sliderBgColor="#"+e),null!=(e=getParameter("troughcolor"))&&6==e.length&&validateColor(e)&&(sliderBgColor="#"+e),cubeColor="black",null!=(e=getParameter("cubecolor"))&&6==e.length&&validateColor(e)&&(cubeColor="#"+e),moveCounter=1,"0"==(e=getParameter("counter"))&&(moveCounter=0),curInfoText=move.length>0&&move[0][0]>=1e3?0:-1,init2(),demo&&startAnimation(-1)}function getParameter(e){var r=searchParams[e];return void 0===r?config[e]:r}function setBorderWidth(e){border[0][0]=border[0][1]=border[1][1]=border[3][0]=e,border[1][0]=border[2][0]=border[2][1]=border[3][1]=1-e}var moveModes=[0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,2,2,2,2,2,2],moveCodes=[0,1,2,3,4,5,1,2,4,5,2,0,5,2,0,0,1,2,3,4,5];function getMove(e,r){if(r){for(var t=e.indexOf("{");-1!=t;)t=e.indexOf("{",t+1);if(null==infoText)curInfoText=0,infoText=[];else{for(var o=[],a=0;a=e.length)return"";for(var o="",a=r;a=e.length)return"";if(e[r]>=1e3)return"";if(-1==e[r])return".";var t=turnSymbol[moveText-1][Math.floor(e[r]/4)%7][Math.floor(e[r]/28)];return"~"==t.charAt(0)?t.substring(1)+modifierStrings[(e[r]+2)%4]:t+modifierStrings[e[r]%4]}var metricChar=["","q","h","s"];function realMoveLength(e){for(var r=0,t=0;t=1e3;)t++;if(o==r)break;t=1e3)return 0;var r=e%4,t=Math.floor(e/4)%7,o=1;switch(metric){case 1:1!=r&&3!=r||(o*=2);case 2:(1==t||t>3)&&(o*=2);case 3:3==t&&(o=0),3!=metric||4!=t&&5!=t||(o*=2)}return o}function initInfoText(e){curInfoText=e.length>0&&e[0]>=1e3?e[0]-1e3:-1}function doMove(e,r,t,o,a){for(var i=a?t+o:t;;){if(a){if(i<=t)break;i--}if(r[i]>=1e3)curInfoText=a?-1:r[i]-1e3;else if(r[i]>=0){var n=r[i]%4+1,s=Math.floor(r[i]/4)%7;4==n&&(n=2),a&&(n=4-n),twistLayers(e,Math.floor(r[i]/28),n,s)}if(!a&&++i>=t+o)break}}var jobNumber=0,nowServing=0;function startAnimation(e){if(stopAnimation(),(demo||0!=move.length&&0!=move[curMove].length)&&(!demo||0!=demoMove.length&&0!=demoMove[0].length)){switch(moveDir=1,moveOne=!1,moveAnimated=!0,e){case 0:break;case 1:moveDir=-1;break;case 2:moveOne=!0;break;case 3:moveDir=-1,moveOne=!0;break;case 4:moveAnimated=!1}run(jobNumber++,moveDir)}}function stopAnimation(){1==animating&&(restarted=!0)}function clear(){movePos=0,natural=!0,mirrored=!1;for(var e=0;e<6;e++)for(var r=0;r<16;r++)cube[e][r]=initialCube[e][r],scube[e][r]=initialSCube[e][r];initialMove.length>0&&void 0!==initialMove[curMove]&&doMove(cube,initialMove[curMove],0,initialMove[curMove].length,!1),initialReversedMove.length>0&&void 0!==initialReversedMove[curMove]&&doMove(cube,initialReversedMove[curMove],0,initialReversedMove[curMove].length,!0),move.length>0&&initInfoText(move[curMove]),scramble>0&&(move=getMove(randMoves(4,randMoveCount),!1)),2==scramble&&doMove(cube,move[0],0,move[0].length,!0);for(e=0;e<3;e++)eye[e]=initialEye[e],eyeX[e]=initialEyeX[e],eyeY[e]=initialEyeY[e];setTimeout(paint,0)}var cubeBlocks=[[[0,4],[0,4]],[[0,4],[0,4]],[[0,4],[0,4]],[[0,4],[0,4]],[[0,4],[0,4]],[[0,4],[0,4]]],topBlocks=[],midBlocks=[],midBlocks2=[],botBlocks=[],topBlockTable=[[[0,0],[0,0]],[[0,4],[0,4]],[[0,4],[0,1]],[[0,1],[0,4]],[[0,4],[3,4]],[[3,4],[0,4]]],topBlockFaceDim=[[1,0,3,3,2,3],[0,1,5,5,4,5],[2,3,1,0,3,2],[4,5,0,1,5,4],[3,2,2,4,1,0],[5,4,4,2,0,1]],botBlockFaceDim=[[0,1,5,5,4,5],[1,0,3,3,2,3],[4,5,0,1,5,4],[2,3,1,0,3,2],[5,4,4,2,0,1],[3,2,2,4,1,0]],midBlockTable=[[[0,0],[0,0]],[[0,4],[2,3]],[[0,4],[1,2]],[[2,3],[0,4]],[[1,2],[0,4]]],midBlockFaceDim=[[0,0,4,4,2,4],[0,0,3,3,1,3],[2,4,0,0,4,2],[1,3,0,0,3,1],[4,2,2,1,0,0],[3,1,1,2,0,0]],midBlockFaceDim2=[[0,0,3,3,1,3],[0,0,4,4,2,4],[1,3,0,0,3,1],[2,4,0,0,4,2],[3,1,1,2,0,0],[4,2,2,1,0,0]];function splitCube(e){for(var r=0;r<6;r++)topBlocks[r]=topBlockTable[topBlockFaceDim[e][r]],botBlocks[r]=topBlockTable[botBlockFaceDim[e][r]],midBlocks[r]=midBlockTable[midBlockFaceDim[e][r]],midBlocks2[r]=midBlockTable[midBlockFaceDim2[e][r]];natural=!1}function twistLayers(e,r,t,o){switch(o){case 3:twistLayer(e,r,4-t,!1),twistLayer(e,r,4-t,!0),twistLayer(e,1^r,t,!0),twistLayer(e,1^r,t,!1);break;case 2:twistLayer(e,r,4-t,!1);case 1:twistLayer(e,r,4-t,!0);break;case 6:twistLayer(e,r,4-t,!0),twistLayer(e,1^r,t,!0);break;case 5:twistLayer(e,1^r,4-t,!1),twistLayer(e,r,4-t,!1);break;case 4:twistLayer(e,1^r,t,!1);default:twistLayer(e,r,4-t,!1)}}var cycleOrder=[0,1,2,3,7,11,15,14,13,12,8,4],cycleOrder2=[5,6,10,9],cycleFactors=[1,4,-1,-4,1,4,-1,-4],cycleOffsets=[0,3,15,12,4,2,11,13],cycleLayerSides=[[3,3,3,0],[2,1,1,1],[3,3,0,0],[2,1,1,2],[3,2,0,0],[2,2,0,1]],cycleCenters=[[7,7,7,4],[6,5,5,5],[7,7,4,4],[6,5,5,6],[7,6,4,4],[6,6,4,5]],twistBuffer=[];function twistLayer(e,r,t,o){twistLayer2(e,r,t,o),1==superCube&&t>0&&t<4&&(twistLayer2(scube,r,t,o),twistSuperLayer(r,t,o))}function twistLayer2(e,r,t,o){if(!o){for(var a=0;a<12;a++)twistBuffer[(a+3*t)%12]=e[r][cycleOrder[a]];for(a=0;a<12;a++)e[r][cycleOrder[a]]=twistBuffer[a];for(a=0;a<4;a++)twistBuffer[(a+t)%4]=e[r][cycleOrder2[a]];for(a=0;a<4;a++)e[r][cycleOrder2[a]]=twistBuffer[a]}var i=4*t;for(a=0;a<4;a++)for(var n=adjacentFaces[r][a],s=o?cycleCenters[r][a]:cycleLayerSides[r][a],l=cycleFactors[s],c=cycleOffsets[s],d=0;d<4;d++,i++)twistBuffer[i%16]=e[n][d*l+c];for(a=0,i=0;a<4;a++)for(n=adjacentFaces[r][a],s=o?cycleCenters[r][a]:cycleLayerSides[r][a],l=cycleFactors[s],c=cycleOffsets[s],d=0;d<4;d++,i++)e[n][d*l+c]=twistBuffer[i]}var superTwistArr=[[[0,1,0],[0,4,1],[0,4,4],[0,1,5]],[[12,1,0],[3,4,1],[3,4,4],[12,1,5]],[[4,1,0],[1,4,1],[1,4,4],[4,1,5]],[[8,1,0],[2,4,1],[2,4,4],[8,1,5]],[[12,1,3],[0,1,1],[0,1,2],[0,4,0]],[[0,1,3],[3,4,0],[12,1,2],[12,1,1]],[[8,1,3],[4,1,1],[4,1,2],[1,4,0]],[[4,1,3],[2,4,0],[8,1,2],[8,1,1]]],width,height,lastX,lastY,lastDragX,lastDragY,dragAreas;function twistSuperLayer(e,r,t){if(0==t){for(var o=0;o<16;o++)scube[e][o]=(scube[e][o]+4-r)%4;2==e&&superTwist2(0,4-r),3==e&&superTwist2(1,r),4==e&&superTwist(4,r),5==e&&superTwist(5,r)}1==t&&(2==e&&superTwist2(2,4-r),3==e&&superTwist2(3,r),4==e&&superTwist(6,r),5==e&&superTwist(7,r))}function superTwist(e,r){superTwist1(superTwistArr[e][0]),superTwist1(superTwistArr[e][r])}function superTwist1(e){for(var r=e[0],t=0;t<4;r+=e[1],t++)scube[e[2]][r]=(scube[e[2]][r]+2)%4}function superTwist2(e,r){for(var t=0;t<4;t++)for(var o=superTwistArr[e][t],a=o[0],i=0;i<4;a+=o[1],i++)scube[o[2]][a]=(scube[o[2]][a]+r)%4}var dragCornersX=[],dragCornersY=[],dragDirsX=[],dragDirsY=[],dragBlocks=[[[0,0],[4,0],[4,1],[0,1]],[[4,0],[4,4],[3,4],[3,0]],[[4,4],[0,4],[0,3],[4,3]],[[0,4],[0,0],[1,0],[1,4]],[[0,1],[4,1],[4,2],[0,2]],[[3,0],[3,4],[2,4],[2,0]],[[0,2],[4,2],[4,3],[0,3]],[[2,0],[2,4],[1,4],[1,0]]],areaDirs=[[1,0],[0,1],[-1,0],[0,-1],[1,0],[0,1],[-1,0],[0,-1]],twistDirs=[[1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1],[1,-1,1,-1,1,-1,1,-1],[1,-1,1,-1,1,-1,1,-1],[-1,1,-1,1,-1,1,-1,1],[1,-1,1,-1,1,-1,1,-1]],dragLayers=[],dragModes=[],dragX,dragY,rotCos=[[[1,0,0],[0,0,0],[0,0,1]],[[1,0,0],[0,1,0],[0,0,0]],[[0,0,0],[0,1,0],[0,0,1]]],rotSin=[[[0,0,1],[0,0,0],[-1,0,0]],[[0,1,0],[-1,0,0],[0,0,0]],[[0,0,0],[0,0,1],[0,-1,0]]],rotVec=[[[0,0,0],[0,1,0],[0,0,0]],[[0,0,0],[0,0,0],[0,0,1]],[[1,0,0],[0,0,0],[0,0,0]]],rotSign=[1,-1,1,-1,1,-1],tempEye=[],tempEyeX=[],tempEyeY=[],tempEye2=[],tempEyeX2=[],tempEyeY2=[],perspEye=[],perspEyeI=[],perspNormal=[],eyeArray=[],eyeArrayX=[],eyeArrayY=[],eyeOrder=[[1,0,0,0],[0,1,0,0],[1,1,0,0],[1,1,1,1],[1,0,0,1],[1,0,0,2],[0,1,1,0]],blockArray=[],blockMode=[[0,2,2,2],[2,1,2,2],[2,2,2,2],[2,2,2,2],[0,2,2,0],[0,2,2,0],[0,2,2,0]],drawOrder=[[0,1,2,3],[0,1,3,2],[0,3,2,1],[3,2,1,0]],sliceNormals=[];function initSliceNormals(){for(var e=[],r=[],t=0;t<6;t++){sliceNormals[t]=[];for(var o=0;o0&&d<0||c<0&&d>0){l=cubeDim-o;break}d=c}if(hint)for(o=0;o0&&2==buttonBar)){if(!demo&&move.length>0){if(move[curMove].length>0){if(progressHeight>0){graphics.lineWidth=lineWidth,graphics.strokeStyle=buttonBorderColor;var h=(width-2)*realMovePos(move[curMove],movePos)/realMoveLength(move[curMove]);graphics.fillStyle=sliderBgColor,graphics.fillRect(dph,height-progressHeight-dph,width-dpr,progressHeight),graphics.fillStyle=sliderColor,graphics.fillRect(dph,height-progressHeight-dph,h,progressHeight),graphics.beginPath(),graphics.rect(dph,height-progressHeight-dph,width-dpr,progressHeight),graphics.stroke()}graphics.font="bold "+textHeight+"px helvetica";var v=moveLength(move[curMove],movePos)+"/"+moveLength(move[curMove],-1)+metricChar[metric],f=graphics.measureText(v).width,m=width-f-2,p=height-progressHeight-Math.floor(4*dpr);moveText>0?(moveCounter&&drawString(graphics,v,outlined?m-dpr:m,p-textHeight),drawMoveTextFunc(graphics,p)):moveCounter&&drawString(graphics,v,outlined?m-dpr:m,p)}if(move.length>1){graphics.font="bold "+textHeight+"px helvetica";v=curMove+1+"/"+move.length,f=graphics.measureText(v).width,m=width-f-2*buttonHeight-Math.floor(5*dpr);drawString(graphics,v,m,adjTextHeight()),drawButton(graphics,7,width-2*buttonHeight,0),drawButton(graphics,8,width-buttonHeight,0)}}curInfoText>=0&&(graphics.font="bold "+textHeight+"px helvetica",drawString(graphics,infoText[curInfoText],outlined?dpr:0,adjTextHeight()))}graphics.restore(),drawButtons&&0!=buttonBar&&drawButtonsFunc(graphics)}function adjTextHeight(){return utextHeight<10?Math.floor(10*dpr):utextHeight<12?Math.floor(12*dpr):utextHeight<14?Math.floor(14*dpr):textHeight}var fillX=[],fillY=[],coordsX=[],coordsY=[],cooX=[[],[],[],[],[],[]],cooY=[[],[],[],[],[],[]],border=[[.1,.1],[.9,.1],[.9,.9],[.1,.9]],factors=[[0,0],[0,1],[1,1],[1,0]],tempNormal=[];function fixBlock(e,r,t,o,a,i,n){for(var s=0;s<8;s++){var l=(u=width0&&v>0)for(var f=0,m=o[s][1][0];f-(1-scale)){h=o[s][0][1]-o[s][0][0],v=o[s][1][1]-o[s][1][0];if(h>0&&v>0)for(f=0,m=o[s][1][0];f0&&v>0){for(d=4==h?0==o[s][1][0]?0:2:0==o[s][0][0]?3:1,y=0;y<4;y++)getCorners(s,y,dragCornersX[dragAreas],dragCornersY[dragAreas],dragBlocks[d][y][0],dragBlocks[d][y][1],!1);dragDirsX[dragAreas]=(M*areaDirs[d][0]+x*areaDirs[d][1])*twistDirs[s][d],dragDirsY[dragAreas]=(C*areaDirs[d][0]+P*areaDirs[d][1])*twistDirs[s][d],dragLayers[dragAreas]=twistedLayer,dragModes[dragAreas]=0,dragAreas++}}else if(1==a&&s!=twistedLayer&&h>0&&v>0){d=4==h?1==o[s][1][0]?0:2:1==o[s][0][0]?3:1;d+=4;for(y=0;y<4;y++)getCorners(s,y,dragCornersX[dragAreas],dragCornersY[dragAreas],dragBlocks[d][y][0],dragBlocks[d][y][1],!1);dragDirsX[dragAreas]=(M*areaDirs[d][0]+x*areaDirs[d][1])*twistDirs[s][d],dragDirsY[dragAreas]=(C*areaDirs[d][0]+P*areaDirs[d][1])*twistDirs[s][d],dragLayers[dragAreas]=twistedLayer,dragModes[dragAreas]=1,dragAreas++}}}}function getCorners(e,r,t,o,a,i,n){a/=4,i/=4;var s=cooX[e][0]+(cooX[e][1]-cooX[e][0])*a,l=cooY[e][0]+(cooY[e][1]-cooY[e][0])*a,c=cooX[e][3]+(cooX[e][2]-cooX[e][3])*a,d=cooY[e][3]+(cooY[e][2]-cooY[e][3])*a;t[r]=Math.floor(.5+s+(c-s)*i),o[r]=Math.floor(.5+l+(d-l)*i),n&&(t[r]=width-t[r])}var textOffsetInit=[1,1,-1,-1,-1,1,1,-1,-1,0,1,0,0,1,0,-1],textOffset=[];function drawString(e,r,t,o){if(outlined){e.fillStyle="black";for(var a=0;awidth&&(c=Math.min(1,width/2-n-s/2),c=Math.max(c,width-n-s-l-2)),s>0&&(e.fillStyle=hlColor,e.lineWidth=2,e.strokeStyle="black",e.beginPath(),utextHeight<=14?e.fillRect(c+n-1,height-progressHeight-textHeight-Math.floor(4*dpr),s+2,textHeight+Math.floor(3*dpr)):e.fillRect(c+n-1,height-progressHeight-textHeight-Math.floor(2*dpr),s+2,textHeight+Math.floor(dpr))),n>0&&drawString(e,o,c,r),s>0&&drawString(e,a,c+n,r),l>0&&drawString(e,i,c+n+s,r)}function selectButton(e,r){if(0==buttonBar)return-1;if(move.length>1&&e>=width-2*buttonHeight&&e=0&&r1&&e>=width-buttonHeight&&e=0&&r=0&&e=height-buttonHeight&&r=t&&e=height&&r0&&6==buttonPressed?darker(buttonBgColor):buttonBgColor,e.fillRect(dph,height-buttonHeight,buttonHeight,buttonHeight),e.lineWidth=lineWidth,e.strokeStyle=buttonBorderColor,e.beginPath(),e.rect(dph,height-buttonHeight-dph,buttonHeight,buttonHeight),e.stroke(),void drawButton(e,0,buttonHeight/2,height-(buttonHeight+1)/2-r);if(1!=buttonBar);else{for(var t=0,o=0;o<7;o++){var a=Math.floor((width-t)/(7-o));e.fillStyle=buttonPressed==o?darker(buttonBgColor):buttonBgColor,e.fillRect(t,height,a,buttonHeight),e.lineWidth=lineWidth,e.strokeStyle=buttonBorderColor,e.beginPath(),0==o?e.rect(t+dph,height-dph,a-dpr,buttonHeight):e.rect(t-dph,height-dph,a,buttonHeight),e.stroke(),e.strokeStyle="black",drawButton(e,o,t+a/2,height+buttonHeight/2-r),t+=a}drawButtons=!1}}var ds=[];function drawButton(e,r,t,o){switch(t=Math.floor(t),o=Math.floor(o),r){case 0:drawRect(e,t-ds[4],o-ds[3],ds[3],ds[6]+1),drawArrow(e,t+ds[4],o,-1);break;case 1:drawRect(e,t+ds[1],o-ds[3],ds[3],ds[6]+1),drawArrow(e,t-ds[1],o,-1);break;case 2:drawArrow(e,t+ds[1],o,-1);break;case 3:animating?drawRect(e,t-ds[4],o-ds[3],ds[7],ds[7]):(drawRect(e,t-ds[4],o-ds[2],ds[7],ds[5]),drawRect(e,t-ds[2],o-ds[4],ds[3],ds[9]));break;case 4:drawArrow(e,t-ds[2],o,1);break;case 5:drawRect(e,t-ds[4],o-ds[3],ds[3],ds[6]+1),drawArrow(e,t,o,1);break;case 6:drawRect(e,t+ds[1],o-ds[3],ds[3],ds[6]+1),drawArrow(e,t-ds[4],o,1);break;case 7:var a=7==buttonPressed?darker(buttonBgColor):buttonBgColor;drawRect2(e,t-2*dpr,o+dpr,buttonHeight,o+buttonHeight,a),drawArrow(e,t+2*dpr+buttonHeight/2-3*dpr,o+buttonHeight/2+dph,-1);break;case 8:a=8==buttonPressed?darker(buttonBgColor):buttonBgColor;drawRect2(e,t-2*dpr,o+dpr,buttonHeight,o+buttonHeight,a),drawArrow(e,t-dpr+buttonHeight/2-3*dpr,o+buttonHeight/2+dph,1)}}function drawArrow(e,r,t,o){var a=3*dpr,i=[],n=[];i[0]=r,i[1]=r+o,i[2]=r+4*dpr*o,i[3]=r+o,i[4]=r,n[0]=t-a,n[1]=t-a,n[2]=t,n[3]=t+a,n[4]=t+a,drawArrow2(e,i,n)}function drawArrow2(e,r,t){e.beginPath(),e.moveTo(r[0]+dph,t[0]+dph);for(var o=1;o<5;o++)e.lineTo(r[o]+dph,t[o]+dph);e.closePath(),e.fillStyle="white",e.strokeStyle="black",e.fill(),e.lineWidth=lineWidth,e.stroke()}function drawRect(e,r,t,o,a){e.lineWidth=lineWidth,e.beginPath(),e.rect(r+dph,t+dph,o-1,a-1),e.fillStyle="white",e.fill(),e.strokeStyle="black",e.stroke()}function drawRect2(e,r,t,o,a,i){e.lineWidth=lineWidth,e.beginPath(),e.rect(r+dph,t+dph,o-1,a-1),e.fillStyle=i,e.fill(),e.strokeStyle="black",e.stroke()}function drawPolygon(e,r,t,o){e.beginPath(),e.moveTo(r[0],t[0]),e.lineTo(r[1],t[1]),e.lineTo(r[2],t[2]),e.lineTo(r[3],t[3]),e.closePath(),e.strokeStyle=o,e.lineWidth=.7*dpr,e.stroke()}function fillPolygon(e,r,t,o){e.beginPath(),e.moveTo(r[0],t[0]),e.lineTo(r[1],t[1]),e.lineTo(r[2],t[2]),e.lineTo(r[3],t[3]),e.closePath(),e.fillStyle=o,e.fill()}var superRotate=[[0,1,2,3],[3,0,1,2],[2,3,0,1],[1,2,3,0]],offsetX,offsetY;function drawSuperArrow(e,r,t,o,a,i){var n=[],s=[];if(2!=scw||"#ffffff"!=i){for(var l=0;l<4;l++)n[l]=Math.floor(r[l]+.05*(r[superRotate[2][l]]-r[l])),s[l]=Math.floor(t[l]+.05*(t[superRotate[2][l]]-t[l]));0==o&&(a=(a+1)%4),4==o&&(a=(a+3)%4);var c=superRotate[a][0],d=superRotate[a][1],u=superRotate[a][2],g=superRotate[a][3],h=.26*(n[u]-n[d]),v=.26*(s[u]-s[d]),f=(n[c]-n[d])/2,m=s[d]+(s[c]-s[d])/2,p=s[u]+(s[g]-s[u])/2,b=(n[g]-n[u])/2,w=1^a;e.fillStyle=i,e.beginPath(),e.moveTo(n[c]+(n[g]-n[c])/2,s[c]+(s[g]-s[c])/2),e.lineTo(n[w]+f,m),e.lineTo(n[w]+h+f,m+v),e.lineTo(n[w]+h,s[w]+v),w=(w+1)%4,e.lineTo(n[w]-h,s[w]-v),e.lineTo(n[w]-h+b,p-v),e.lineTo(n[w]+b,p),e.closePath(),e.fill(),e.lineWidth=.6*dpr,e.strokeStyle="black",e.stroke()}}function colorToHex(e){return"white"==e?"#FFFFFF":"black"==e?"#000000":"#808080"}function colorAverage(e){return"#"!=e.substring(0,1)&&(e=colorToHex(e)),(299*parseInt(e.substring(1,3),16)+587*parseInt(e.substring(3,5),16)+114*parseInt(e.substring(5,7),16))/1e3}function brighter(e){"#"!=e.substring(0,1)&&(e=colorToHex(e));var r=parseInt(e.substring(1,3),16),t=parseInt(e.substring(3,5),16),o=parseInt(e.substring(5,7),16);return rgbToHex((r=Math.floor(1.3*r))>255?255:r,(t=Math.floor(1.3*t))>255?255:t,(o=Math.floor(1.3*o))>255?255:o)}function darker(e){"#"!=e.substring(0,1)&&(e=colorToHex(e));var r=parseInt(e.substring(1,3),16),t=parseInt(e.substring(3,5),16),o=parseInt(e.substring(5,7),16);return rgbToHex(r=Math.floor(.7*r),t=Math.floor(.7*t),o=Math.floor(.7*o))}function run(e,r){if(e>nowServing)setTimeout(run,0,e,r);else{if(!demo&&(0==move.length||0==move[curMove].length))return animating=!1,drawButtons=!0,void paint();if(!moveAnimated){for(var t=move[curMove];movePos=1e3)curInfoText=t[movePos]-1e3;else if(-1!=t[movePos]){var o=t[movePos]%4+1,a=Math.floor(t[movePos]/4)%7,i=Math.floor(t[movePos]/28);twistLayers(cube,i,4==o?2:o,a)}movePos++}return animating=!1,drawButtons=!0,paint(),void nowServing++}var n,s,l,c,d,u,g=r;interrupted=!1,requestAnimationFrame((function e(){if(v){if(v=!1,u=!1,repeatable)g>0?movePos>=t.length&&(movePos=0,initInfoText(t)):(curInfoText=-1,0==movePos&&(movePos=t.length));else if(g>0&&movePos>=t.length||g<0&&0==movePos)return restarted=!1,animating=!1,nowServing++,drawButtons=!0,void paint();animating=!0,drawButtons=!0}if(f&&(f=!1,g<0&&(b=!1,0==movePos?(b=!0,p=!0):movePos--),!b)){if(h=!1,-1==t[movePos]){if(paint(),!moveOne)for(n=Date.now();Date.now()-n<1e3;);}else t[movePos]>=1e3?curInfoText=g>0?t[movePos]-1e3:-1:h=!0;if(h){o=t[movePos]%4+1,a=Math.floor(t[movePos]/4)%7;var r=o<3;if(4==o&&(o=2),g<0&&(r=!r,o=4-o),i=Math.floor(t[movePos]/28),twisting=!1,natural=!0,spinning=!0,originalAngle=0,faceTwistDirs[i]>0&&(r=!r),moveAnimated){d=Math.PI/2,c=r?1:-1;var w=67*speed;2==o&&(d=Math.PI,w=67*doubleSpeed),twisting=!0,twistedLayer=i,twistedMode=a,splitCube(i),n=Date.now(),s=n,l=c*d/w,currentAngle=0}}else m=!0}b||(h&&(moveAnimated&¤tAngle*c0?(++movePos=1e3&&(curInfoText=t[movePos]-1e3,movePos++),movePos==t.length&&(demo?clearDemo(t):p=!0)):curInfoText=-1,(interrupted||restarted||u)&&(p=!0)));if(p)return p=!1,v=!0,jobNumber<=nowServing+1&&(animating=!1),drawButtons=!0,(0==buttonPressed||buttonPressed>6)&&clear(),paint(),demo&&(clear(),demo=!1),restarted=!1,nowServing++,void(1==movePos&&t[0]>=1e3&&movePos--);requestAnimationFrame(e)}));var h=!1,v=!0,f=!0,m=!1,p=!1,b=!1;t=demo?demoMove[0]:move[curMove]}}function clearDemo(e){movePos=0;for(var r=0;r<6;r++)for(var t=0;t<16;t++)cube[r][t]=initialCube[r][t],scube[r][t]=initialSCube[r][t];initialMove.length>0&&void 0!==initialMove[curMove]&&doMove(cube,initialMove[curMove],0,initialMove[curMove].length,!1),initialReversedMove.length>0&&void 0!==initialReversedMove[curMove]&&doMove(cube,initialReversedMove[curMove],0,initialReversedMove[curMove].length,!0),initInfoText(e)}document.addEventListener("touchstart",mousedown),document.addEventListener("touchmove",mousemove,{passive:!1}),document.addEventListener("touchend",mouseup),document.addEventListener("mousedown",mousedown),document.addEventListener("mousemove",mousemove),document.addEventListener("mouseup",mouseup),document.addEventListener("contextmenu",contextmenu);var mouseIsDown=!1,showContextMenu=!0,divs=document.getElementsByTagName("div"),wrapDiv=divs.length>0&&"wrap"==divs[0].className;function touchfunc(e){wrapDiv?divs[0].style.overflow=e:document.body.style.overflow=e}function mouseup(e){if(mouseIsDown&&void 0!==e.touches&&(e.preventDefault(),touchfunc("auto")),mouseIsDown?setTimeout((function(){showContextMenu=!0}),100):showContextMenu=!0,mouseIsDown=!1,dragging=!1,pushed)pushed=!1,drawButtons=!0,paint();else if(twisting&&!spinning){twisting=!1,originalAngle+=currentAngle,currentAngle=0;for(var r=originalAngle;r<0;)r+=32*Math.PI;var t=Math.floor(8*r/Math.PI)%16;(snap||t%4==0||t%4==3)&&(t=Math.floor((t+2)/4),faceTwistDirs[twistedLayer]>0&&(t=(4-t)%4),originalAngle=0,natural=!0,twistLayers(cube,twistedLayer,t,twistedMode)),paint()}}function mousedown(e){var r=canvas.getBoundingClientRect(),t=Math.floor(r.left),o=Math.floor(r.top);if(void 0===e.touches)var a=e.clientX,i=e.clientY;else a=e.touches[0].clientX,i=e.touches[0].clientY;at+width/dpr||io+(height+buttonHeight)/dpr||(e.preventDefault(),mouseIsDown=!0,showContextMenu=!1,void 0!==e.touches&&touchfunc("hidden"),offsetX=t,offsetY=o,lastDragX=lastX=getX(e),lastDragY=lastY=getY(e),toTwist=!1,(buttonPressed=selectButton(lastX,lastY))>=0?button():progressHeight>0&&move.length>0&&move[curMove].length>0&&lastY>height-progressHeight&&lastY<=height?clickProgress&&(stopAnimation(),progress(jobNumber++)):(mirrored&&(lastDragX=lastX=width-lastX),void 0===e.touches?!editable||animating||0!=e.button||e.shiftKey||(toTwist=!0):editable&&!animating&&(toTwist=!0)))}function button(){pushed=!0,3==buttonPressed?animating?stopAnimation():mirrored=!mirrored:0==buttonPressed?scramble>0&&2==buttonBar?1==scrambleToggle?(scrambleToggle=!1,stopAnimation(),clear()):(scrambleToggle=!0,startAnimation(buttonAction[buttonPressed=6])):(stopAnimation(),clear()):7==buttonPressed||8==buttonPressed?(stopAnimation(),setTimeout(clear,0),curMove=7==buttonPressed?curMove>0?curMove-1:move.length-1:curMovenowServing)setTimeout(progress,0,e);else{var r=realMoveLength(move[curMove]),t=Math.floor(((lastX-1)*r*2/(width-2)+1)/2);(t=Math.max(0,Math.min(r,t)))>0&&(t=arrayMovePos(move[curMove],t)),t>movePos&&doMove(cube,move[curMove],movePos,t-movePos,!1),t0&&(t=arrayMovePos(move[curMove],t)),t>movePos&&doMove(cube,move[curMove],movePos,t-movePos,!1),t0&&v<1&&f>0&&f<1){if(i*i+n*n<144)return;if(dragX=dragDirsX[s],dragY=dragDirsY[s],Math.abs(dragX*i+dragY*n)/Math.sqrt((dragX*dragX+dragY*dragY)*(i*i+n*n))>.75){twisting=!0,twistedLayer=dragLayers[s],twistedMode=dragModes[s];break}}}toTwist=!1,lastX=lastDragX,lastY=lastDragY}i=(o-lastX)/dpr,n=(a-lastY)/dpr,!twisting||animating?rotateAllowed&&(vNorm(vAdd(eye,vScale(vCopy(eyeD,eyeX),-.016*i))),vNorm(vMul(eyeX,eyeY,eye)),vNorm(vAdd(eye,vScale(vCopy(eyeD,eyeY),.016*n))),vNorm(vMul(eyeY,eye,eyeX)),lastX=o,lastY=a):(natural&&splitCube(twistedLayer),currentAngle=.03*(dragX*i+dragY*n)/Math.sqrt(dragX*dragX+dragY*dragY)),paint()}}function getX(e){return void 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I(...e).pipe(oe(o=>d(o,"change").pipe(m(()=>o))),Q(e[r]),m(o=>({index:e.indexOf(o),color:{media:o.getAttribute("data-md-color-media"),scheme:o.getAttribute("data-md-color-scheme"),primary:o.getAttribute("data-md-color-primary"),accent:o.getAttribute("data-md-color-accent")}})),B(1))}function Kn(e){let t=$("input",e),r=x("meta",{name:"theme-color"});document.head.appendChild(r);let o=x("meta",{name:"color-scheme"});document.head.appendChild(o);let n=$t("(prefers-color-scheme: light)");return C(()=>{let i=new g;return i.subscribe(a=>{if(document.body.setAttribute("data-md-color-switching",""),a.color.media==="(prefers-color-scheme)"){let s=matchMedia("(prefers-color-scheme: light)"),p=document.querySelector(s.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: 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Gr=Vt(Yr());function Ga(e){e.setAttribute("data-md-copying","");let t=e.closest("[data-copy]"),r=t?t.getAttribute("data-copy"):e.innerText;return e.removeAttribute("data-md-copying"),r.trimEnd()}function Bn({alert$:e}){Gr.default.isSupported()&&new F(t=>{new Gr.default("[data-clipboard-target], [data-clipboard-text]",{text:r=>r.getAttribute("data-clipboard-text")||Ga(P(r.getAttribute("data-clipboard-target")))}).on("success",r=>t.next(r))}).pipe(E(t=>{t.trigger.focus()}),m(()=>Ee("clipboard.copied"))).subscribe(e)}function Gn(e,t){return e.protocol=t.protocol,e.hostname=t.hostname,e}function Ja(e,t){let r=new Map;for(let o of $("url",e)){let n=P("loc",o),i=[Gn(new URL(n.textContent),t)];r.set(`${i[0]}`,i);for(let a of $("[rel=alternate]",o)){let s=a.getAttribute("href");s!=null&&i.push(Gn(new URL(s),t))}}return r}function ur(e){return mn(new URL("sitemap.xml",e)).pipe(m(t=>Ja(t,new URL(e))),ve(()=>I(new Map)))}function Xa(e,t){if(!(e.target instanceof Element))return M;let 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g,o=r.pipe(X(),ne(!0));z([t.pipe(Ae(It)),r],(i,a)=>a).pipe(Z("value")).subscribe(({value:i})=>t.next({type:2,data:i})),r.pipe(Z("focus")).subscribe(({focus:i})=>{i&&Je("search",i)}),d(e.form,"reset").pipe(U(o)).subscribe(()=>e.focus());let n=P("header [for=__search]");return d(n,"click").subscribe(()=>e.focus()),ns(e,{worker$:t}).pipe(E(i=>r.next(i)),L(()=>r.complete()),m(i=>R({ref:e},i)),B(1))}function si(e,{worker$:t,query$:r}){let o=new g,n=tn(e.parentElement).pipe(b(Boolean)),i=e.parentElement,a=P(":scope > :first-child",e),s=P(":scope > :last-child",e);Ve("search").subscribe(l=>s.setAttribute("role",l?"list":"presentation")),o.pipe(ee(r),Ur(t.pipe(Ae(It)))).subscribe(([{items:l},{value:f}])=>{switch(l.length){case 0:a.textContent=f.length?Ee("search.result.none"):Ee("search.result.placeholder");break;case 1:a.textContent=Ee("search.result.one");break;default:let u=sr(l.length);a.textContent=Ee("search.result.other",u)}});let p=o.pipe(E(()=>s.innerHTML=""),v(({items:l})=>S(I(...l.slice(0,10)),I(...l.slice(10)).pipe(Ye(4),Vr(n),v(([f])=>f)))),m(Tn),pe());return p.subscribe(l=>s.appendChild(l)),p.pipe(oe(l=>{let f=fe("details",l);return typeof f=="undefined"?M:d(f,"toggle").pipe(U(o),m(()=>f))})).subscribe(l=>{l.open===!1&&l.offsetTop<=i.scrollTop&&i.scrollTo({top:l.offsetTop})}),t.pipe(b(dr),m(({data:l})=>l)).pipe(E(l=>o.next(l)),L(()=>o.complete()),m(l=>R({ref:e},l)))}function is(e,{query$:t}){return t.pipe(m(({value:r})=>{let o=xe();return o.hash="",r=r.replace(/\s+/g,"+").replace(/&/g,"%26").replace(/=/g,"%3D"),o.search=`q=${r}`,{url:o}}))}function ci(e,t){let r=new g,o=r.pipe(X(),ne(!0));return r.subscribe(({url:n})=>{e.setAttribute("data-clipboard-text",e.href),e.href=`${n}`}),d(e,"click").pipe(U(o)).subscribe(n=>n.preventDefault()),is(e,t).pipe(E(n=>r.next(n)),L(()=>r.complete()),m(n=>R({ref:e},n)))}function pi(e,{worker$:t,keyboard$:r}){let o=new g,n=Se("search-query"),i=S(d(n,"keydown"),d(n,"focus")).pipe(be(se),m(()=>n.value),K());return o.pipe(We(i),m(([{suggest:s},p])=>{let c=p.split(/([\s-]+)/);if(s!=null&&s.length&&c[c.length-1]){let l=s[s.length-1];l.startsWith(c[c.length-1])&&(c[c.length-1]=l)}else c.length=0;return c})).subscribe(s=>e.innerHTML=s.join("").replace(/\s/g," ")),r.pipe(b(({mode:s})=>s==="search")).subscribe(s=>{switch(s.type){case"ArrowRight":e.innerText.length&&n.selectionStart===n.value.length&&(n.value=e.innerText);break}}),t.pipe(b(dr),m(({data:s})=>s)).pipe(E(s=>o.next(s)),L(()=>o.complete()),m(()=>({ref:e})))}function li(e,{index$:t,keyboard$:r}){let o=ye();try{let n=ni(o.search,t),i=Se("search-query",e),a=Se("search-result",e);d(e,"click").pipe(b(({target:p})=>p instanceof Element&&!!p.closest("a"))).subscribe(()=>Je("search",!1)),r.pipe(b(({mode:p})=>p==="search")).subscribe(p=>{let c=Re();switch(p.type){case"Enter":if(c===i){let l=new Map;for(let f of $(":first-child [href]",a)){let u=f.firstElementChild;l.set(f,parseFloat(u.getAttribute("data-md-score")))}if(l.size){let[[f]]=[...l].sort(([,u],[,h])=>h-u);f.click()}p.claim()}break;case"Escape":case"Tab":Je("search",!1),i.blur();break;case"ArrowUp":case"ArrowDown":if(typeof c=="undefined")i.focus();else{let l=[i,...$(":not(details) > [href], summary, details[open] [href]",a)],f=Math.max(0,(Math.max(0,l.indexOf(c))+l.length+(p.type==="ArrowUp"?-1:1))%l.length);l[f].focus()}p.claim();break;default:i!==Re()&&i.focus()}}),r.pipe(b(({mode:p})=>p==="global")).subscribe(p=>{switch(p.type){case"f":case"s":case"/":i.focus(),i.select(),p.claim();break}});let s=ai(i,{worker$:n});return S(s,si(a,{worker$:n,query$:s})).pipe(Pe(...ae("search-share",e).map(p=>ci(p,{query$:s})),...ae("search-suggest",e).map(p=>pi(p,{worker$:n,keyboard$:r}))))}catch(n){return e.hidden=!0,Ke}}function mi(e,{index$:t,location$:r}){return z([t,r.pipe(Q(xe()),b(o=>!!o.searchParams.get("h")))]).pipe(m(([o,n])=>oi(o.config)(n.searchParams.get("h"))),m(o=>{var a;let n=new Map,i=document.createNodeIterator(e,NodeFilter.SHOW_TEXT);for(let s=i.nextNode();s;s=i.nextNode())if((a=s.parentElement)!=null&&a.offsetHeight){let p=s.textContent,c=o(p);c.length>p.length&&n.set(s,c)}for(let[s,p]of n){let{childNodes:c}=x("span",null,p);s.replaceWith(...Array.from(c))}return{ref:e,nodes:n}}))}function as(e,{viewport$:t,main$:r}){let o=e.closest(".md-grid"),n=o.offsetTop-o.parentElement.offsetTop;return z([r,t]).pipe(m(([{offset:i,height:a},{offset:{y:s}}])=>(a=a+Math.min(n,Math.max(0,s-i))-n,{height:a,locked:s>=i+n})),K((i,a)=>i.height===a.height&&i.locked===a.locked))}function Jr(e,o){var n=o,{header$:t}=n,r=io(n,["header$"]);let i=P(".md-sidebar__scrollwrap",e),{y:a}=Ue(i);return C(()=>{let s=new g,p=s.pipe(X(),ne(!0)),c=s.pipe(Le(0,me));return c.pipe(ee(t)).subscribe({next([{height:l},{height:f}]){i.style.height=`${l-2*a}px`,e.style.top=`${f}px`},complete(){i.style.height="",e.style.top=""}}),c.pipe(Ae()).subscribe(()=>{for(let l of $(".md-nav__link--active[href]",e)){if(!l.clientHeight)continue;let f=l.closest(".md-sidebar__scrollwrap");if(typeof f!="undefined"){let u=l.offsetTop-f.offsetTop,{height:h}=ce(f);f.scrollTo({top:u-h/2})}}}),ue($("label[tabindex]",e)).pipe(oe(l=>d(l,"click").pipe(be(se),m(()=>l),U(p)))).subscribe(l=>{let f=P(`[id="${l.htmlFor}"]`);P(`[aria-labelledby="${l.id}"]`).setAttribute("aria-expanded",`${f.checked}`)}),as(e,r).pipe(E(l=>s.next(l)),L(()=>s.complete()),m(l=>R({ref:e},l)))})}function fi(e,t){if(typeof t!="undefined"){let r=`https://api.github.com/repos/${e}/${t}`;return Ct(Ne(`${r}/releases/latest`).pipe(ve(()=>M),m(o=>({version:o.tag_name})),Be({})),Ne(r).pipe(ve(()=>M),m(o=>({stars:o.stargazers_count,forks:o.forks_count})),Be({}))).pipe(m(([o,n])=>R(R({},o),n)))}else{let r=`https://api.github.com/users/${e}`;return Ne(r).pipe(m(o=>({repositories:o.public_repos})),Be({}))}}function ui(e,t){let r=`https://${e}/api/v4/projects/${encodeURIComponent(t)}`;return Ne(r).pipe(ve(()=>M),m(({star_count:o,forks_count:n})=>({stars:o,forks:n})),Be({}))}function di(e){let t=e.match(/^.+github\.com\/([^/]+)\/?([^/]+)?/i);if(t){let[,r,o]=t;return fi(r,o)}if(t=e.match(/^.+?([^/]*gitlab[^/]+)\/(.+?)\/?$/i),t){let[,r,o]=t;return ui(r,o)}return M}var ss;function cs(e){return ss||(ss=C(()=>{let t=__md_get("__source",sessionStorage);if(t)return I(t);if(ae("consent").length){let o=__md_get("__consent");if(!(o&&o.github))return M}return di(e.href).pipe(E(o=>__md_set("__source",o,sessionStorage)))}).pipe(ve(()=>M),b(t=>Object.keys(t).length>0),m(t=>({facts:t})),B(1)))}function hi(e){let t=P(":scope > :last-child",e);return C(()=>{let r=new g;return 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n.title=o,lt(n,{viewport$:t}).pipe(U(e.pipe(Ce(1))),L(()=>n.removeAttribute("title")))})).subscribe(),e.pipe(v(()=>$(".md-status")),oe(r=>lt(r,{viewport$:t}))).subscribe()}function yi({document$:e,tablet$:t}){e.pipe(v(()=>$(".md-toggle--indeterminate")),E(r=>{r.indeterminate=!0,r.checked=!1}),oe(r=>d(r,"change").pipe(Dr(()=>r.classList.contains("md-toggle--indeterminate")),m(()=>r))),ee(t)).subscribe(([r,o])=>{r.classList.remove("md-toggle--indeterminate"),o&&(r.checked=!1)})}function fs(){return/(iPad|iPhone|iPod)/.test(navigator.userAgent)}function Ei({document$:e}){e.pipe(v(()=>$("[data-md-scrollfix]")),E(t=>t.removeAttribute("data-md-scrollfix")),b(fs),oe(t=>d(t,"touchstart").pipe(m(()=>t)))).subscribe(t=>{let r=t.scrollTop;r===0?t.scrollTop=1:r+t.offsetHeight===t.scrollHeight&&(t.scrollTop=r-1)})}function wi({viewport$:e,tablet$:t}){z([Ve("search"),t]).pipe(m(([r,o])=>r&&!o),v(r=>I(r).pipe(Ge(r?400:100))),ee(e)).subscribe(([r,{offset:{y:o}}])=>{if(r)document.body.setAttribute("data-md-scrolllock",""),document.body.style.top=`-${o}px`;else{let n=-1*parseInt(document.body.style.top,10);document.body.removeAttribute("data-md-scrolllock"),document.body.style.top="",n&&window.scrollTo(0,n)}})}Object.entries||(Object.entries=function(e){let t=[];for(let r of Object.keys(e))t.push([r,e[r]]);return t});Object.values||(Object.values=function(e){let t=[];for(let r of Object.keys(e))t.push(e[r]);return t});typeof Element!="undefined"&&(Element.prototype.scrollTo||(Element.prototype.scrollTo=function(e,t){typeof e=="object"?(this.scrollLeft=e.left,this.scrollTop=e.top):(this.scrollLeft=e,this.scrollTop=t)}),Element.prototype.replaceWith||(Element.prototype.replaceWith=function(...e){let t=this.parentNode;if(t){e.length===0&&t.removeChild(this);for(let r=e.length-1;r>=0;r--){let o=e[r];typeof o=="string"?o=document.createTextNode(o):o.parentNode&&o.parentNode.removeChild(o),r?t.insertBefore(this.previousSibling,o):t.replaceChild(o,this)}}}));function us(){return location.protocol==="file:"?wt(`${new URL("search/search_index.js",Xr.base)}`).pipe(m(()=>__index),B(1)):Ne(new URL("search/search_index.json",Xr.base))}document.documentElement.classList.remove("no-js");document.documentElement.classList.add("js");var ot=Yo(),jt=nn(),Ot=cn(jt),Zr=on(),Oe=bn(),hr=$t("(min-width: 960px)"),Si=$t("(min-width: 1220px)"),Oi=pn(),Xr=ye(),Mi=document.forms.namedItem("search")?us():Ke,eo=new g;Bn({alert$:eo});var to=new g;G("navigation.instant")&&Zn({location$:jt,viewport$:Oe,progress$:to}).subscribe(ot);var Ti;((Ti=Xr.version)==null?void 0:Ti.provider)==="mike"&&ii({document$:ot});S(jt,Ot).pipe(Ge(125)).subscribe(()=>{Je("drawer",!1),Je("search",!1)});Zr.pipe(b(({mode:e})=>e==="global")).subscribe(e=>{switch(e.type){case"p":case",":let t=fe("link[rel=prev]");typeof t!="undefined"&&pt(t);break;case"n":case".":let r=fe("link[rel=next]");typeof r!="undefined"&&pt(r);break;case"Enter":let o=Re();o instanceof HTMLLabelElement&&o.click()}});xi({viewport$:Oe,document$:ot});yi({document$:ot,tablet$:hr});Ei({document$:ot});wi({viewport$:Oe,tablet$:hr});var rt=Nn(Se("header"),{viewport$:Oe}),Ft=ot.pipe(m(()=>Se("main")),v(e=>Qn(e,{viewport$:Oe,header$:rt})),B(1)),ds=S(...ae("consent").map(e=>xn(e,{target$:Ot})),...ae("dialog").map(e=>Dn(e,{alert$:eo})),...ae("header").map(e=>zn(e,{viewport$:Oe,header$:rt,main$:Ft})),...ae("palette").map(e=>Kn(e)),...ae("progress").map(e=>Yn(e,{progress$:to})),...ae("search").map(e=>li(e,{index$:Mi,keyboard$:Zr})),...ae("source").map(e=>hi(e))),hs=C(()=>S(...ae("announce").map(e=>gn(e)),...ae("content").map(e=>Un(e,{viewport$:Oe,target$:Ot,print$:Oi})),...ae("content").map(e=>G("search.highlight")?mi(e,{index$:Mi,location$:jt}):M),...ae("header-title").map(e=>qn(e,{viewport$:Oe,header$:rt})),...ae("sidebar").map(e=>e.getAttribute("data-md-type")==="navigation"?Nr(Si,()=>Jr(e,{viewport$:Oe,header$:rt,main$:Ft})):Nr(hr,()=>Jr(e,{viewport$:Oe,header$:rt,main$:Ft}))),...ae("tabs").map(e=>bi(e,{viewport$:Oe,header$:rt})),...ae("toc").map(e=>vi(e,{viewport$:Oe,header$:rt,main$:Ft,target$:Ot})),...ae("top").map(e=>gi(e,{viewport$:Oe,header$:rt,main$:Ft,target$:Ot})))),Li=ot.pipe(v(()=>hs),Pe(ds),B(1));Li.subscribe();window.document$=ot;window.location$=jt;window.target$=Ot;window.keyboard$=Zr;window.viewport$=Oe;window.tablet$=hr;window.screen$=Si;window.print$=Oi;window.alert$=eo;window.progress$=to;window.component$=Li;})(); 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"node_modules/rxjs/src/internal/observable/of.ts", "node_modules/rxjs/src/internal/observable/throwError.ts", "node_modules/rxjs/src/internal/util/EmptyError.ts", "node_modules/rxjs/src/internal/util/isDate.ts", "node_modules/rxjs/src/internal/operators/map.ts", "node_modules/rxjs/src/internal/util/mapOneOrManyArgs.ts", "node_modules/rxjs/src/internal/util/argsArgArrayOrObject.ts", "node_modules/rxjs/src/internal/util/createObject.ts", "node_modules/rxjs/src/internal/observable/combineLatest.ts", "node_modules/rxjs/src/internal/operators/mergeInternals.ts", "node_modules/rxjs/src/internal/operators/mergeMap.ts", "node_modules/rxjs/src/internal/operators/mergeAll.ts", "node_modules/rxjs/src/internal/operators/concatAll.ts", "node_modules/rxjs/src/internal/observable/concat.ts", "node_modules/rxjs/src/internal/observable/defer.ts", "node_modules/rxjs/src/internal/observable/fromEvent.ts", "node_modules/rxjs/src/internal/observable/fromEventPattern.ts", "node_modules/rxjs/src/internal/observable/timer.ts", "node_modules/rxjs/src/internal/observable/merge.ts", "node_modules/rxjs/src/internal/observable/never.ts", "node_modules/rxjs/src/internal/util/argsOrArgArray.ts", "node_modules/rxjs/src/internal/operators/filter.ts", "node_modules/rxjs/src/internal/observable/zip.ts", "node_modules/rxjs/src/internal/operators/audit.ts", "node_modules/rxjs/src/internal/operators/auditTime.ts", "node_modules/rxjs/src/internal/operators/bufferCount.ts", "node_modules/rxjs/src/internal/operators/catchError.ts", "node_modules/rxjs/src/internal/operators/scanInternals.ts", "node_modules/rxjs/src/internal/operators/combineLatest.ts", "node_modules/rxjs/src/internal/operators/combineLatestWith.ts", "node_modules/rxjs/src/internal/operators/debounce.ts", "node_modules/rxjs/src/internal/operators/debounceTime.ts", "node_modules/rxjs/src/internal/operators/defaultIfEmpty.ts", "node_modules/rxjs/src/internal/operators/take.ts", "node_modules/rxjs/src/internal/operators/ignoreElements.ts", "node_modules/rxjs/src/internal/operators/mapTo.ts", "node_modules/rxjs/src/internal/operators/delayWhen.ts", "node_modules/rxjs/src/internal/operators/delay.ts", "node_modules/rxjs/src/internal/operators/distinctUntilChanged.ts", "node_modules/rxjs/src/internal/operators/distinctUntilKeyChanged.ts", "node_modules/rxjs/src/internal/operators/throwIfEmpty.ts", "node_modules/rxjs/src/internal/operators/endWith.ts", "node_modules/rxjs/src/internal/operators/finalize.ts", "node_modules/rxjs/src/internal/operators/first.ts", "node_modules/rxjs/src/internal/operators/takeLast.ts", "node_modules/rxjs/src/internal/operators/merge.ts", "node_modules/rxjs/src/internal/operators/mergeWith.ts", "node_modules/rxjs/src/internal/operators/repeat.ts", "node_modules/rxjs/src/internal/operators/scan.ts", "node_modules/rxjs/src/internal/operators/share.ts", "node_modules/rxjs/src/internal/operators/shareReplay.ts", 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"sourcesContent": ["(function (global, factory) {\n typeof exports === 'object' && typeof module !== 'undefined' ? factory() :\n typeof define === 'function' && define.amd ? define(factory) :\n (factory());\n}(this, (function () { 'use strict';\n\n /**\n * Applies the :focus-visible polyfill at the given scope.\n * A scope in this case is either the top-level Document or a Shadow Root.\n *\n * @param {(Document|ShadowRoot)} scope\n * @see https://github.com/WICG/focus-visible\n */\n function applyFocusVisiblePolyfill(scope) {\n var hadKeyboardEvent = true;\n var hadFocusVisibleRecently = false;\n var hadFocusVisibleRecentlyTimeout = null;\n\n var inputTypesAllowlist = {\n text: true,\n search: true,\n url: true,\n tel: true,\n email: true,\n password: true,\n number: true,\n date: true,\n month: true,\n week: true,\n time: true,\n datetime: true,\n 'datetime-local': true\n };\n\n /**\n * Helper function for legacy browsers and iframes which sometimes focus\n * elements like document, body, and non-interactive SVG.\n * @param {Element} el\n */\n function isValidFocusTarget(el) {\n if (\n el &&\n el !== document &&\n el.nodeName !== 'HTML' &&\n el.nodeName !== 'BODY' &&\n 'classList' in el &&\n 'contains' in el.classList\n ) {\n return true;\n }\n return false;\n }\n\n /**\n * Computes whether the given element should automatically trigger the\n * `focus-visible` class being added, i.e. whether it should always match\n * `:focus-visible` when focused.\n * @param {Element} el\n * @return {boolean}\n */\n function focusTriggersKeyboardModality(el) {\n var type = el.type;\n var tagName = el.tagName;\n\n if (tagName === 'INPUT' && inputTypesAllowlist[type] && !el.readOnly) {\n return true;\n }\n\n if (tagName === 'TEXTAREA' && !el.readOnly) {\n return true;\n }\n\n if (el.isContentEditable) {\n return true;\n }\n\n return false;\n }\n\n /**\n * Add the `focus-visible` class to the given element if it was not added by\n * the author.\n * @param {Element} el\n */\n function addFocusVisibleClass(el) {\n if (el.classList.contains('focus-visible')) {\n return;\n }\n el.classList.add('focus-visible');\n el.setAttribute('data-focus-visible-added', '');\n }\n\n /**\n * Remove the `focus-visible` class from the given element if it was not\n * originally added by the author.\n * @param {Element} el\n */\n function removeFocusVisibleClass(el) {\n if (!el.hasAttribute('data-focus-visible-added')) {\n return;\n }\n el.classList.remove('focus-visible');\n el.removeAttribute('data-focus-visible-added');\n }\n\n /**\n * If the most recent user interaction was via the keyboard;\n * and the key press did not include a meta, alt/option, or control key;\n * then the modality is keyboard. Otherwise, the modality is not keyboard.\n * Apply `focus-visible` to any current active element and keep track\n * of our keyboard modality state with `hadKeyboardEvent`.\n * @param {KeyboardEvent} e\n */\n function onKeyDown(e) {\n if (e.metaKey || e.altKey || e.ctrlKey) {\n return;\n }\n\n if (isValidFocusTarget(scope.activeElement)) {\n addFocusVisibleClass(scope.activeElement);\n }\n\n hadKeyboardEvent = true;\n }\n\n /**\n * If at any point a user clicks with a pointing device, ensure that we change\n * the modality away from keyboard.\n * This avoids the situation where a user presses a key on an already focused\n * element, and then clicks on a different element, focusing it with a\n * pointing device, while we still think we're in keyboard modality.\n * @param {Event} e\n */\n function onPointerDown(e) {\n hadKeyboardEvent = false;\n }\n\n /**\n * On `focus`, add the `focus-visible` class to the target if:\n * - the target received focus as a result of keyboard navigation, or\n * - the event target is an element that will likely require interaction\n * via the keyboard (e.g. a text box)\n * @param {Event} e\n */\n function onFocus(e) {\n // Prevent IE from focusing the document or HTML element.\n if (!isValidFocusTarget(e.target)) {\n return;\n }\n\n if (hadKeyboardEvent || focusTriggersKeyboardModality(e.target)) {\n addFocusVisibleClass(e.target);\n }\n }\n\n /**\n * On `blur`, remove the `focus-visible` class from the target.\n * @param {Event} e\n */\n function onBlur(e) {\n if (!isValidFocusTarget(e.target)) {\n return;\n }\n\n if (\n e.target.classList.contains('focus-visible') ||\n e.target.hasAttribute('data-focus-visible-added')\n ) {\n // To detect a tab/window switch, we look for a blur event followed\n // rapidly by a visibility change.\n // If we don't see a visibility change within 100ms, it's probably a\n // regular focus change.\n hadFocusVisibleRecently = true;\n window.clearTimeout(hadFocusVisibleRecentlyTimeout);\n hadFocusVisibleRecentlyTimeout = window.setTimeout(function() {\n hadFocusVisibleRecently = false;\n }, 100);\n removeFocusVisibleClass(e.target);\n }\n }\n\n /**\n * If the user changes tabs, keep track of whether or not the previously\n * focused element had .focus-visible.\n * @param {Event} e\n */\n function onVisibilityChange(e) {\n if (document.visibilityState === 'hidden') {\n // If the tab becomes active again, the browser will handle calling focus\n // on the element (Safari actually calls it twice).\n // If this tab change caused a blur on an element with focus-visible,\n // re-apply the class when the user switches back to the tab.\n if (hadFocusVisibleRecently) {\n hadKeyboardEvent = true;\n }\n addInitialPointerMoveListeners();\n }\n }\n\n /**\n * Add a group of listeners to detect usage of any pointing devices.\n * These listeners will be added when the polyfill first loads, and anytime\n * the window is blurred, so that they are active when the window regains\n * focus.\n */\n function addInitialPointerMoveListeners() {\n document.addEventListener('mousemove', onInitialPointerMove);\n document.addEventListener('mousedown', onInitialPointerMove);\n document.addEventListener('mouseup', onInitialPointerMove);\n document.addEventListener('pointermove', onInitialPointerMove);\n document.addEventListener('pointerdown', onInitialPointerMove);\n document.addEventListener('pointerup', onInitialPointerMove);\n document.addEventListener('touchmove', onInitialPointerMove);\n document.addEventListener('touchstart', onInitialPointerMove);\n document.addEventListener('touchend', onInitialPointerMove);\n }\n\n function removeInitialPointerMoveListeners() {\n document.removeEventListener('mousemove', onInitialPointerMove);\n document.removeEventListener('mousedown', onInitialPointerMove);\n document.removeEventListener('mouseup', onInitialPointerMove);\n document.removeEventListener('pointermove', onInitialPointerMove);\n document.removeEventListener('pointerdown', onInitialPointerMove);\n document.removeEventListener('pointerup', onInitialPointerMove);\n document.removeEventListener('touchmove', onInitialPointerMove);\n document.removeEventListener('touchstart', onInitialPointerMove);\n document.removeEventListener('touchend', onInitialPointerMove);\n }\n\n /**\n * When the polfyill first loads, assume the user is in keyboard modality.\n * If any event is received from a pointing device (e.g. mouse, pointer,\n * touch), turn off keyboard modality.\n * This accounts for situations where focus enters the page from the URL bar.\n * @param {Event} e\n */\n function onInitialPointerMove(e) {\n // Work around a Safari quirk that fires a mousemove on whenever the\n // window blurs, even if you're tabbing out of the page. \u00AF\\_(\u30C4)_/\u00AF\n if (e.target.nodeName && e.target.nodeName.toLowerCase() === 'html') {\n return;\n }\n\n hadKeyboardEvent = false;\n removeInitialPointerMoveListeners();\n }\n\n // For some kinds of state, we are interested in changes at the global scope\n // only. For example, global pointer input, global key presses and global\n // visibility change should affect the state at every scope:\n document.addEventListener('keydown', onKeyDown, true);\n document.addEventListener('mousedown', onPointerDown, true);\n document.addEventListener('pointerdown', onPointerDown, true);\n document.addEventListener('touchstart', onPointerDown, true);\n document.addEventListener('visibilitychange', onVisibilityChange, true);\n\n addInitialPointerMoveListeners();\n\n // For focus and blur, we specifically care about state changes in the local\n // scope. This is because focus / blur events that originate from within a\n // shadow root are not re-dispatched from the host element if it was already\n // the active element in its own scope:\n scope.addEventListener('focus', onFocus, true);\n scope.addEventListener('blur', onBlur, true);\n\n // We detect that a node is a ShadowRoot by ensuring that it is a\n // DocumentFragment and also has a host property. This check covers native\n // implementation and polyfill implementation transparently. If we only cared\n // about the native implementation, we could just check if the scope was\n // an instance of a ShadowRoot.\n if (scope.nodeType === Node.DOCUMENT_FRAGMENT_NODE && scope.host) {\n // Since a ShadowRoot is a special kind of DocumentFragment, it does not\n // have a root element to add a class to. So, we add this attribute to the\n // host element instead:\n scope.host.setAttribute('data-js-focus-visible', '');\n } else if (scope.nodeType === Node.DOCUMENT_NODE) {\n document.documentElement.classList.add('js-focus-visible');\n document.documentElement.setAttribute('data-js-focus-visible', '');\n }\n }\n\n // It is important to wrap all references to global window and document in\n // these checks to support server-side rendering use cases\n // @see https://github.com/WICG/focus-visible/issues/199\n if (typeof window !== 'undefined' && typeof document !== 'undefined') {\n // Make the polyfill helper globally available. This can be used as a signal\n // to interested libraries that wish to coordinate with the polyfill for e.g.,\n // applying the polyfill to a shadow root:\n window.applyFocusVisiblePolyfill = applyFocusVisiblePolyfill;\n\n // Notify interested libraries of the polyfill's presence, in case the\n // polyfill was loaded lazily:\n var event;\n\n try {\n event = new CustomEvent('focus-visible-polyfill-ready');\n } catch (error) {\n // IE11 does not support using CustomEvent as a constructor directly:\n event = document.createEvent('CustomEvent');\n event.initCustomEvent('focus-visible-polyfill-ready', false, false, {});\n }\n\n window.dispatchEvent(event);\n }\n\n if (typeof document !== 'undefined') {\n // Apply the polyfill to the global document, so that no JavaScript\n // coordination is required to use the polyfill in the top-level document:\n applyFocusVisiblePolyfill(document);\n }\n\n})));\n", "/*!\n * clipboard.js v2.0.11\n * https://clipboardjs.com/\n *\n * Licensed MIT \u00A9 Zeno Rocha\n */\n(function webpackUniversalModuleDefinition(root, factory) {\n\tif(typeof exports === 'object' && typeof module === 'object')\n\t\tmodule.exports = factory();\n\telse if(typeof define === 'function' && define.amd)\n\t\tdefine([], factory);\n\telse if(typeof exports === 'object')\n\t\texports[\"ClipboardJS\"] = factory();\n\telse\n\t\troot[\"ClipboardJS\"] = factory();\n})(this, function() {\nreturn /******/ (function() { // webpackBootstrap\n/******/ \tvar __webpack_modules__ = ({\n\n/***/ 686:\n/***/ (function(__unused_webpack_module, __webpack_exports__, __webpack_require__) {\n\n\"use strict\";\n\n// EXPORTS\n__webpack_require__.d(__webpack_exports__, {\n \"default\": function() { return /* binding */ clipboard; }\n});\n\n// EXTERNAL MODULE: ./node_modules/tiny-emitter/index.js\nvar tiny_emitter = __webpack_require__(279);\nvar tiny_emitter_default = /*#__PURE__*/__webpack_require__.n(tiny_emitter);\n// EXTERNAL MODULE: ./node_modules/good-listener/src/listen.js\nvar listen = __webpack_require__(370);\nvar listen_default = /*#__PURE__*/__webpack_require__.n(listen);\n// EXTERNAL MODULE: ./node_modules/select/src/select.js\nvar src_select = __webpack_require__(817);\nvar select_default = /*#__PURE__*/__webpack_require__.n(src_select);\n;// CONCATENATED MODULE: ./src/common/command.js\n/**\n * Executes a given operation type.\n * @param {String} type\n * @return {Boolean}\n */\nfunction command(type) {\n try {\n return document.execCommand(type);\n } catch (err) {\n return false;\n }\n}\n;// CONCATENATED MODULE: ./src/actions/cut.js\n\n\n/**\n * Cut action wrapper.\n * @param {String|HTMLElement} target\n * @return {String}\n */\n\nvar ClipboardActionCut = function ClipboardActionCut(target) {\n var selectedText = select_default()(target);\n command('cut');\n return selectedText;\n};\n\n/* harmony default export */ var actions_cut = (ClipboardActionCut);\n;// CONCATENATED MODULE: ./src/common/create-fake-element.js\n/**\n * Creates a fake textarea element with a value.\n * @param {String} value\n * @return {HTMLElement}\n */\nfunction createFakeElement(value) {\n var isRTL = document.documentElement.getAttribute('dir') === 'rtl';\n var fakeElement = document.createElement('textarea'); // Prevent zooming on iOS\n\n fakeElement.style.fontSize = '12pt'; // Reset box model\n\n fakeElement.style.border = '0';\n fakeElement.style.padding = '0';\n fakeElement.style.margin = '0'; // Move element out of screen horizontally\n\n fakeElement.style.position = 'absolute';\n fakeElement.style[isRTL ? 'right' : 'left'] = '-9999px'; // Move element to the same position vertically\n\n var yPosition = window.pageYOffset || document.documentElement.scrollTop;\n fakeElement.style.top = \"\".concat(yPosition, \"px\");\n fakeElement.setAttribute('readonly', '');\n fakeElement.value = value;\n return fakeElement;\n}\n;// CONCATENATED MODULE: ./src/actions/copy.js\n\n\n\n/**\n * Create fake copy action wrapper using a fake element.\n * @param {String} target\n * @param {Object} options\n * @return {String}\n */\n\nvar fakeCopyAction = function fakeCopyAction(value, options) {\n var fakeElement = createFakeElement(value);\n options.container.appendChild(fakeElement);\n var selectedText = select_default()(fakeElement);\n command('copy');\n fakeElement.remove();\n return selectedText;\n};\n/**\n * Copy action wrapper.\n * @param {String|HTMLElement} target\n * @param {Object} options\n * @return {String}\n */\n\n\nvar ClipboardActionCopy = function ClipboardActionCopy(target) {\n var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n container: document.body\n };\n var selectedText = '';\n\n if (typeof target === 'string') {\n selectedText = fakeCopyAction(target, options);\n } else if (target instanceof HTMLInputElement && !['text', 'search', 'url', 'tel', 'password'].includes(target === null || target === void 0 ? void 0 : target.type)) {\n // If input type doesn't support `setSelectionRange`. Simulate it. https://developer.mozilla.org/en-US/docs/Web/API/HTMLInputElement/setSelectionRange\n selectedText = fakeCopyAction(target.value, options);\n } else {\n selectedText = select_default()(target);\n command('copy');\n }\n\n return selectedText;\n};\n\n/* harmony default export */ var actions_copy = (ClipboardActionCopy);\n;// CONCATENATED MODULE: ./src/actions/default.js\nfunction _typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { _typeof = function _typeof(obj) { return typeof obj; }; } else { _typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return _typeof(obj); }\n\n\n\n/**\n * Inner function which performs selection from either `text` or `target`\n * properties and then executes copy or cut operations.\n * @param {Object} options\n */\n\nvar ClipboardActionDefault = function ClipboardActionDefault() {\n var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n // Defines base properties passed from constructor.\n var _options$action = options.action,\n action = _options$action === void 0 ? 'copy' : _options$action,\n container = options.container,\n target = options.target,\n text = options.text; // Sets the `action` to be performed which can be either 'copy' or 'cut'.\n\n if (action !== 'copy' && action !== 'cut') {\n throw new Error('Invalid \"action\" value, use either \"copy\" or \"cut\"');\n } // Sets the `target` property using an element that will be have its content copied.\n\n\n if (target !== undefined) {\n if (target && _typeof(target) === 'object' && target.nodeType === 1) {\n if (action === 'copy' && target.hasAttribute('disabled')) {\n throw new Error('Invalid \"target\" attribute. Please use \"readonly\" instead of \"disabled\" attribute');\n }\n\n if (action === 'cut' && (target.hasAttribute('readonly') || target.hasAttribute('disabled'))) {\n throw new Error('Invalid \"target\" attribute. You can\\'t cut text from elements with \"readonly\" or \"disabled\" attributes');\n }\n } else {\n throw new Error('Invalid \"target\" value, use a valid Element');\n }\n } // Define selection strategy based on `text` property.\n\n\n if (text) {\n return actions_copy(text, {\n container: container\n });\n } // Defines which selection strategy based on `target` property.\n\n\n if (target) {\n return action === 'cut' ? actions_cut(target) : actions_copy(target, {\n container: container\n });\n }\n};\n\n/* harmony default export */ var actions_default = (ClipboardActionDefault);\n;// CONCATENATED MODULE: ./src/clipboard.js\nfunction clipboard_typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { clipboard_typeof = function _typeof(obj) { return typeof obj; }; } else { clipboard_typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return clipboard_typeof(obj); }\n\nfunction _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError(\"Cannot call a class as a function\"); } }\n\nfunction _defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if (\"value\" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } }\n\nfunction _createClass(Constructor, protoProps, staticProps) { if (protoProps) _defineProperties(Constructor.prototype, protoProps); if (staticProps) _defineProperties(Constructor, staticProps); return Constructor; }\n\nfunction _inherits(subClass, superClass) { if (typeof superClass !== \"function\" && superClass !== null) { throw new TypeError(\"Super expression must either be null or a function\"); } subClass.prototype = Object.create(superClass && superClass.prototype, { constructor: { value: subClass, writable: true, configurable: true } }); if (superClass) _setPrototypeOf(subClass, superClass); }\n\nfunction _setPrototypeOf(o, p) { _setPrototypeOf = Object.setPrototypeOf || function _setPrototypeOf(o, p) { o.__proto__ = p; return o; }; return _setPrototypeOf(o, p); }\n\nfunction _createSuper(Derived) { var hasNativeReflectConstruct = _isNativeReflectConstruct(); return function _createSuperInternal() { var Super = _getPrototypeOf(Derived), result; if (hasNativeReflectConstruct) { var NewTarget = _getPrototypeOf(this).constructor; result = Reflect.construct(Super, arguments, NewTarget); } else { result = Super.apply(this, arguments); } return _possibleConstructorReturn(this, result); }; }\n\nfunction _possibleConstructorReturn(self, call) { if (call && (clipboard_typeof(call) === \"object\" || typeof call === \"function\")) { return call; } return _assertThisInitialized(self); }\n\nfunction _assertThisInitialized(self) { if (self === void 0) { throw new ReferenceError(\"this hasn't been initialised - super() hasn't been called\"); } return self; }\n\nfunction _isNativeReflectConstruct() { if (typeof Reflect === \"undefined\" || !Reflect.construct) return false; if (Reflect.construct.sham) return false; if (typeof Proxy === \"function\") return true; try { Date.prototype.toString.call(Reflect.construct(Date, [], function () {})); return true; } catch (e) { return false; } }\n\nfunction _getPrototypeOf(o) { _getPrototypeOf = Object.setPrototypeOf ? Object.getPrototypeOf : function _getPrototypeOf(o) { return o.__proto__ || Object.getPrototypeOf(o); }; return _getPrototypeOf(o); }\n\n\n\n\n\n\n/**\n * Helper function to retrieve attribute value.\n * @param {String} suffix\n * @param {Element} element\n */\n\nfunction getAttributeValue(suffix, element) {\n var attribute = \"data-clipboard-\".concat(suffix);\n\n if (!element.hasAttribute(attribute)) {\n return;\n }\n\n return element.getAttribute(attribute);\n}\n/**\n * Base class which takes one or more elements, adds event listeners to them,\n * and instantiates a new `ClipboardAction` on each click.\n */\n\n\nvar Clipboard = /*#__PURE__*/function (_Emitter) {\n _inherits(Clipboard, _Emitter);\n\n var _super = _createSuper(Clipboard);\n\n /**\n * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n * @param {Object} options\n */\n function Clipboard(trigger, options) {\n var _this;\n\n _classCallCheck(this, Clipboard);\n\n _this = _super.call(this);\n\n _this.resolveOptions(options);\n\n _this.listenClick(trigger);\n\n return _this;\n }\n /**\n * Defines if attributes would be resolved using internal setter functions\n * or custom functions that were passed in the constructor.\n * @param {Object} options\n */\n\n\n _createClass(Clipboard, [{\n key: \"resolveOptions\",\n value: function resolveOptions() {\n var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n this.action = typeof options.action === 'function' ? options.action : this.defaultAction;\n this.target = typeof options.target === 'function' ? options.target : this.defaultTarget;\n this.text = typeof options.text === 'function' ? options.text : this.defaultText;\n this.container = clipboard_typeof(options.container) === 'object' ? options.container : document.body;\n }\n /**\n * Adds a click event listener to the passed trigger.\n * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n */\n\n }, {\n key: \"listenClick\",\n value: function listenClick(trigger) {\n var _this2 = this;\n\n this.listener = listen_default()(trigger, 'click', function (e) {\n return _this2.onClick(e);\n });\n }\n /**\n * Defines a new `ClipboardAction` on each click event.\n * @param {Event} e\n */\n\n }, {\n key: \"onClick\",\n value: function onClick(e) {\n var trigger = e.delegateTarget || e.currentTarget;\n var action = this.action(trigger) || 'copy';\n var text = actions_default({\n action: action,\n container: this.container,\n target: this.target(trigger),\n text: this.text(trigger)\n }); // Fires an event based on the copy operation result.\n\n this.emit(text ? 'success' : 'error', {\n action: action,\n text: text,\n trigger: trigger,\n clearSelection: function clearSelection() {\n if (trigger) {\n trigger.focus();\n }\n\n window.getSelection().removeAllRanges();\n }\n });\n }\n /**\n * Default `action` lookup function.\n * @param {Element} trigger\n */\n\n }, {\n key: \"defaultAction\",\n value: function defaultAction(trigger) {\n return getAttributeValue('action', trigger);\n }\n /**\n * Default `target` lookup function.\n * @param {Element} trigger\n */\n\n }, {\n key: \"defaultTarget\",\n value: function defaultTarget(trigger) {\n var selector = getAttributeValue('target', trigger);\n\n if (selector) {\n return document.querySelector(selector);\n }\n }\n /**\n * Allow fire programmatically a copy action\n * @param {String|HTMLElement} target\n * @param {Object} options\n * @returns Text copied.\n */\n\n }, {\n key: \"defaultText\",\n\n /**\n * Default `text` lookup function.\n * @param {Element} trigger\n */\n value: function defaultText(trigger) {\n return getAttributeValue('text', trigger);\n }\n /**\n * Destroy lifecycle.\n */\n\n }, {\n key: \"destroy\",\n value: function destroy() {\n this.listener.destroy();\n }\n }], [{\n key: \"copy\",\n value: function copy(target) {\n var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n container: document.body\n };\n return actions_copy(target, options);\n }\n /**\n * Allow fire programmatically a cut action\n * @param {String|HTMLElement} target\n * @returns Text cutted.\n */\n\n }, {\n key: \"cut\",\n value: function cut(target) {\n return actions_cut(target);\n }\n /**\n * Returns the support of the given action, or all actions if no action is\n * given.\n * @param {String} [action]\n */\n\n }, {\n key: \"isSupported\",\n value: function isSupported() {\n var action = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : ['copy', 'cut'];\n var actions = typeof action === 'string' ? [action] : action;\n var support = !!document.queryCommandSupported;\n actions.forEach(function (action) {\n support = support && !!document.queryCommandSupported(action);\n });\n return support;\n }\n }]);\n\n return Clipboard;\n}((tiny_emitter_default()));\n\n/* harmony default export */ var clipboard = (Clipboard);\n\n/***/ }),\n\n/***/ 828:\n/***/ (function(module) {\n\nvar DOCUMENT_NODE_TYPE = 9;\n\n/**\n * A polyfill for Element.matches()\n */\nif (typeof Element !== 'undefined' && !Element.prototype.matches) {\n var proto = Element.prototype;\n\n proto.matches = proto.matchesSelector ||\n proto.mozMatchesSelector ||\n proto.msMatchesSelector ||\n proto.oMatchesSelector ||\n proto.webkitMatchesSelector;\n}\n\n/**\n * Finds the closest parent that matches a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @return {Function}\n */\nfunction closest (element, selector) {\n while (element && element.nodeType !== DOCUMENT_NODE_TYPE) {\n if (typeof element.matches === 'function' &&\n element.matches(selector)) {\n return element;\n }\n element = element.parentNode;\n }\n}\n\nmodule.exports = closest;\n\n\n/***/ }),\n\n/***/ 438:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar closest = __webpack_require__(828);\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction _delegate(element, selector, type, callback, useCapture) {\n var listenerFn = listener.apply(this, arguments);\n\n element.addEventListener(type, listenerFn, useCapture);\n\n return {\n destroy: function() {\n element.removeEventListener(type, listenerFn, useCapture);\n }\n }\n}\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element|String|Array} [elements]\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction delegate(elements, selector, type, callback, useCapture) {\n // Handle the regular Element usage\n if (typeof elements.addEventListener === 'function') {\n return _delegate.apply(null, arguments);\n }\n\n // Handle Element-less usage, it defaults to global delegation\n if (typeof type === 'function') {\n // Use `document` as the first parameter, then apply arguments\n // This is a short way to .unshift `arguments` without running into deoptimizations\n return _delegate.bind(null, document).apply(null, arguments);\n }\n\n // Handle Selector-based usage\n if (typeof elements === 'string') {\n elements = document.querySelectorAll(elements);\n }\n\n // Handle Array-like based usage\n return Array.prototype.map.call(elements, function (element) {\n return _delegate(element, selector, type, callback, useCapture);\n });\n}\n\n/**\n * Finds closest match and invokes callback.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Function}\n */\nfunction listener(element, selector, type, callback) {\n return function(e) {\n e.delegateTarget = closest(e.target, selector);\n\n if (e.delegateTarget) {\n callback.call(element, e);\n }\n }\n}\n\nmodule.exports = delegate;\n\n\n/***/ }),\n\n/***/ 879:\n/***/ (function(__unused_webpack_module, exports) {\n\n/**\n * Check if argument is a HTML element.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.node = function(value) {\n return value !== undefined\n && value instanceof HTMLElement\n && value.nodeType === 1;\n};\n\n/**\n * Check if argument is a list of HTML elements.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.nodeList = function(value) {\n var type = Object.prototype.toString.call(value);\n\n return value !== undefined\n && (type === '[object NodeList]' || type === '[object HTMLCollection]')\n && ('length' in value)\n && (value.length === 0 || exports.node(value[0]));\n};\n\n/**\n * Check if argument is a string.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.string = function(value) {\n return typeof value === 'string'\n || value instanceof String;\n};\n\n/**\n * Check if argument is a function.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.fn = function(value) {\n var type = Object.prototype.toString.call(value);\n\n return type === '[object Function]';\n};\n\n\n/***/ }),\n\n/***/ 370:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar is = __webpack_require__(879);\nvar delegate = __webpack_require__(438);\n\n/**\n * Validates all params and calls the right\n * listener function based on its target type.\n *\n * @param {String|HTMLElement|HTMLCollection|NodeList} target\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listen(target, type, callback) {\n if (!target && !type && !callback) {\n throw new Error('Missing required arguments');\n }\n\n if (!is.string(type)) {\n throw new TypeError('Second argument must be a String');\n }\n\n if (!is.fn(callback)) {\n throw new TypeError('Third argument must be a Function');\n }\n\n if (is.node(target)) {\n return listenNode(target, type, callback);\n }\n else if (is.nodeList(target)) {\n return listenNodeList(target, type, callback);\n }\n else if (is.string(target)) {\n return listenSelector(target, type, callback);\n }\n else {\n throw new TypeError('First argument must be a String, HTMLElement, HTMLCollection, or NodeList');\n }\n}\n\n/**\n * Adds an event listener to a HTML element\n * and returns a remove listener function.\n *\n * @param {HTMLElement} node\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNode(node, type, callback) {\n node.addEventListener(type, callback);\n\n return {\n destroy: function() {\n node.removeEventListener(type, callback);\n }\n }\n}\n\n/**\n * Add an event listener to a list of HTML elements\n * and returns a remove listener function.\n *\n * @param {NodeList|HTMLCollection} nodeList\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNodeList(nodeList, type, callback) {\n Array.prototype.forEach.call(nodeList, function(node) {\n node.addEventListener(type, callback);\n });\n\n return {\n destroy: function() {\n Array.prototype.forEach.call(nodeList, function(node) {\n node.removeEventListener(type, callback);\n });\n }\n }\n}\n\n/**\n * Add an event listener to a selector\n * and returns a remove listener function.\n *\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenSelector(selector, type, callback) {\n return delegate(document.body, selector, type, callback);\n}\n\nmodule.exports = listen;\n\n\n/***/ }),\n\n/***/ 817:\n/***/ (function(module) {\n\nfunction select(element) {\n var selectedText;\n\n if (element.nodeName === 'SELECT') {\n element.focus();\n\n selectedText = element.value;\n }\n else if (element.nodeName === 'INPUT' || element.nodeName === 'TEXTAREA') {\n var isReadOnly = element.hasAttribute('readonly');\n\n if (!isReadOnly) {\n element.setAttribute('readonly', '');\n }\n\n element.select();\n element.setSelectionRange(0, element.value.length);\n\n if (!isReadOnly) {\n element.removeAttribute('readonly');\n }\n\n selectedText = element.value;\n }\n else {\n if (element.hasAttribute('contenteditable')) {\n element.focus();\n }\n\n var selection = window.getSelection();\n var range = document.createRange();\n\n range.selectNodeContents(element);\n selection.removeAllRanges();\n selection.addRange(range);\n\n selectedText = selection.toString();\n }\n\n return selectedText;\n}\n\nmodule.exports = select;\n\n\n/***/ }),\n\n/***/ 279:\n/***/ (function(module) {\n\nfunction E () {\n // Keep this empty so it's easier to inherit from\n // (via https://github.com/lipsmack from https://github.com/scottcorgan/tiny-emitter/issues/3)\n}\n\nE.prototype = {\n on: function (name, callback, ctx) {\n var e = this.e || (this.e = {});\n\n (e[name] || (e[name] = [])).push({\n fn: callback,\n ctx: ctx\n });\n\n return this;\n },\n\n once: function (name, callback, ctx) {\n var self = this;\n function listener () {\n self.off(name, listener);\n callback.apply(ctx, arguments);\n };\n\n listener._ = callback\n return this.on(name, listener, ctx);\n },\n\n emit: function (name) {\n var data = [].slice.call(arguments, 1);\n var evtArr = ((this.e || (this.e = {}))[name] || []).slice();\n var i = 0;\n var len = evtArr.length;\n\n for (i; i < len; i++) {\n evtArr[i].fn.apply(evtArr[i].ctx, data);\n }\n\n return this;\n },\n\n off: function (name, callback) {\n var e = this.e || (this.e = {});\n var evts = e[name];\n var liveEvents = [];\n\n if (evts && callback) {\n for (var i = 0, len = evts.length; i < len; i++) {\n if (evts[i].fn !== callback && evts[i].fn._ !== callback)\n liveEvents.push(evts[i]);\n }\n }\n\n // Remove event from queue to prevent memory leak\n // Suggested by https://github.com/lazd\n // Ref: https://github.com/scottcorgan/tiny-emitter/commit/c6ebfaa9bc973b33d110a84a307742b7cf94c953#commitcomment-5024910\n\n (liveEvents.length)\n ? e[name] = liveEvents\n : delete e[name];\n\n return this;\n }\n};\n\nmodule.exports = E;\nmodule.exports.TinyEmitter = E;\n\n\n/***/ })\n\n/******/ \t});\n/************************************************************************/\n/******/ \t// The module cache\n/******/ \tvar __webpack_module_cache__ = {};\n/******/ \t\n/******/ \t// The require function\n/******/ \tfunction __webpack_require__(moduleId) {\n/******/ \t\t// Check if module is in cache\n/******/ \t\tif(__webpack_module_cache__[moduleId]) {\n/******/ \t\t\treturn __webpack_module_cache__[moduleId].exports;\n/******/ \t\t}\n/******/ \t\t// Create a new module (and put it into the cache)\n/******/ \t\tvar module = __webpack_module_cache__[moduleId] = {\n/******/ \t\t\t// no module.id needed\n/******/ \t\t\t// no module.loaded needed\n/******/ \t\t\texports: {}\n/******/ \t\t};\n/******/ \t\n/******/ \t\t// Execute the module function\n/******/ \t\t__webpack_modules__[moduleId](module, module.exports, __webpack_require__);\n/******/ \t\n/******/ \t\t// Return the exports of the module\n/******/ \t\treturn module.exports;\n/******/ \t}\n/******/ \t\n/************************************************************************/\n/******/ \t/* webpack/runtime/compat get default export */\n/******/ \t!function() {\n/******/ \t\t// getDefaultExport function for compatibility with non-harmony modules\n/******/ \t\t__webpack_require__.n = function(module) {\n/******/ \t\t\tvar getter = module && module.__esModule ?\n/******/ \t\t\t\tfunction() { return module['default']; } :\n/******/ \t\t\t\tfunction() { return module; };\n/******/ \t\t\t__webpack_require__.d(getter, { a: getter });\n/******/ \t\t\treturn getter;\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/define property getters */\n/******/ \t!function() {\n/******/ \t\t// define getter functions for harmony exports\n/******/ \t\t__webpack_require__.d = function(exports, definition) {\n/******/ \t\t\tfor(var key in definition) {\n/******/ \t\t\t\tif(__webpack_require__.o(definition, key) && !__webpack_require__.o(exports, key)) {\n/******/ \t\t\t\t\tObject.defineProperty(exports, key, { enumerable: true, get: definition[key] });\n/******/ \t\t\t\t}\n/******/ \t\t\t}\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/hasOwnProperty shorthand */\n/******/ \t!function() {\n/******/ \t\t__webpack_require__.o = function(obj, prop) { return Object.prototype.hasOwnProperty.call(obj, prop); }\n/******/ \t}();\n/******/ \t\n/************************************************************************/\n/******/ \t// module exports must be returned from runtime so entry inlining is disabled\n/******/ \t// startup\n/******/ \t// Load entry module and return exports\n/******/ \treturn __webpack_require__(686);\n/******/ })()\n.default;\n});", "/*!\n * escape-html\n * Copyright(c) 2012-2013 TJ Holowaychuk\n * Copyright(c) 2015 Andreas Lubbe\n * Copyright(c) 2015 Tiancheng \"Timothy\" Gu\n * MIT Licensed\n */\n\n'use strict';\n\n/**\n * Module variables.\n * @private\n */\n\nvar matchHtmlRegExp = /[\"'&<>]/;\n\n/**\n * Module exports.\n * @public\n */\n\nmodule.exports = escapeHtml;\n\n/**\n * Escape special characters in the given string of html.\n *\n * @param {string} string The string to escape for inserting into HTML\n * @return {string}\n * @public\n */\n\nfunction escapeHtml(string) {\n var str = '' + string;\n var match = matchHtmlRegExp.exec(str);\n\n if (!match) {\n return str;\n }\n\n var escape;\n var html = '';\n var index = 0;\n var lastIndex = 0;\n\n for (index = match.index; index < str.length; index++) {\n switch (str.charCodeAt(index)) {\n case 34: // \"\n escape = '"';\n break;\n case 38: // &\n escape = '&';\n break;\n case 39: // '\n escape = ''';\n break;\n case 60: // <\n escape = '<';\n break;\n case 62: // >\n escape = '>';\n break;\n default:\n continue;\n }\n\n if (lastIndex !== index) {\n html += str.substring(lastIndex, index);\n }\n\n lastIndex = index + 1;\n html += escape;\n }\n\n return lastIndex !== index\n ? html + str.substring(lastIndex, index)\n : html;\n}\n", "/*\n * Copyright (c) 2016-2024 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport \"focus-visible\"\n\nimport {\n EMPTY,\n NEVER,\n Observable,\n Subject,\n defer,\n delay,\n filter,\n map,\n merge,\n mergeWith,\n shareReplay,\n switchMap\n} from \"rxjs\"\n\nimport { configuration, feature } from \"./_\"\nimport {\n at,\n getActiveElement,\n getOptionalElement,\n requestJSON,\n setLocation,\n setToggle,\n watchDocument,\n watchKeyboard,\n watchLocation,\n watchLocationTarget,\n watchMedia,\n watchPrint,\n watchScript,\n watchViewport\n} from \"./browser\"\nimport {\n getComponentElement,\n getComponentElements,\n mountAnnounce,\n mountBackToTop,\n mountConsent,\n mountContent,\n mountDialog,\n mountHeader,\n mountHeaderTitle,\n mountPalette,\n mountProgress,\n mountSearch,\n mountSearchHiglight,\n mountSidebar,\n mountSource,\n mountTableOfContents,\n mountTabs,\n watchHeader,\n watchMain\n} from \"./components\"\nimport {\n SearchIndex,\n setupClipboardJS,\n setupInstantNavigation,\n setupVersionSelector\n} from \"./integrations\"\nimport {\n patchEllipsis,\n patchIndeterminate,\n patchScrollfix,\n patchScrolllock\n} from \"./patches\"\nimport \"./polyfills\"\n\n/* ----------------------------------------------------------------------------\n * Functions - @todo refactor\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch search index\n *\n * @returns Search index observable\n */\nfunction fetchSearchIndex(): Observable {\n if (location.protocol === \"file:\") {\n return watchScript(\n `${new URL(\"search/search_index.js\", config.base)}`\n )\n .pipe(\n // @ts-ignore - @todo fix typings\n map(() => __index),\n shareReplay(1)\n )\n } else {\n return requestJSON(\n new URL(\"search/search_index.json\", config.base)\n )\n }\n}\n\n/* ----------------------------------------------------------------------------\n * Application\n * ------------------------------------------------------------------------- */\n\n/* Yay, JavaScript is available */\ndocument.documentElement.classList.remove(\"no-js\")\ndocument.documentElement.classList.add(\"js\")\n\n/* Set up navigation observables and subjects */\nconst document$ = watchDocument()\nconst location$ = watchLocation()\nconst target$ = watchLocationTarget(location$)\nconst keyboard$ = watchKeyboard()\n\n/* Set up media observables */\nconst viewport$ = watchViewport()\nconst tablet$ = watchMedia(\"(min-width: 960px)\")\nconst screen$ = watchMedia(\"(min-width: 1220px)\")\nconst print$ = watchPrint()\n\n/* Retrieve search index, if search is enabled */\nconst config = configuration()\nconst index$ = document.forms.namedItem(\"search\")\n ? fetchSearchIndex()\n : NEVER\n\n/* Set up Clipboard.js integration */\nconst alert$ = new Subject()\nsetupClipboardJS({ alert$ })\n\n/* Set up progress indicator */\nconst progress$ = new Subject()\n\n/* Set up instant navigation, if enabled */\nif (feature(\"navigation.instant\"))\n setupInstantNavigation({ location$, viewport$, progress$ })\n .subscribe(document$)\n\n/* Set up version selector */\nif (config.version?.provider === \"mike\")\n setupVersionSelector({ document$ })\n\n/* Always close drawer and search on navigation */\nmerge(location$, target$)\n .pipe(\n delay(125)\n )\n .subscribe(() => {\n setToggle(\"drawer\", false)\n setToggle(\"search\", false)\n })\n\n/* Set up global keyboard handlers */\nkeyboard$\n .pipe(\n filter(({ mode }) => mode === \"global\")\n )\n .subscribe(key => {\n switch (key.type) {\n\n /* Go to previous page */\n case \"p\":\n case \",\":\n const prev = getOptionalElement(\"link[rel=prev]\")\n if (typeof prev !== \"undefined\")\n setLocation(prev)\n break\n\n /* Go to next page */\n case \"n\":\n case \".\":\n const next = getOptionalElement(\"link[rel=next]\")\n if (typeof next !== \"undefined\")\n setLocation(next)\n break\n\n /* Expand navigation, see https://bit.ly/3ZjG5io */\n case \"Enter\":\n const active = getActiveElement()\n if (active instanceof HTMLLabelElement)\n active.click()\n }\n })\n\n/* Set up patches */\npatchEllipsis({ viewport$, document$ })\npatchIndeterminate({ document$, tablet$ })\npatchScrollfix({ document$ })\npatchScrolllock({ viewport$, tablet$ })\n\n/* Set up header and main area observable */\nconst header$ = watchHeader(getComponentElement(\"header\"), { viewport$ })\nconst main$ = document$\n .pipe(\n map(() => getComponentElement(\"main\")),\n switchMap(el => watchMain(el, { viewport$, header$ })),\n shareReplay(1)\n )\n\n/* Set up control component observables */\nconst control$ = merge(\n\n /* Consent */\n ...getComponentElements(\"consent\")\n .map(el => mountConsent(el, { target$ })),\n\n /* Dialog */\n ...getComponentElements(\"dialog\")\n .map(el => mountDialog(el, { alert$ })),\n\n /* Header */\n ...getComponentElements(\"header\")\n .map(el => mountHeader(el, { viewport$, header$, main$ })),\n\n /* Color palette */\n ...getComponentElements(\"palette\")\n .map(el => mountPalette(el)),\n\n /* Progress bar */\n ...getComponentElements(\"progress\")\n .map(el => mountProgress(el, { progress$ })),\n\n /* Search */\n ...getComponentElements(\"search\")\n .map(el => mountSearch(el, { index$, keyboard$ })),\n\n /* Repository information */\n ...getComponentElements(\"source\")\n .map(el => mountSource(el))\n)\n\n/* Set up content component observables */\nconst content$ = defer(() => merge(\n\n /* Announcement bar */\n ...getComponentElements(\"announce\")\n .map(el => mountAnnounce(el)),\n\n /* Content */\n ...getComponentElements(\"content\")\n .map(el => mountContent(el, { viewport$, target$, print$ })),\n\n /* Search highlighting */\n ...getComponentElements(\"content\")\n .map(el => feature(\"search.highlight\")\n ? mountSearchHiglight(el, { index$, location$ })\n : EMPTY\n ),\n\n /* Header title */\n ...getComponentElements(\"header-title\")\n .map(el => mountHeaderTitle(el, { viewport$, header$ })),\n\n /* Sidebar */\n ...getComponentElements(\"sidebar\")\n .map(el => el.getAttribute(\"data-md-type\") === \"navigation\"\n ? at(screen$, () => mountSidebar(el, { viewport$, header$, main$ }))\n : at(tablet$, () => mountSidebar(el, { viewport$, header$, main$ }))\n ),\n\n /* Navigation tabs */\n ...getComponentElements(\"tabs\")\n .map(el => mountTabs(el, { viewport$, header$ })),\n\n /* Table of contents */\n ...getComponentElements(\"toc\")\n .map(el => mountTableOfContents(el, {\n viewport$, header$, main$, target$\n })),\n\n /* Back-to-top button */\n ...getComponentElements(\"top\")\n .map(el => mountBackToTop(el, { viewport$, header$, main$, target$ }))\n))\n\n/* Set up component observables */\nconst component$ = document$\n .pipe(\n switchMap(() => content$),\n mergeWith(control$),\n shareReplay(1)\n )\n\n/* Subscribe to all components */\ncomponent$.subscribe()\n\n/* ----------------------------------------------------------------------------\n * Exports\n * ------------------------------------------------------------------------- */\n\nwindow.document$ = document$ /* Document observable */\nwindow.location$ = location$ /* Location subject */\nwindow.target$ = target$ /* Location target observable */\nwindow.keyboard$ = keyboard$ /* Keyboard observable */\nwindow.viewport$ = viewport$ /* Viewport observable */\nwindow.tablet$ = tablet$ /* Media tablet observable */\nwindow.screen$ = screen$ /* Media screen observable */\nwindow.print$ = print$ /* Media print observable */\nwindow.alert$ = alert$ /* Alert subject */\nwindow.progress$ = progress$ /* Progress indicator subject */\nwindow.component$ = component$ /* Component observable */\n", "/*! *****************************************************************************\r\nCopyright (c) Microsoft Corporation.\r\n\r\nPermission to use, copy, modify, and/or distribute this software for any\r\npurpose with or without fee is hereby granted.\r\n\r\nTHE SOFTWARE IS PROVIDED \"AS IS\" AND THE AUTHOR DISCLAIMS ALL WARRANTIES WITH\r\nREGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY\r\nAND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT,\r\nINDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM\r\nLOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR\r\nOTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR\r\nPERFORMANCE OF THIS SOFTWARE.\r\n***************************************************************************** */\r\n/* global Reflect, Promise */\r\n\r\nvar extendStatics = function(d, b) {\r\n extendStatics = Object.setPrototypeOf ||\r\n ({ __proto__: [] } instanceof Array && function (d, b) { d.__proto__ = b; }) ||\r\n function (d, b) { for (var p in b) if (Object.prototype.hasOwnProperty.call(b, p)) d[p] = b[p]; };\r\n return extendStatics(d, b);\r\n};\r\n\r\nexport function __extends(d, b) {\r\n if (typeof b !== \"function\" && b !== null)\r\n throw new TypeError(\"Class extends value \" + String(b) + \" is not a constructor or null\");\r\n extendStatics(d, b);\r\n function __() { this.constructor = d; }\r\n d.prototype = b === null ? Object.create(b) : (__.prototype = b.prototype, new __());\r\n}\r\n\r\nexport var __assign = function() {\r\n __assign = Object.assign || function __assign(t) {\r\n for (var s, i = 1, n = arguments.length; i < n; i++) {\r\n s = arguments[i];\r\n for (var p in s) if (Object.prototype.hasOwnProperty.call(s, p)) t[p] = s[p];\r\n }\r\n return t;\r\n }\r\n return __assign.apply(this, arguments);\r\n}\r\n\r\nexport function __rest(s, e) {\r\n var t = {};\r\n for (var p in s) if (Object.prototype.hasOwnProperty.call(s, p) && e.indexOf(p) < 0)\r\n t[p] = s[p];\r\n if (s != null && typeof Object.getOwnPropertySymbols === \"function\")\r\n for (var i = 0, p = Object.getOwnPropertySymbols(s); i < p.length; i++) {\r\n if (e.indexOf(p[i]) < 0 && Object.prototype.propertyIsEnumerable.call(s, p[i]))\r\n t[p[i]] = s[p[i]];\r\n }\r\n return t;\r\n}\r\n\r\nexport function __decorate(decorators, target, key, desc) {\r\n var c = arguments.length, r = c < 3 ? target : desc === null ? desc = Object.getOwnPropertyDescriptor(target, key) : desc, d;\r\n if (typeof Reflect === \"object\" && typeof Reflect.decorate === \"function\") r = Reflect.decorate(decorators, target, key, desc);\r\n else for (var i = decorators.length - 1; i >= 0; i--) if (d = decorators[i]) r = (c < 3 ? d(r) : c > 3 ? d(target, key, r) : d(target, key)) || r;\r\n return c > 3 && r && Object.defineProperty(target, key, r), r;\r\n}\r\n\r\nexport function __param(paramIndex, decorator) {\r\n return function (target, key) { decorator(target, key, paramIndex); }\r\n}\r\n\r\nexport function __metadata(metadataKey, metadataValue) {\r\n if (typeof Reflect === \"object\" && typeof Reflect.metadata === \"function\") return Reflect.metadata(metadataKey, metadataValue);\r\n}\r\n\r\nexport function __awaiter(thisArg, _arguments, P, generator) {\r\n function adopt(value) { return value instanceof P ? value : new P(function (resolve) { resolve(value); }); }\r\n return new (P || (P = Promise))(function (resolve, reject) {\r\n function fulfilled(value) { try { step(generator.next(value)); } catch (e) { reject(e); } }\r\n function rejected(value) { try { step(generator[\"throw\"](value)); } catch (e) { reject(e); } }\r\n function step(result) { result.done ? resolve(result.value) : adopt(result.value).then(fulfilled, rejected); }\r\n step((generator = generator.apply(thisArg, _arguments || [])).next());\r\n });\r\n}\r\n\r\nexport function __generator(thisArg, body) {\r\n var _ = { label: 0, sent: function() { if (t[0] & 1) throw t[1]; return t[1]; }, trys: [], ops: [] }, f, y, t, g;\r\n return g = { next: verb(0), \"throw\": verb(1), \"return\": verb(2) }, typeof Symbol === \"function\" && (g[Symbol.iterator] = function() { return this; }), g;\r\n function verb(n) { return function (v) { return step([n, v]); }; }\r\n function step(op) {\r\n if (f) throw new TypeError(\"Generator is already executing.\");\r\n while (_) try {\r\n if (f = 1, y && (t = op[0] & 2 ? y[\"return\"] : op[0] ? y[\"throw\"] || ((t = y[\"return\"]) && t.call(y), 0) : y.next) && !(t = t.call(y, op[1])).done) return t;\r\n if (y = 0, t) op = [op[0] & 2, t.value];\r\n switch (op[0]) {\r\n case 0: case 1: t = op; break;\r\n case 4: _.label++; return { value: op[1], done: false };\r\n case 5: _.label++; y = op[1]; op = [0]; continue;\r\n case 7: op = _.ops.pop(); _.trys.pop(); continue;\r\n default:\r\n if (!(t = _.trys, t = t.length > 0 && t[t.length - 1]) && (op[0] === 6 || op[0] === 2)) { _ = 0; continue; }\r\n if (op[0] === 3 && (!t || (op[1] > t[0] && op[1] < t[3]))) { _.label = op[1]; break; }\r\n if (op[0] === 6 && _.label < t[1]) { _.label = t[1]; t = op; break; }\r\n if (t && _.label < t[2]) { _.label = t[2]; _.ops.push(op); break; }\r\n if (t[2]) _.ops.pop();\r\n _.trys.pop(); continue;\r\n }\r\n op = body.call(thisArg, _);\r\n } catch (e) { op = [6, e]; y = 0; } finally { f = t = 0; }\r\n if (op[0] & 5) throw op[1]; return { value: op[0] ? op[1] : void 0, done: true };\r\n }\r\n}\r\n\r\nexport var __createBinding = Object.create ? (function(o, m, k, k2) {\r\n if (k2 === undefined) k2 = k;\r\n Object.defineProperty(o, k2, { enumerable: true, get: function() { return m[k]; } });\r\n}) : (function(o, m, k, k2) {\r\n if (k2 === undefined) k2 = k;\r\n o[k2] = m[k];\r\n});\r\n\r\nexport function __exportStar(m, o) {\r\n for (var p in m) if (p !== \"default\" && !Object.prototype.hasOwnProperty.call(o, p)) __createBinding(o, m, p);\r\n}\r\n\r\nexport function __values(o) {\r\n var s = typeof Symbol === \"function\" && Symbol.iterator, m = s && o[s], i = 0;\r\n if (m) return m.call(o);\r\n if (o && typeof o.length === \"number\") return {\r\n next: function () {\r\n if (o && i >= o.length) o = void 0;\r\n return { value: o && o[i++], done: !o };\r\n }\r\n };\r\n throw new TypeError(s ? \"Object is not iterable.\" : \"Symbol.iterator is not defined.\");\r\n}\r\n\r\nexport function __read(o, n) {\r\n var m = typeof Symbol === \"function\" && o[Symbol.iterator];\r\n if (!m) return o;\r\n var i = m.call(o), r, ar = [], e;\r\n try {\r\n while ((n === void 0 || n-- > 0) && !(r = i.next()).done) ar.push(r.value);\r\n }\r\n catch (error) { e = { error: error }; }\r\n finally {\r\n try {\r\n if (r && !r.done && (m = i[\"return\"])) m.call(i);\r\n }\r\n finally { if (e) throw e.error; }\r\n }\r\n return ar;\r\n}\r\n\r\n/** @deprecated */\r\nexport function __spread() {\r\n for (var ar = [], i = 0; i < arguments.length; i++)\r\n ar = ar.concat(__read(arguments[i]));\r\n return ar;\r\n}\r\n\r\n/** @deprecated */\r\nexport function __spreadArrays() {\r\n for (var s = 0, i = 0, il = arguments.length; i < il; i++) s += arguments[i].length;\r\n for (var r = Array(s), k = 0, i = 0; i < il; i++)\r\n for (var a = arguments[i], j = 0, jl = a.length; j < jl; j++, k++)\r\n r[k] = a[j];\r\n return r;\r\n}\r\n\r\nexport function __spreadArray(to, from, pack) {\r\n if (pack || arguments.length === 2) for (var i = 0, l = from.length, ar; i < l; i++) {\r\n if (ar || !(i in from)) {\r\n if (!ar) ar = Array.prototype.slice.call(from, 0, i);\r\n ar[i] = from[i];\r\n }\r\n }\r\n return to.concat(ar || Array.prototype.slice.call(from));\r\n}\r\n\r\nexport function __await(v) {\r\n return this instanceof __await ? (this.v = v, this) : new __await(v);\r\n}\r\n\r\nexport function __asyncGenerator(thisArg, _arguments, generator) {\r\n if (!Symbol.asyncIterator) throw new TypeError(\"Symbol.asyncIterator is not defined.\");\r\n var g = generator.apply(thisArg, _arguments || []), i, q = [];\r\n return i = {}, verb(\"next\"), verb(\"throw\"), verb(\"return\"), i[Symbol.asyncIterator] = function () { return this; }, i;\r\n function verb(n) { if (g[n]) i[n] = function (v) { return new Promise(function (a, b) { q.push([n, v, a, b]) > 1 || resume(n, v); }); }; }\r\n function resume(n, v) { try { step(g[n](v)); } catch (e) { settle(q[0][3], e); } }\r\n function step(r) { r.value instanceof __await ? Promise.resolve(r.value.v).then(fulfill, reject) : settle(q[0][2], r); }\r\n function fulfill(value) { resume(\"next\", value); }\r\n function reject(value) { resume(\"throw\", value); }\r\n function settle(f, v) { if (f(v), q.shift(), q.length) resume(q[0][0], q[0][1]); }\r\n}\r\n\r\nexport function __asyncDelegator(o) {\r\n var i, p;\r\n return i = {}, verb(\"next\"), verb(\"throw\", function (e) { throw e; }), verb(\"return\"), i[Symbol.iterator] = function () { return this; }, i;\r\n function verb(n, f) { i[n] = o[n] ? function (v) { return (p = !p) ? { value: __await(o[n](v)), done: n === \"return\" } : f ? f(v) : v; } : f; }\r\n}\r\n\r\nexport function __asyncValues(o) {\r\n if (!Symbol.asyncIterator) throw new TypeError(\"Symbol.asyncIterator is not defined.\");\r\n var m = o[Symbol.asyncIterator], i;\r\n return m ? m.call(o) : (o = typeof __values === \"function\" ? __values(o) : o[Symbol.iterator](), i = {}, verb(\"next\"), verb(\"throw\"), verb(\"return\"), i[Symbol.asyncIterator] = function () { return this; }, i);\r\n function verb(n) { i[n] = o[n] && function (v) { return new Promise(function (resolve, reject) { v = o[n](v), settle(resolve, reject, v.done, v.value); }); }; }\r\n function settle(resolve, reject, d, v) { Promise.resolve(v).then(function(v) { resolve({ value: v, done: d }); }, reject); }\r\n}\r\n\r\nexport function __makeTemplateObject(cooked, raw) {\r\n if (Object.defineProperty) { Object.defineProperty(cooked, \"raw\", { value: raw }); } else { cooked.raw = raw; }\r\n return cooked;\r\n};\r\n\r\nvar __setModuleDefault = Object.create ? (function(o, v) {\r\n Object.defineProperty(o, \"default\", { enumerable: true, value: v });\r\n}) : function(o, v) {\r\n o[\"default\"] = v;\r\n};\r\n\r\nexport function __importStar(mod) {\r\n if (mod && mod.__esModule) return mod;\r\n var result = {};\r\n if (mod != null) for (var k in mod) if (k !== \"default\" && Object.prototype.hasOwnProperty.call(mod, k)) __createBinding(result, mod, k);\r\n __setModuleDefault(result, mod);\r\n return result;\r\n}\r\n\r\nexport function __importDefault(mod) {\r\n return (mod && mod.__esModule) ? mod : { default: mod };\r\n}\r\n\r\nexport function __classPrivateFieldGet(receiver, state, kind, f) {\r\n if (kind === \"a\" && !f) throw new TypeError(\"Private accessor was defined without a getter\");\r\n if (typeof state === \"function\" ? receiver !== state || !f : !state.has(receiver)) throw new TypeError(\"Cannot read private member from an object whose class did not declare it\");\r\n return kind === \"m\" ? f : kind === \"a\" ? f.call(receiver) : f ? f.value : state.get(receiver);\r\n}\r\n\r\nexport function __classPrivateFieldSet(receiver, state, value, kind, f) {\r\n if (kind === \"m\") throw new TypeError(\"Private method is not writable\");\r\n if (kind === \"a\" && !f) throw new TypeError(\"Private accessor was defined without a setter\");\r\n if (typeof state === \"function\" ? receiver !== state || !f : !state.has(receiver)) throw new TypeError(\"Cannot write private member to an object whose class did not declare it\");\r\n return (kind === \"a\" ? f.call(receiver, value) : f ? f.value = value : state.set(receiver, value)), value;\r\n}\r\n", "/**\n * Returns true if the object is a function.\n * @param value The value to check\n */\nexport function isFunction(value: any): value is (...args: any[]) => any {\n return typeof value === 'function';\n}\n", "/**\n * Used to create Error subclasses until the community moves away from ES5.\n *\n * This is because compiling from TypeScript down to ES5 has issues with subclassing Errors\n * as well as other built-in types: https://github.com/Microsoft/TypeScript/issues/12123\n *\n * @param createImpl A factory function to create the actual constructor implementation. The returned\n * function should be a named function that calls `_super` internally.\n */\nexport function createErrorClass(createImpl: (_super: any) => any): T {\n const _super = (instance: any) => {\n Error.call(instance);\n instance.stack = new Error().stack;\n };\n\n const ctorFunc = createImpl(_super);\n ctorFunc.prototype = Object.create(Error.prototype);\n ctorFunc.prototype.constructor = ctorFunc;\n return ctorFunc;\n}\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface UnsubscriptionError extends Error {\n readonly errors: any[];\n}\n\nexport interface UnsubscriptionErrorCtor {\n /**\n * @deprecated Internal implementation detail. Do not construct error instances.\n * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n */\n new (errors: any[]): UnsubscriptionError;\n}\n\n/**\n * An error thrown when one or more errors have occurred during the\n * `unsubscribe` of a {@link Subscription}.\n */\nexport const UnsubscriptionError: UnsubscriptionErrorCtor = createErrorClass(\n (_super) =>\n function UnsubscriptionErrorImpl(this: any, errors: (Error | string)[]) {\n _super(this);\n this.message = errors\n ? `${errors.length} errors occurred during unsubscription:\n${errors.map((err, i) => `${i + 1}) ${err.toString()}`).join('\\n ')}`\n : '';\n this.name = 'UnsubscriptionError';\n this.errors = errors;\n }\n);\n", "/**\n * Removes an item from an array, mutating it.\n * @param arr The array to remove the item from\n * @param item The item to remove\n */\nexport function arrRemove(arr: T[] | undefined | null, item: T) {\n if (arr) {\n const index = arr.indexOf(item);\n 0 <= index && arr.splice(index, 1);\n }\n}\n", "import { isFunction } from './util/isFunction';\nimport { UnsubscriptionError } from './util/UnsubscriptionError';\nimport { SubscriptionLike, TeardownLogic, Unsubscribable } from './types';\nimport { arrRemove } from './util/arrRemove';\n\n/**\n * Represents a disposable resource, such as the execution of an Observable. A\n * Subscription has one important method, `unsubscribe`, that takes no argument\n * and just disposes the resource held by the subscription.\n *\n * Additionally, subscriptions may be grouped together through the `add()`\n * method, which will attach a child Subscription to the current Subscription.\n * When a Subscription is unsubscribed, all its children (and its grandchildren)\n * will be unsubscribed as well.\n *\n * @class Subscription\n */\nexport class Subscription implements SubscriptionLike {\n /** @nocollapse */\n public static EMPTY = (() => {\n const empty = new Subscription();\n empty.closed = true;\n return empty;\n })();\n\n /**\n * A flag to indicate whether this Subscription has already been unsubscribed.\n */\n public closed = false;\n\n private _parentage: Subscription[] | Subscription | null = null;\n\n /**\n * The list of registered finalizers to execute upon unsubscription. Adding and removing from this\n * list occurs in the {@link #add} and {@link #remove} methods.\n */\n private _finalizers: Exclude[] | null = null;\n\n /**\n * @param initialTeardown A function executed first as part of the finalization\n * process that is kicked off when {@link #unsubscribe} is called.\n */\n constructor(private initialTeardown?: () => void) {}\n\n /**\n * Disposes the resources held by the subscription. May, for instance, cancel\n * an ongoing Observable execution or cancel any other type of work that\n * started when the Subscription was created.\n * @return {void}\n */\n unsubscribe(): void {\n let errors: any[] | undefined;\n\n if (!this.closed) {\n this.closed = true;\n\n // Remove this from it's parents.\n const { _parentage } = this;\n if (_parentage) {\n this._parentage = null;\n if (Array.isArray(_parentage)) {\n for (const parent of _parentage) {\n parent.remove(this);\n }\n } else {\n _parentage.remove(this);\n }\n }\n\n const { initialTeardown: initialFinalizer } = this;\n if (isFunction(initialFinalizer)) {\n try {\n initialFinalizer();\n } catch (e) {\n errors = e instanceof UnsubscriptionError ? e.errors : [e];\n }\n }\n\n const { _finalizers } = this;\n if (_finalizers) {\n this._finalizers = null;\n for (const finalizer of _finalizers) {\n try {\n execFinalizer(finalizer);\n } catch (err) {\n errors = errors ?? [];\n if (err instanceof UnsubscriptionError) {\n errors = [...errors, ...err.errors];\n } else {\n errors.push(err);\n }\n }\n }\n }\n\n if (errors) {\n throw new UnsubscriptionError(errors);\n }\n }\n }\n\n /**\n * Adds a finalizer to this subscription, so that finalization will be unsubscribed/called\n * when this subscription is unsubscribed. If this subscription is already {@link #closed},\n * because it has already been unsubscribed, then whatever finalizer is passed to it\n * will automatically be executed (unless the finalizer itself is also a closed subscription).\n *\n * Closed Subscriptions cannot be added as finalizers to any subscription. Adding a closed\n * subscription to a any subscription will result in no operation. (A noop).\n *\n * Adding a subscription to itself, or adding `null` or `undefined` will not perform any\n * operation at all. (A noop).\n *\n * `Subscription` instances that are added to this instance will automatically remove themselves\n * if they are unsubscribed. Functions and {@link Unsubscribable} objects that you wish to remove\n * will need to be removed manually with {@link #remove}\n *\n * @param teardown The finalization logic to add to this subscription.\n */\n add(teardown: TeardownLogic): void {\n // Only add the finalizer if it's not undefined\n // and don't add a subscription to itself.\n if (teardown && teardown !== this) {\n if (this.closed) {\n // If this subscription is already closed,\n // execute whatever finalizer is handed to it automatically.\n execFinalizer(teardown);\n } else {\n if (teardown instanceof Subscription) {\n // We don't add closed subscriptions, and we don't add the same subscription\n // twice. Subscription unsubscribe is idempotent.\n if (teardown.closed || teardown._hasParent(this)) {\n return;\n }\n teardown._addParent(this);\n }\n (this._finalizers = this._finalizers ?? []).push(teardown);\n }\n }\n }\n\n /**\n * Checks to see if a this subscription already has a particular parent.\n * This will signal that this subscription has already been added to the parent in question.\n * @param parent the parent to check for\n */\n private _hasParent(parent: Subscription) {\n const { _parentage } = this;\n return _parentage === parent || (Array.isArray(_parentage) && _parentage.includes(parent));\n }\n\n /**\n * Adds a parent to this subscription so it can be removed from the parent if it\n * unsubscribes on it's own.\n *\n * NOTE: THIS ASSUMES THAT {@link _hasParent} HAS ALREADY BEEN CHECKED.\n * @param parent The parent subscription to add\n */\n private _addParent(parent: Subscription) {\n const { _parentage } = this;\n this._parentage = Array.isArray(_parentage) ? (_parentage.push(parent), _parentage) : _parentage ? [_parentage, parent] : parent;\n }\n\n /**\n * Called on a child when it is removed via {@link #remove}.\n * @param parent The parent to remove\n */\n private _removeParent(parent: Subscription) {\n const { _parentage } = this;\n if (_parentage === parent) {\n this._parentage = null;\n } else if (Array.isArray(_parentage)) {\n arrRemove(_parentage, parent);\n }\n }\n\n /**\n * Removes a finalizer from this subscription that was previously added with the {@link #add} method.\n *\n * Note that `Subscription` instances, when unsubscribed, will automatically remove themselves\n * from every other `Subscription` they have been added to. This means that using the `remove` method\n * is not a common thing and should be used thoughtfully.\n *\n * If you add the same finalizer instance of a function or an unsubscribable object to a `Subscription` instance\n * more than once, you will need to call `remove` the same number of times to remove all instances.\n *\n * All finalizer instances are removed to free up memory upon unsubscription.\n *\n * @param teardown The finalizer to remove from this subscription\n */\n remove(teardown: Exclude): void {\n const { _finalizers } = this;\n _finalizers && arrRemove(_finalizers, teardown);\n\n if (teardown instanceof Subscription) {\n teardown._removeParent(this);\n }\n }\n}\n\nexport const EMPTY_SUBSCRIPTION = Subscription.EMPTY;\n\nexport function isSubscription(value: any): value is Subscription {\n return (\n value instanceof Subscription ||\n (value && 'closed' in value && isFunction(value.remove) && isFunction(value.add) && isFunction(value.unsubscribe))\n );\n}\n\nfunction execFinalizer(finalizer: Unsubscribable | (() => void)) {\n if (isFunction(finalizer)) {\n finalizer();\n } else {\n finalizer.unsubscribe();\n }\n}\n", "import { Subscriber } from './Subscriber';\nimport { ObservableNotification } from './types';\n\n/**\n * The {@link GlobalConfig} object for RxJS. It is used to configure things\n * like how to react on unhandled errors.\n */\nexport const config: GlobalConfig = {\n onUnhandledError: null,\n onStoppedNotification: null,\n Promise: undefined,\n useDeprecatedSynchronousErrorHandling: false,\n useDeprecatedNextContext: false,\n};\n\n/**\n * The global configuration object for RxJS, used to configure things\n * like how to react on unhandled errors. Accessible via {@link config}\n * object.\n */\nexport interface GlobalConfig {\n /**\n * A registration point for unhandled errors from RxJS. These are errors that\n * cannot were not handled by consuming code in the usual subscription path. For\n * example, if you have this configured, and you subscribe to an observable without\n * providing an error handler, errors from that subscription will end up here. This\n * will _always_ be called asynchronously on another job in the runtime. This is because\n * we do not want errors thrown in this user-configured handler to interfere with the\n * behavior of the library.\n */\n onUnhandledError: ((err: any) => void) | null;\n\n /**\n * A registration point for notifications that cannot be sent to subscribers because they\n * have completed, errored or have been explicitly unsubscribed. By default, next, complete\n * and error notifications sent to stopped subscribers are noops. However, sometimes callers\n * might want a different behavior. For example, with sources that attempt to report errors\n * to stopped subscribers, a caller can configure RxJS to throw an unhandled error instead.\n * This will _always_ be called asynchronously on another job in the runtime. This is because\n * we do not want errors thrown in this user-configured handler to interfere with the\n * behavior of the library.\n */\n onStoppedNotification: ((notification: ObservableNotification, subscriber: Subscriber) => void) | null;\n\n /**\n * The promise constructor used by default for {@link Observable#toPromise toPromise} and {@link Observable#forEach forEach}\n * methods.\n *\n * @deprecated As of version 8, RxJS will no longer support this sort of injection of a\n * Promise constructor. If you need a Promise implementation other than native promises,\n * please polyfill/patch Promise as you see appropriate. Will be removed in v8.\n */\n Promise?: PromiseConstructorLike;\n\n /**\n * If true, turns on synchronous error rethrowing, which is a deprecated behavior\n * in v6 and higher. This behavior enables bad patterns like wrapping a subscribe\n * call in a try/catch block. It also enables producer interference, a nasty bug\n * where a multicast can be broken for all observers by a downstream consumer with\n * an unhandled error. DO NOT USE THIS FLAG UNLESS IT'S NEEDED TO BUY TIME\n * FOR MIGRATION REASONS.\n *\n * @deprecated As of version 8, RxJS will no longer support synchronous throwing\n * of unhandled errors. All errors will be thrown on a separate call stack to prevent bad\n * behaviors described above. Will be removed in v8.\n */\n useDeprecatedSynchronousErrorHandling: boolean;\n\n /**\n * If true, enables an as-of-yet undocumented feature from v5: The ability to access\n * `unsubscribe()` via `this` context in `next` functions created in observers passed\n * to `subscribe`.\n *\n * This is being removed because the performance was severely problematic, and it could also cause\n * issues when types other than POJOs are passed to subscribe as subscribers, as they will likely have\n * their `this` context overwritten.\n *\n * @deprecated As of version 8, RxJS will no longer support altering the\n * context of next functions provided as part of an observer to Subscribe. Instead,\n * you will have access to a subscription or a signal or token that will allow you to do things like\n * unsubscribe and test closed status. Will be removed in v8.\n */\n useDeprecatedNextContext: boolean;\n}\n", "import type { TimerHandle } from './timerHandle';\ntype SetTimeoutFunction = (handler: () => void, timeout?: number, ...args: any[]) => TimerHandle;\ntype ClearTimeoutFunction = (handle: TimerHandle) => void;\n\ninterface TimeoutProvider {\n setTimeout: SetTimeoutFunction;\n clearTimeout: ClearTimeoutFunction;\n delegate:\n | {\n setTimeout: SetTimeoutFunction;\n clearTimeout: ClearTimeoutFunction;\n }\n | undefined;\n}\n\nexport const timeoutProvider: TimeoutProvider = {\n // When accessing the delegate, use the variable rather than `this` so that\n // the functions can be called without being bound to the provider.\n setTimeout(handler: () => void, timeout?: number, ...args) {\n const { delegate } = timeoutProvider;\n if (delegate?.setTimeout) {\n return delegate.setTimeout(handler, timeout, ...args);\n }\n return setTimeout(handler, timeout, ...args);\n },\n clearTimeout(handle) {\n const { delegate } = timeoutProvider;\n return (delegate?.clearTimeout || clearTimeout)(handle as any);\n },\n delegate: undefined,\n};\n", "import { config } from '../config';\nimport { timeoutProvider } from '../scheduler/timeoutProvider';\n\n/**\n * Handles an error on another job either with the user-configured {@link onUnhandledError},\n * or by throwing it on that new job so it can be picked up by `window.onerror`, `process.on('error')`, etc.\n *\n * This should be called whenever there is an error that is out-of-band with the subscription\n * or when an error hits a terminal boundary of the subscription and no error handler was provided.\n *\n * @param err the error to report\n */\nexport function reportUnhandledError(err: any) {\n timeoutProvider.setTimeout(() => {\n const { onUnhandledError } = config;\n if (onUnhandledError) {\n // Execute the user-configured error handler.\n onUnhandledError(err);\n } else {\n // Throw so it is picked up by the runtime's uncaught error mechanism.\n throw err;\n }\n });\n}\n", "/* tslint:disable:no-empty */\nexport function noop() { }\n", "import { CompleteNotification, NextNotification, ErrorNotification } from './types';\n\n/**\n * A completion object optimized for memory use and created to be the\n * same \"shape\" as other notifications in v8.\n * @internal\n */\nexport const COMPLETE_NOTIFICATION = (() => createNotification('C', undefined, undefined) as CompleteNotification)();\n\n/**\n * Internal use only. Creates an optimized error notification that is the same \"shape\"\n * as other notifications.\n * @internal\n */\nexport function errorNotification(error: any): ErrorNotification {\n return createNotification('E', undefined, error) as any;\n}\n\n/**\n * Internal use only. Creates an optimized next notification that is the same \"shape\"\n * as other notifications.\n * @internal\n */\nexport function nextNotification(value: T) {\n return createNotification('N', value, undefined) as NextNotification;\n}\n\n/**\n * Ensures that all notifications created internally have the same \"shape\" in v8.\n *\n * TODO: This is only exported to support a crazy legacy test in `groupBy`.\n * @internal\n */\nexport function createNotification(kind: 'N' | 'E' | 'C', value: any, error: any) {\n return {\n kind,\n value,\n error,\n };\n}\n", "import { config } from '../config';\n\nlet context: { errorThrown: boolean; error: any } | null = null;\n\n/**\n * Handles dealing with errors for super-gross mode. Creates a context, in which\n * any synchronously thrown errors will be passed to {@link captureError}. Which\n * will record the error such that it will be rethrown after the call back is complete.\n * TODO: Remove in v8\n * @param cb An immediately executed function.\n */\nexport function errorContext(cb: () => void) {\n if (config.useDeprecatedSynchronousErrorHandling) {\n const isRoot = !context;\n if (isRoot) {\n context = { errorThrown: false, error: null };\n }\n cb();\n if (isRoot) {\n const { errorThrown, error } = context!;\n context = null;\n if (errorThrown) {\n throw error;\n }\n }\n } else {\n // This is the general non-deprecated path for everyone that\n // isn't crazy enough to use super-gross mode (useDeprecatedSynchronousErrorHandling)\n cb();\n }\n}\n\n/**\n * Captures errors only in super-gross mode.\n * @param err the error to capture\n */\nexport function captureError(err: any) {\n if (config.useDeprecatedSynchronousErrorHandling && context) {\n context.errorThrown = true;\n context.error = err;\n }\n}\n", "import { isFunction } from './util/isFunction';\nimport { Observer, ObservableNotification } from './types';\nimport { isSubscription, Subscription } from './Subscription';\nimport { config } from './config';\nimport { reportUnhandledError } from './util/reportUnhandledError';\nimport { noop } from './util/noop';\nimport { nextNotification, errorNotification, COMPLETE_NOTIFICATION } from './NotificationFactories';\nimport { timeoutProvider } from './scheduler/timeoutProvider';\nimport { captureError } from './util/errorContext';\n\n/**\n * Implements the {@link Observer} interface and extends the\n * {@link Subscription} class. While the {@link Observer} is the public API for\n * consuming the values of an {@link Observable}, all Observers get converted to\n * a Subscriber, in order to provide Subscription-like capabilities such as\n * `unsubscribe`. Subscriber is a common type in RxJS, and crucial for\n * implementing operators, but it is rarely used as a public API.\n *\n * @class Subscriber\n */\nexport class Subscriber extends Subscription implements Observer {\n /**\n * A static factory for a Subscriber, given a (potentially partial) definition\n * of an Observer.\n * @param next The `next` callback of an Observer.\n * @param error The `error` callback of an\n * Observer.\n * @param complete The `complete` callback of an\n * Observer.\n * @return A Subscriber wrapping the (partially defined)\n * Observer represented by the given arguments.\n * @nocollapse\n * @deprecated Do not use. Will be removed in v8. There is no replacement for this\n * method, and there is no reason to be creating instances of `Subscriber` directly.\n * If you have a specific use case, please file an issue.\n */\n static create(next?: (x?: T) => void, error?: (e?: any) => void, complete?: () => void): Subscriber {\n return new SafeSubscriber(next, error, complete);\n }\n\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n protected isStopped: boolean = false;\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n protected destination: Subscriber | Observer; // this `any` is the escape hatch to erase extra type param (e.g. R)\n\n /**\n * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n * There is no reason to directly create an instance of Subscriber. This type is exported for typings reasons.\n */\n constructor(destination?: Subscriber | Observer) {\n super();\n if (destination) {\n this.destination = destination;\n // Automatically chain subscriptions together here.\n // if destination is a Subscription, then it is a Subscriber.\n if (isSubscription(destination)) {\n destination.add(this);\n }\n } else {\n this.destination = EMPTY_OBSERVER;\n }\n }\n\n /**\n * The {@link Observer} callback to receive notifications of type `next` from\n * the Observable, with a value. The Observable may call this method 0 or more\n * times.\n * @param {T} [value] The `next` value.\n * @return {void}\n */\n next(value?: T): void {\n if (this.isStopped) {\n handleStoppedNotification(nextNotification(value), this);\n } else {\n this._next(value!);\n }\n }\n\n /**\n * The {@link Observer} callback to receive notifications of type `error` from\n * the Observable, with an attached `Error`. Notifies the Observer that\n * the Observable has experienced an error condition.\n * @param {any} [err] The `error` exception.\n * @return {void}\n */\n error(err?: any): void {\n if (this.isStopped) {\n handleStoppedNotification(errorNotification(err), this);\n } else {\n this.isStopped = true;\n this._error(err);\n }\n }\n\n /**\n * The {@link Observer} callback to receive a valueless notification of type\n * `complete` from the Observable. Notifies the Observer that the Observable\n * has finished sending push-based notifications.\n * @return {void}\n */\n complete(): void {\n if (this.isStopped) {\n handleStoppedNotification(COMPLETE_NOTIFICATION, this);\n } else {\n this.isStopped = true;\n this._complete();\n }\n }\n\n unsubscribe(): void {\n if (!this.closed) {\n this.isStopped = true;\n super.unsubscribe();\n this.destination = null!;\n }\n }\n\n protected _next(value: T): void {\n this.destination.next(value);\n }\n\n protected _error(err: any): void {\n try {\n this.destination.error(err);\n } finally {\n this.unsubscribe();\n }\n }\n\n protected _complete(): void {\n try {\n this.destination.complete();\n } finally {\n this.unsubscribe();\n }\n }\n}\n\n/**\n * This bind is captured here because we want to be able to have\n * compatibility with monoid libraries that tend to use a method named\n * `bind`. In particular, a library called Monio requires this.\n */\nconst _bind = Function.prototype.bind;\n\nfunction bind any>(fn: Fn, thisArg: any): Fn {\n return _bind.call(fn, thisArg);\n}\n\n/**\n * Internal optimization only, DO NOT EXPOSE.\n * @internal\n */\nclass ConsumerObserver implements Observer {\n constructor(private partialObserver: Partial>) {}\n\n next(value: T): void {\n const { partialObserver } = this;\n if (partialObserver.next) {\n try {\n partialObserver.next(value);\n } catch (error) {\n handleUnhandledError(error);\n }\n }\n }\n\n error(err: any): void {\n const { partialObserver } = this;\n if (partialObserver.error) {\n try {\n partialObserver.error(err);\n } catch (error) {\n handleUnhandledError(error);\n }\n } else {\n handleUnhandledError(err);\n }\n }\n\n complete(): void {\n const { partialObserver } = this;\n if (partialObserver.complete) {\n try {\n partialObserver.complete();\n } catch (error) {\n handleUnhandledError(error);\n }\n }\n }\n}\n\nexport class SafeSubscriber extends Subscriber {\n constructor(\n observerOrNext?: Partial> | ((value: T) => void) | null,\n error?: ((e?: any) => void) | null,\n complete?: (() => void) | null\n ) {\n super();\n\n let partialObserver: Partial>;\n if (isFunction(observerOrNext) || !observerOrNext) {\n // The first argument is a function, not an observer. The next\n // two arguments *could* be observers, or they could be empty.\n partialObserver = {\n next: (observerOrNext ?? undefined) as (((value: T) => void) | undefined),\n error: error ?? undefined,\n complete: complete ?? undefined,\n };\n } else {\n // The first argument is a partial observer.\n let context: any;\n if (this && config.useDeprecatedNextContext) {\n // This is a deprecated path that made `this.unsubscribe()` available in\n // next handler functions passed to subscribe. This only exists behind a flag\n // now, as it is *very* slow.\n context = Object.create(observerOrNext);\n context.unsubscribe = () => this.unsubscribe();\n partialObserver = {\n next: observerOrNext.next && bind(observerOrNext.next, context),\n error: observerOrNext.error && bind(observerOrNext.error, context),\n complete: observerOrNext.complete && bind(observerOrNext.complete, context),\n };\n } else {\n // The \"normal\" path. Just use the partial observer directly.\n partialObserver = observerOrNext;\n }\n }\n\n // Wrap the partial observer to ensure it's a full observer, and\n // make sure proper error handling is accounted for.\n this.destination = new ConsumerObserver(partialObserver);\n }\n}\n\nfunction handleUnhandledError(error: any) {\n if (config.useDeprecatedSynchronousErrorHandling) {\n captureError(error);\n } else {\n // Ideal path, we report this as an unhandled error,\n // which is thrown on a new call stack.\n reportUnhandledError(error);\n }\n}\n\n/**\n * An error handler used when no error handler was supplied\n * to the SafeSubscriber -- meaning no error handler was supplied\n * do the `subscribe` call on our observable.\n * @param err The error to handle\n */\nfunction defaultErrorHandler(err: any) {\n throw err;\n}\n\n/**\n * A handler for notifications that cannot be sent to a stopped subscriber.\n * @param notification The notification being sent\n * @param subscriber The stopped subscriber\n */\nfunction handleStoppedNotification(notification: ObservableNotification, subscriber: Subscriber) {\n const { onStoppedNotification } = config;\n onStoppedNotification && timeoutProvider.setTimeout(() => onStoppedNotification(notification, subscriber));\n}\n\n/**\n * The observer used as a stub for subscriptions where the user did not\n * pass any arguments to `subscribe`. Comes with the default error handling\n * behavior.\n */\nexport const EMPTY_OBSERVER: Readonly> & { closed: true } = {\n closed: true,\n next: noop,\n error: defaultErrorHandler,\n complete: noop,\n};\n", "/**\n * Symbol.observable or a string \"@@observable\". Used for interop\n *\n * @deprecated We will no longer be exporting this symbol in upcoming versions of RxJS.\n * Instead polyfill and use Symbol.observable directly *or* use https://www.npmjs.com/package/symbol-observable\n */\nexport const observable: string | symbol = (() => (typeof Symbol === 'function' && Symbol.observable) || '@@observable')();\n", "/**\n * This function takes one parameter and just returns it. Simply put,\n * this is like `(x: T): T => x`.\n *\n * ## Examples\n *\n * This is useful in some cases when using things like `mergeMap`\n *\n * ```ts\n * import { interval, take, map, range, mergeMap, identity } from 'rxjs';\n *\n * const source$ = interval(1000).pipe(take(5));\n *\n * const result$ = source$.pipe(\n * map(i => range(i)),\n * mergeMap(identity) // same as mergeMap(x => x)\n * );\n *\n * result$.subscribe({\n * next: console.log\n * });\n * ```\n *\n * Or when you want to selectively apply an operator\n *\n * ```ts\n * import { interval, take, identity } from 'rxjs';\n *\n * const shouldLimit = () => Math.random() < 0.5;\n *\n * const source$ = interval(1000);\n *\n * const result$ = source$.pipe(shouldLimit() ? take(5) : identity);\n *\n * result$.subscribe({\n * next: console.log\n * });\n * ```\n *\n * @param x Any value that is returned by this function\n * @returns The value passed as the first parameter to this function\n */\nexport function identity(x: T): T {\n return x;\n}\n", "import { identity } from './identity';\nimport { UnaryFunction } from '../types';\n\nexport function pipe(): typeof identity;\nexport function pipe(fn1: UnaryFunction): UnaryFunction;\nexport function pipe(fn1: UnaryFunction, fn2: UnaryFunction): UnaryFunction;\nexport function pipe(fn1: UnaryFunction, fn2: UnaryFunction, fn3: UnaryFunction): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction,\n fn7: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction,\n fn7: UnaryFunction,\n fn8: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction,\n fn7: UnaryFunction,\n fn8: UnaryFunction,\n fn9: UnaryFunction\n): UnaryFunction;\nexport function pipe(\n fn1: UnaryFunction,\n fn2: UnaryFunction,\n fn3: UnaryFunction,\n fn4: UnaryFunction,\n fn5: UnaryFunction,\n fn6: UnaryFunction,\n fn7: UnaryFunction,\n fn8: UnaryFunction,\n fn9: UnaryFunction,\n ...fns: UnaryFunction[]\n): UnaryFunction;\n\n/**\n * pipe() can be called on one or more functions, each of which can take one argument (\"UnaryFunction\")\n * and uses it to return a value.\n * It returns a function that takes one argument, passes it to the first UnaryFunction, and then\n * passes the result to the next one, passes that result to the next one, and so on. \n */\nexport function pipe(...fns: Array>): UnaryFunction {\n return pipeFromArray(fns);\n}\n\n/** @internal */\nexport function pipeFromArray(fns: Array>): UnaryFunction {\n if (fns.length === 0) {\n return identity as UnaryFunction;\n }\n\n if (fns.length === 1) {\n return fns[0];\n }\n\n return function piped(input: T): R {\n return fns.reduce((prev: any, fn: UnaryFunction) => fn(prev), input as any);\n };\n}\n", "import { Operator } from './Operator';\nimport { SafeSubscriber, Subscriber } from './Subscriber';\nimport { isSubscription, Subscription } from './Subscription';\nimport { TeardownLogic, OperatorFunction, Subscribable, Observer } from './types';\nimport { observable as Symbol_observable } from './symbol/observable';\nimport { pipeFromArray } from './util/pipe';\nimport { config } from './config';\nimport { isFunction } from './util/isFunction';\nimport { errorContext } from './util/errorContext';\n\n/**\n * A representation of any set of values over any amount of time. This is the most basic building block\n * of RxJS.\n *\n * @class Observable\n */\nexport class Observable implements Subscribable {\n /**\n * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n */\n source: Observable | undefined;\n\n /**\n * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n */\n operator: Operator | undefined;\n\n /**\n * @constructor\n * @param {Function} subscribe the function that is called when the Observable is\n * initially subscribed to. This function is given a Subscriber, to which new values\n * can be `next`ed, or an `error` method can be called to raise an error, or\n * `complete` can be called to notify of a successful completion.\n */\n constructor(subscribe?: (this: Observable, subscriber: Subscriber) => TeardownLogic) {\n if (subscribe) {\n this._subscribe = subscribe;\n }\n }\n\n // HACK: Since TypeScript inherits static properties too, we have to\n // fight against TypeScript here so Subject can have a different static create signature\n /**\n * Creates a new Observable by calling the Observable constructor\n * @owner Observable\n * @method create\n * @param {Function} subscribe? the subscriber function to be passed to the Observable constructor\n * @return {Observable} a new observable\n * @nocollapse\n * @deprecated Use `new Observable()` instead. Will be removed in v8.\n */\n static create: (...args: any[]) => any = (subscribe?: (subscriber: Subscriber) => TeardownLogic) => {\n return new Observable(subscribe);\n };\n\n /**\n * Creates a new Observable, with this Observable instance as the source, and the passed\n * operator defined as the new observable's operator.\n * @method lift\n * @param operator the operator defining the operation to take on the observable\n * @return a new observable with the Operator applied\n * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n * If you have implemented an operator using `lift`, it is recommended that you create an\n * operator by simply returning `new Observable()` directly. See \"Creating new operators from\n * scratch\" section here: https://rxjs.dev/guide/operators\n */\n lift(operator?: Operator): Observable {\n const observable = new Observable();\n observable.source = this;\n observable.operator = operator;\n return observable;\n }\n\n subscribe(observerOrNext?: Partial> | ((value: T) => void)): Subscription;\n /** @deprecated Instead of passing separate callback arguments, use an observer argument. Signatures taking separate callback arguments will be removed in v8. Details: https://rxjs.dev/deprecations/subscribe-arguments */\n subscribe(next?: ((value: T) => void) | null, error?: ((error: any) => void) | null, complete?: (() => void) | null): Subscription;\n /**\n * Invokes an execution of an Observable and registers Observer handlers for notifications it will emit.\n *\n * Use it when you have all these Observables, but still nothing is happening.\n *\n * `subscribe` is not a regular operator, but a method that calls Observable's internal `subscribe` function. It\n * might be for example a function that you passed to Observable's constructor, but most of the time it is\n * a library implementation, which defines what will be emitted by an Observable, and when it be will emitted. This means\n * that calling `subscribe` is actually the moment when Observable starts its work, not when it is created, as it is often\n * the thought.\n *\n * Apart from starting the execution of an Observable, this method allows you to listen for values\n * that an Observable emits, as well as for when it completes or errors. You can achieve this in two\n * of the following ways.\n *\n * The first way is creating an object that implements {@link Observer} interface. It should have methods\n * defined by that interface, but note that it should be just a regular JavaScript object, which you can create\n * yourself in any way you want (ES6 class, classic function constructor, object literal etc.). In particular, do\n * not attempt to use any RxJS implementation details to create Observers - you don't need them. Remember also\n * that your object does not have to implement all methods. If you find yourself creating a method that doesn't\n * do anything, you can simply omit it. Note however, if the `error` method is not provided and an error happens,\n * it will be thrown asynchronously. Errors thrown asynchronously cannot be caught using `try`/`catch`. Instead,\n * use the {@link onUnhandledError} configuration option or use a runtime handler (like `window.onerror` or\n * `process.on('error)`) to be notified of unhandled errors. Because of this, it's recommended that you provide\n * an `error` method to avoid missing thrown errors.\n *\n * The second way is to give up on Observer object altogether and simply provide callback functions in place of its methods.\n * This means you can provide three functions as arguments to `subscribe`, where the first function is equivalent\n * of a `next` method, the second of an `error` method and the third of a `complete` method. Just as in case of an Observer,\n * if you do not need to listen for something, you can omit a function by passing `undefined` or `null`,\n * since `subscribe` recognizes these functions by where they were placed in function call. When it comes\n * to the `error` function, as with an Observer, if not provided, errors emitted by an Observable will be thrown asynchronously.\n *\n * You can, however, subscribe with no parameters at all. This may be the case where you're not interested in terminal events\n * and you also handled emissions internally by using operators (e.g. using `tap`).\n *\n * Whichever style of calling `subscribe` you use, in both cases it returns a Subscription object.\n * This object allows you to call `unsubscribe` on it, which in turn will stop the work that an Observable does and will clean\n * up all resources that an Observable used. Note that cancelling a subscription will not call `complete` callback\n * provided to `subscribe` function, which is reserved for a regular completion signal that comes from an Observable.\n *\n * Remember that callbacks provided to `subscribe` are not guaranteed to be called asynchronously.\n * It is an Observable itself that decides when these functions will be called. For example {@link of}\n * by default emits all its values synchronously. Always check documentation for how given Observable\n * will behave when subscribed and if its default behavior can be modified with a `scheduler`.\n *\n * #### Examples\n *\n * Subscribe with an {@link guide/observer Observer}\n *\n * ```ts\n * import { of } from 'rxjs';\n *\n * const sumObserver = {\n * sum: 0,\n * next(value) {\n * console.log('Adding: ' + value);\n * this.sum = this.sum + value;\n * },\n * error() {\n * // We actually could just remove this method,\n * // since we do not really care about errors right now.\n * },\n * complete() {\n * console.log('Sum equals: ' + this.sum);\n * }\n * };\n *\n * of(1, 2, 3) // Synchronously emits 1, 2, 3 and then completes.\n * .subscribe(sumObserver);\n *\n * // Logs:\n * // 'Adding: 1'\n * // 'Adding: 2'\n * // 'Adding: 3'\n * // 'Sum equals: 6'\n * ```\n *\n * Subscribe with functions ({@link deprecations/subscribe-arguments deprecated})\n *\n * ```ts\n * import { of } from 'rxjs'\n *\n * let sum = 0;\n *\n * of(1, 2, 3).subscribe(\n * value => {\n * console.log('Adding: ' + value);\n * sum = sum + value;\n * },\n * undefined,\n * () => console.log('Sum equals: ' + sum)\n * );\n *\n * // Logs:\n * // 'Adding: 1'\n * // 'Adding: 2'\n * // 'Adding: 3'\n * // 'Sum equals: 6'\n * ```\n *\n * Cancel a subscription\n *\n * ```ts\n * import { interval } from 'rxjs';\n *\n * const subscription = interval(1000).subscribe({\n * next(num) {\n * console.log(num)\n * },\n * complete() {\n * // Will not be called, even when cancelling subscription.\n * console.log('completed!');\n * }\n * });\n *\n * setTimeout(() => {\n * subscription.unsubscribe();\n * console.log('unsubscribed!');\n * }, 2500);\n *\n * // Logs:\n * // 0 after 1s\n * // 1 after 2s\n * // 'unsubscribed!' after 2.5s\n * ```\n *\n * @param {Observer|Function} observerOrNext (optional) Either an observer with methods to be called,\n * or the first of three possible handlers, which is the handler for each value emitted from the subscribed\n * Observable.\n * @param {Function} error (optional) A handler for a terminal event resulting from an error. If no error handler is provided,\n * the error will be thrown asynchronously as unhandled.\n * @param {Function} complete (optional) A handler for a terminal event resulting from successful completion.\n * @return {Subscription} a subscription reference to the registered handlers\n * @method subscribe\n */\n subscribe(\n observerOrNext?: Partial> | ((value: T) => void) | null,\n error?: ((error: any) => void) | null,\n complete?: (() => void) | null\n ): Subscription {\n const subscriber = isSubscriber(observerOrNext) ? observerOrNext : new SafeSubscriber(observerOrNext, error, complete);\n\n errorContext(() => {\n const { operator, source } = this;\n subscriber.add(\n operator\n ? // We're dealing with a subscription in the\n // operator chain to one of our lifted operators.\n operator.call(subscriber, source)\n : source\n ? // If `source` has a value, but `operator` does not, something that\n // had intimate knowledge of our API, like our `Subject`, must have\n // set it. We're going to just call `_subscribe` directly.\n this._subscribe(subscriber)\n : // In all other cases, we're likely wrapping a user-provided initializer\n // function, so we need to catch errors and handle them appropriately.\n this._trySubscribe(subscriber)\n );\n });\n\n return subscriber;\n }\n\n /** @internal */\n protected _trySubscribe(sink: Subscriber): TeardownLogic {\n try {\n return this._subscribe(sink);\n } catch (err) {\n // We don't need to return anything in this case,\n // because it's just going to try to `add()` to a subscription\n // above.\n sink.error(err);\n }\n }\n\n /**\n * Used as a NON-CANCELLABLE means of subscribing to an observable, for use with\n * APIs that expect promises, like `async/await`. You cannot unsubscribe from this.\n *\n * **WARNING**: Only use this with observables you *know* will complete. If the source\n * observable does not complete, you will end up with a promise that is hung up, and\n * potentially all of the state of an async function hanging out in memory. To avoid\n * this situation, look into adding something like {@link timeout}, {@link take},\n * {@link takeWhile}, or {@link takeUntil} amongst others.\n *\n * #### Example\n *\n * ```ts\n * import { interval, take } from 'rxjs';\n *\n * const source$ = interval(1000).pipe(take(4));\n *\n * async function getTotal() {\n * let total = 0;\n *\n * await source$.forEach(value => {\n * total += value;\n * console.log('observable -> ' + value);\n * });\n *\n * return total;\n * }\n *\n * getTotal().then(\n * total => console.log('Total: ' + total)\n * );\n *\n * // Expected:\n * // 'observable -> 0'\n * // 'observable -> 1'\n * // 'observable -> 2'\n * // 'observable -> 3'\n * // 'Total: 6'\n * ```\n *\n * @param next a handler for each value emitted by the observable\n * @return a promise that either resolves on observable completion or\n * rejects with the handled error\n */\n forEach(next: (value: T) => void): Promise;\n\n /**\n * @param next a handler for each value emitted by the observable\n * @param promiseCtor a constructor function used to instantiate the Promise\n * @return a promise that either resolves on observable completion or\n * rejects with the handled error\n * @deprecated Passing a Promise constructor will no longer be available\n * in upcoming versions of RxJS. This is because it adds weight to the library, for very\n * little benefit. If you need this functionality, it is recommended that you either\n * polyfill Promise, or you create an adapter to convert the returned native promise\n * to whatever promise implementation you wanted. Will be removed in v8.\n */\n forEach(next: (value: T) => void, promiseCtor: PromiseConstructorLike): Promise;\n\n forEach(next: (value: T) => void, promiseCtor?: PromiseConstructorLike): Promise {\n promiseCtor = getPromiseCtor(promiseCtor);\n\n return new promiseCtor((resolve, reject) => {\n const subscriber = new SafeSubscriber({\n next: (value) => {\n try {\n next(value);\n } catch (err) {\n reject(err);\n subscriber.unsubscribe();\n }\n },\n error: reject,\n complete: resolve,\n });\n this.subscribe(subscriber);\n }) as Promise;\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): TeardownLogic {\n return this.source?.subscribe(subscriber);\n }\n\n /**\n * An interop point defined by the es7-observable spec https://github.com/zenparsing/es-observable\n * @method Symbol.observable\n * @return {Observable} this instance of the observable\n */\n [Symbol_observable]() {\n return this;\n }\n\n /* tslint:disable:max-line-length */\n pipe(): Observable;\n pipe(op1: OperatorFunction): Observable;\n pipe(op1: OperatorFunction, op2: OperatorFunction): Observable;\n pipe(op1: OperatorFunction, op2: OperatorFunction, op3: OperatorFunction): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction,\n op7: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction,\n op7: OperatorFunction,\n op8: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction,\n op7: OperatorFunction,\n op8: OperatorFunction,\n op9: OperatorFunction\n ): Observable;\n pipe(\n op1: OperatorFunction,\n op2: OperatorFunction,\n op3: OperatorFunction,\n op4: OperatorFunction,\n op5: OperatorFunction,\n op6: OperatorFunction,\n op7: OperatorFunction,\n op8: OperatorFunction,\n op9: OperatorFunction,\n ...operations: OperatorFunction[]\n ): Observable;\n /* tslint:enable:max-line-length */\n\n /**\n * Used to stitch together functional operators into a chain.\n * @method pipe\n * @return {Observable} the Observable result of all of the operators having\n * been called in the order they were passed in.\n *\n * ## Example\n *\n * ```ts\n * import { interval, filter, map, scan } from 'rxjs';\n *\n * interval(1000)\n * .pipe(\n * filter(x => x % 2 === 0),\n * map(x => x + x),\n * scan((acc, x) => acc + x)\n * )\n * .subscribe(x => console.log(x));\n * ```\n */\n pipe(...operations: OperatorFunction[]): Observable {\n return pipeFromArray(operations)(this);\n }\n\n /* tslint:disable:max-line-length */\n /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n toPromise(): Promise;\n /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n toPromise(PromiseCtor: typeof Promise): Promise;\n /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n toPromise(PromiseCtor: PromiseConstructorLike): Promise;\n /* tslint:enable:max-line-length */\n\n /**\n * Subscribe to this Observable and get a Promise resolving on\n * `complete` with the last emission (if any).\n *\n * **WARNING**: Only use this with observables you *know* will complete. If the source\n * observable does not complete, you will end up with a promise that is hung up, and\n * potentially all of the state of an async function hanging out in memory. To avoid\n * this situation, look into adding something like {@link timeout}, {@link take},\n * {@link takeWhile}, or {@link takeUntil} amongst others.\n *\n * @method toPromise\n * @param [promiseCtor] a constructor function used to instantiate\n * the Promise\n * @return A Promise that resolves with the last value emit, or\n * rejects on an error. If there were no emissions, Promise\n * resolves with undefined.\n * @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise\n */\n toPromise(promiseCtor?: PromiseConstructorLike): Promise {\n promiseCtor = getPromiseCtor(promiseCtor);\n\n return new promiseCtor((resolve, reject) => {\n let value: T | undefined;\n this.subscribe(\n (x: T) => (value = x),\n (err: any) => reject(err),\n () => resolve(value)\n );\n }) as Promise;\n }\n}\n\n/**\n * Decides between a passed promise constructor from consuming code,\n * A default configured promise constructor, and the native promise\n * constructor and returns it. If nothing can be found, it will throw\n * an error.\n * @param promiseCtor The optional promise constructor to passed by consuming code\n */\nfunction getPromiseCtor(promiseCtor: PromiseConstructorLike | undefined) {\n return promiseCtor ?? config.Promise ?? Promise;\n}\n\nfunction isObserver(value: any): value is Observer {\n return value && isFunction(value.next) && isFunction(value.error) && isFunction(value.complete);\n}\n\nfunction isSubscriber(value: any): value is Subscriber {\n return (value && value instanceof Subscriber) || (isObserver(value) && isSubscription(value));\n}\n", "import { Observable } from '../Observable';\nimport { Subscriber } from '../Subscriber';\nimport { OperatorFunction } from '../types';\nimport { isFunction } from './isFunction';\n\n/**\n * Used to determine if an object is an Observable with a lift function.\n */\nexport function hasLift(source: any): source is { lift: InstanceType['lift'] } {\n return isFunction(source?.lift);\n}\n\n/**\n * Creates an `OperatorFunction`. Used to define operators throughout the library in a concise way.\n * @param init The logic to connect the liftedSource to the subscriber at the moment of subscription.\n */\nexport function operate(\n init: (liftedSource: Observable, subscriber: Subscriber) => (() => void) | void\n): OperatorFunction {\n return (source: Observable) => {\n if (hasLift(source)) {\n return source.lift(function (this: Subscriber, liftedSource: Observable) {\n try {\n return init(liftedSource, this);\n } catch (err) {\n this.error(err);\n }\n });\n }\n throw new TypeError('Unable to lift unknown Observable type');\n };\n}\n", "import { Subscriber } from '../Subscriber';\n\n/**\n * Creates an instance of an `OperatorSubscriber`.\n * @param destination The downstream subscriber.\n * @param onNext Handles next values, only called if this subscriber is not stopped or closed. Any\n * error that occurs in this function is caught and sent to the `error` method of this subscriber.\n * @param onError Handles errors from the subscription, any errors that occur in this handler are caught\n * and send to the `destination` error handler.\n * @param onComplete Handles completion notification from the subscription. Any errors that occur in\n * this handler are sent to the `destination` error handler.\n * @param onFinalize Additional teardown logic here. This will only be called on teardown if the\n * subscriber itself is not already closed. This is called after all other teardown logic is executed.\n */\nexport function createOperatorSubscriber(\n destination: Subscriber,\n onNext?: (value: T) => void,\n onComplete?: () => void,\n onError?: (err: any) => void,\n onFinalize?: () => void\n): Subscriber {\n return new OperatorSubscriber(destination, onNext, onComplete, onError, onFinalize);\n}\n\n/**\n * A generic helper for allowing operators to be created with a Subscriber and\n * use closures to capture necessary state from the operator function itself.\n */\nexport class OperatorSubscriber extends Subscriber {\n /**\n * Creates an instance of an `OperatorSubscriber`.\n * @param destination The downstream subscriber.\n * @param onNext Handles next values, only called if this subscriber is not stopped or closed. Any\n * error that occurs in this function is caught and sent to the `error` method of this subscriber.\n * @param onError Handles errors from the subscription, any errors that occur in this handler are caught\n * and send to the `destination` error handler.\n * @param onComplete Handles completion notification from the subscription. Any errors that occur in\n * this handler are sent to the `destination` error handler.\n * @param onFinalize Additional finalization logic here. This will only be called on finalization if the\n * subscriber itself is not already closed. This is called after all other finalization logic is executed.\n * @param shouldUnsubscribe An optional check to see if an unsubscribe call should truly unsubscribe.\n * NOTE: This currently **ONLY** exists to support the strange behavior of {@link groupBy}, where unsubscription\n * to the resulting observable does not actually disconnect from the source if there are active subscriptions\n * to any grouped observable. (DO NOT EXPOSE OR USE EXTERNALLY!!!)\n */\n constructor(\n destination: Subscriber,\n onNext?: (value: T) => void,\n onComplete?: () => void,\n onError?: (err: any) => void,\n private onFinalize?: () => void,\n private shouldUnsubscribe?: () => boolean\n ) {\n // It's important - for performance reasons - that all of this class's\n // members are initialized and that they are always initialized in the same\n // order. This will ensure that all OperatorSubscriber instances have the\n // same hidden class in V8. This, in turn, will help keep the number of\n // hidden classes involved in property accesses within the base class as\n // low as possible. If the number of hidden classes involved exceeds four,\n // the property accesses will become megamorphic and performance penalties\n // will be incurred - i.e. inline caches won't be used.\n //\n // The reasons for ensuring all instances have the same hidden class are\n // further discussed in this blog post from Benedikt Meurer:\n // https://benediktmeurer.de/2018/03/23/impact-of-polymorphism-on-component-based-frameworks-like-react/\n super(destination);\n this._next = onNext\n ? function (this: OperatorSubscriber, value: T) {\n try {\n onNext(value);\n } catch (err) {\n destination.error(err);\n }\n }\n : super._next;\n this._error = onError\n ? function (this: OperatorSubscriber, err: any) {\n try {\n onError(err);\n } catch (err) {\n // Send any errors that occur down stream.\n destination.error(err);\n } finally {\n // Ensure finalization.\n this.unsubscribe();\n }\n }\n : super._error;\n this._complete = onComplete\n ? function (this: OperatorSubscriber) {\n try {\n onComplete();\n } catch (err) {\n // Send any errors that occur down stream.\n destination.error(err);\n } finally {\n // Ensure finalization.\n this.unsubscribe();\n }\n }\n : super._complete;\n }\n\n unsubscribe() {\n if (!this.shouldUnsubscribe || this.shouldUnsubscribe()) {\n const { closed } = this;\n super.unsubscribe();\n // Execute additional teardown if we have any and we didn't already do so.\n !closed && this.onFinalize?.();\n }\n }\n}\n", "import { Subscription } from '../Subscription';\n\ninterface AnimationFrameProvider {\n schedule(callback: FrameRequestCallback): Subscription;\n requestAnimationFrame: typeof requestAnimationFrame;\n cancelAnimationFrame: typeof cancelAnimationFrame;\n delegate:\n | {\n requestAnimationFrame: typeof requestAnimationFrame;\n cancelAnimationFrame: typeof cancelAnimationFrame;\n }\n | undefined;\n}\n\nexport const animationFrameProvider: AnimationFrameProvider = {\n // When accessing the delegate, use the variable rather than `this` so that\n // the functions can be called without being bound to the provider.\n schedule(callback) {\n let request = requestAnimationFrame;\n let cancel: typeof cancelAnimationFrame | undefined = cancelAnimationFrame;\n const { delegate } = animationFrameProvider;\n if (delegate) {\n request = delegate.requestAnimationFrame;\n cancel = delegate.cancelAnimationFrame;\n }\n const handle = request((timestamp) => {\n // Clear the cancel function. The request has been fulfilled, so\n // attempting to cancel the request upon unsubscription would be\n // pointless.\n cancel = undefined;\n callback(timestamp);\n });\n return new Subscription(() => cancel?.(handle));\n },\n requestAnimationFrame(...args) {\n const { delegate } = animationFrameProvider;\n return (delegate?.requestAnimationFrame || requestAnimationFrame)(...args);\n },\n cancelAnimationFrame(...args) {\n const { delegate } = animationFrameProvider;\n return (delegate?.cancelAnimationFrame || cancelAnimationFrame)(...args);\n },\n delegate: undefined,\n};\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface ObjectUnsubscribedError extends Error {}\n\nexport interface ObjectUnsubscribedErrorCtor {\n /**\n * @deprecated Internal implementation detail. Do not construct error instances.\n * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n */\n new (): ObjectUnsubscribedError;\n}\n\n/**\n * An error thrown when an action is invalid because the object has been\n * unsubscribed.\n *\n * @see {@link Subject}\n * @see {@link BehaviorSubject}\n *\n * @class ObjectUnsubscribedError\n */\nexport const ObjectUnsubscribedError: ObjectUnsubscribedErrorCtor = createErrorClass(\n (_super) =>\n function ObjectUnsubscribedErrorImpl(this: any) {\n _super(this);\n this.name = 'ObjectUnsubscribedError';\n this.message = 'object unsubscribed';\n }\n);\n", "import { Operator } from './Operator';\nimport { Observable } from './Observable';\nimport { Subscriber } from './Subscriber';\nimport { Subscription, EMPTY_SUBSCRIPTION } from './Subscription';\nimport { Observer, SubscriptionLike, TeardownLogic } from './types';\nimport { ObjectUnsubscribedError } from './util/ObjectUnsubscribedError';\nimport { arrRemove } from './util/arrRemove';\nimport { errorContext } from './util/errorContext';\n\n/**\n * A Subject is a special type of Observable that allows values to be\n * multicasted to many Observers. Subjects are like EventEmitters.\n *\n * Every Subject is an Observable and an Observer. You can subscribe to a\n * Subject, and you can call next to feed values as well as error and complete.\n */\nexport class Subject extends Observable implements SubscriptionLike {\n closed = false;\n\n private currentObservers: Observer[] | null = null;\n\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n observers: Observer[] = [];\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n isStopped = false;\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n hasError = false;\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n thrownError: any = null;\n\n /**\n * Creates a \"subject\" by basically gluing an observer to an observable.\n *\n * @nocollapse\n * @deprecated Recommended you do not use. Will be removed at some point in the future. Plans for replacement still under discussion.\n */\n static create: (...args: any[]) => any = (destination: Observer, source: Observable): AnonymousSubject => {\n return new AnonymousSubject(destination, source);\n };\n\n constructor() {\n // NOTE: This must be here to obscure Observable's constructor.\n super();\n }\n\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n lift(operator: Operator): Observable {\n const subject = new AnonymousSubject(this, this);\n subject.operator = operator as any;\n return subject as any;\n }\n\n /** @internal */\n protected _throwIfClosed() {\n if (this.closed) {\n throw new ObjectUnsubscribedError();\n }\n }\n\n next(value: T) {\n errorContext(() => {\n this._throwIfClosed();\n if (!this.isStopped) {\n if (!this.currentObservers) {\n this.currentObservers = Array.from(this.observers);\n }\n for (const observer of this.currentObservers) {\n observer.next(value);\n }\n }\n });\n }\n\n error(err: any) {\n errorContext(() => {\n this._throwIfClosed();\n if (!this.isStopped) {\n this.hasError = this.isStopped = true;\n this.thrownError = err;\n const { observers } = this;\n while (observers.length) {\n observers.shift()!.error(err);\n }\n }\n });\n }\n\n complete() {\n errorContext(() => {\n this._throwIfClosed();\n if (!this.isStopped) {\n this.isStopped = true;\n const { observers } = this;\n while (observers.length) {\n observers.shift()!.complete();\n }\n }\n });\n }\n\n unsubscribe() {\n this.isStopped = this.closed = true;\n this.observers = this.currentObservers = null!;\n }\n\n get observed() {\n return this.observers?.length > 0;\n }\n\n /** @internal */\n protected _trySubscribe(subscriber: Subscriber): TeardownLogic {\n this._throwIfClosed();\n return super._trySubscribe(subscriber);\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): Subscription {\n this._throwIfClosed();\n this._checkFinalizedStatuses(subscriber);\n return this._innerSubscribe(subscriber);\n }\n\n /** @internal */\n protected _innerSubscribe(subscriber: Subscriber) {\n const { hasError, isStopped, observers } = this;\n if (hasError || isStopped) {\n return EMPTY_SUBSCRIPTION;\n }\n this.currentObservers = null;\n observers.push(subscriber);\n return new Subscription(() => {\n this.currentObservers = null;\n arrRemove(observers, subscriber);\n });\n }\n\n /** @internal */\n protected _checkFinalizedStatuses(subscriber: Subscriber) {\n const { hasError, thrownError, isStopped } = this;\n if (hasError) {\n subscriber.error(thrownError);\n } else if (isStopped) {\n subscriber.complete();\n }\n }\n\n /**\n * Creates a new Observable with this Subject as the source. You can do this\n * to create custom Observer-side logic of the Subject and conceal it from\n * code that uses the Observable.\n * @return {Observable} Observable that the Subject casts to\n */\n asObservable(): Observable {\n const observable: any = new Observable();\n observable.source = this;\n return observable;\n }\n}\n\n/**\n * @class AnonymousSubject\n */\nexport class AnonymousSubject extends Subject {\n constructor(\n /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n public destination?: Observer,\n source?: Observable\n ) {\n super();\n this.source = source;\n }\n\n next(value: T) {\n this.destination?.next?.(value);\n }\n\n error(err: any) {\n this.destination?.error?.(err);\n }\n\n complete() {\n this.destination?.complete?.();\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): Subscription {\n return this.source?.subscribe(subscriber) ?? EMPTY_SUBSCRIPTION;\n }\n}\n", "import { Subject } from './Subject';\nimport { Subscriber } from './Subscriber';\nimport { Subscription } from './Subscription';\n\n/**\n * A variant of Subject that requires an initial value and emits its current\n * value whenever it is subscribed to.\n *\n * @class BehaviorSubject\n */\nexport class BehaviorSubject extends Subject {\n constructor(private _value: T) {\n super();\n }\n\n get value(): T {\n return this.getValue();\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): Subscription {\n const subscription = super._subscribe(subscriber);\n !subscription.closed && subscriber.next(this._value);\n return subscription;\n }\n\n getValue(): T {\n const { hasError, thrownError, _value } = this;\n if (hasError) {\n throw thrownError;\n }\n this._throwIfClosed();\n return _value;\n }\n\n next(value: T): void {\n super.next((this._value = value));\n }\n}\n", "import { TimestampProvider } from '../types';\n\ninterface DateTimestampProvider extends TimestampProvider {\n delegate: TimestampProvider | undefined;\n}\n\nexport const dateTimestampProvider: DateTimestampProvider = {\n now() {\n // Use the variable rather than `this` so that the function can be called\n // without being bound to the provider.\n return (dateTimestampProvider.delegate || Date).now();\n },\n delegate: undefined,\n};\n", "import { Subject } from './Subject';\nimport { TimestampProvider } from './types';\nimport { Subscriber } from './Subscriber';\nimport { Subscription } from './Subscription';\nimport { dateTimestampProvider } from './scheduler/dateTimestampProvider';\n\n/**\n * A variant of {@link Subject} that \"replays\" old values to new subscribers by emitting them when they first subscribe.\n *\n * `ReplaySubject` has an internal buffer that will store a specified number of values that it has observed. Like `Subject`,\n * `ReplaySubject` \"observes\" values by having them passed to its `next` method. When it observes a value, it will store that\n * value for a time determined by the configuration of the `ReplaySubject`, as passed to its constructor.\n *\n * When a new subscriber subscribes to the `ReplaySubject` instance, it will synchronously emit all values in its buffer in\n * a First-In-First-Out (FIFO) manner. The `ReplaySubject` will also complete, if it has observed completion; and it will\n * error if it has observed an error.\n *\n * There are two main configuration items to be concerned with:\n *\n * 1. `bufferSize` - This will determine how many items are stored in the buffer, defaults to infinite.\n * 2. `windowTime` - The amount of time to hold a value in the buffer before removing it from the buffer.\n *\n * Both configurations may exist simultaneously. So if you would like to buffer a maximum of 3 values, as long as the values\n * are less than 2 seconds old, you could do so with a `new ReplaySubject(3, 2000)`.\n *\n * ### Differences with BehaviorSubject\n *\n * `BehaviorSubject` is similar to `new ReplaySubject(1)`, with a couple of exceptions:\n *\n * 1. `BehaviorSubject` comes \"primed\" with a single value upon construction.\n * 2. `ReplaySubject` will replay values, even after observing an error, where `BehaviorSubject` will not.\n *\n * @see {@link Subject}\n * @see {@link BehaviorSubject}\n * @see {@link shareReplay}\n */\nexport class ReplaySubject extends Subject {\n private _buffer: (T | number)[] = [];\n private _infiniteTimeWindow = true;\n\n /**\n * @param bufferSize The size of the buffer to replay on subscription\n * @param windowTime The amount of time the buffered items will stay buffered\n * @param timestampProvider An object with a `now()` method that provides the current timestamp. This is used to\n * calculate the amount of time something has been buffered.\n */\n constructor(\n private _bufferSize = Infinity,\n private _windowTime = Infinity,\n private _timestampProvider: TimestampProvider = dateTimestampProvider\n ) {\n super();\n this._infiniteTimeWindow = _windowTime === Infinity;\n this._bufferSize = Math.max(1, _bufferSize);\n this._windowTime = Math.max(1, _windowTime);\n }\n\n next(value: T): void {\n const { isStopped, _buffer, _infiniteTimeWindow, _timestampProvider, _windowTime } = this;\n if (!isStopped) {\n _buffer.push(value);\n !_infiniteTimeWindow && _buffer.push(_timestampProvider.now() + _windowTime);\n }\n this._trimBuffer();\n super.next(value);\n }\n\n /** @internal */\n protected _subscribe(subscriber: Subscriber): Subscription {\n this._throwIfClosed();\n this._trimBuffer();\n\n const subscription = this._innerSubscribe(subscriber);\n\n const { _infiniteTimeWindow, _buffer } = this;\n // We use a copy here, so reentrant code does not mutate our array while we're\n // emitting it to a new subscriber.\n const copy = _buffer.slice();\n for (let i = 0; i < copy.length && !subscriber.closed; i += _infiniteTimeWindow ? 1 : 2) {\n subscriber.next(copy[i] as T);\n }\n\n this._checkFinalizedStatuses(subscriber);\n\n return subscription;\n }\n\n private _trimBuffer() {\n const { _bufferSize, _timestampProvider, _buffer, _infiniteTimeWindow } = this;\n // If we don't have an infinite buffer size, and we're over the length,\n // use splice to truncate the old buffer values off. Note that we have to\n // double the size for instances where we're not using an infinite time window\n // because we're storing the values and the timestamps in the same array.\n const adjustedBufferSize = (_infiniteTimeWindow ? 1 : 2) * _bufferSize;\n _bufferSize < Infinity && adjustedBufferSize < _buffer.length && _buffer.splice(0, _buffer.length - adjustedBufferSize);\n\n // Now, if we're not in an infinite time window, remove all values where the time is\n // older than what is allowed.\n if (!_infiniteTimeWindow) {\n const now = _timestampProvider.now();\n let last = 0;\n // Search the array for the first timestamp that isn't expired and\n // truncate the buffer up to that point.\n for (let i = 1; i < _buffer.length && (_buffer[i] as number) <= now; i += 2) {\n last = i;\n }\n last && _buffer.splice(0, last + 1);\n }\n }\n}\n", "import { Scheduler } from '../Scheduler';\nimport { Subscription } from '../Subscription';\nimport { SchedulerAction } from '../types';\n\n/**\n * A unit of work to be executed in a `scheduler`. An action is typically\n * created from within a {@link SchedulerLike} and an RxJS user does not need to concern\n * themselves about creating and manipulating an Action.\n *\n * ```ts\n * class Action extends Subscription {\n * new (scheduler: Scheduler, work: (state?: T) => void);\n * schedule(state?: T, delay: number = 0): Subscription;\n * }\n * ```\n *\n * @class Action\n */\nexport class Action extends Subscription {\n constructor(scheduler: Scheduler, work: (this: SchedulerAction, state?: T) => void) {\n super();\n }\n /**\n * Schedules this action on its parent {@link SchedulerLike} for execution. May be passed\n * some context object, `state`. May happen at some point in the future,\n * according to the `delay` parameter, if specified.\n * @param {T} [state] Some contextual data that the `work` function uses when\n * called by the Scheduler.\n * @param {number} [delay] Time to wait before executing the work, where the\n * time unit is implicit and defined by the Scheduler.\n * @return {void}\n */\n public schedule(state?: T, delay: number = 0): Subscription {\n return this;\n }\n}\n", "import type { TimerHandle } from './timerHandle';\ntype SetIntervalFunction = (handler: () => void, timeout?: number, ...args: any[]) => TimerHandle;\ntype ClearIntervalFunction = (handle: TimerHandle) => void;\n\ninterface IntervalProvider {\n setInterval: SetIntervalFunction;\n clearInterval: ClearIntervalFunction;\n delegate:\n | {\n setInterval: SetIntervalFunction;\n clearInterval: ClearIntervalFunction;\n }\n | undefined;\n}\n\nexport const intervalProvider: IntervalProvider = {\n // When accessing the delegate, use the variable rather than `this` so that\n // the functions can be called without being bound to the provider.\n setInterval(handler: () => void, timeout?: number, ...args) {\n const { delegate } = intervalProvider;\n if (delegate?.setInterval) {\n return delegate.setInterval(handler, timeout, ...args);\n }\n return setInterval(handler, timeout, ...args);\n },\n clearInterval(handle) {\n const { delegate } = intervalProvider;\n return (delegate?.clearInterval || clearInterval)(handle as any);\n },\n delegate: undefined,\n};\n", "import { Action } from './Action';\nimport { SchedulerAction } from '../types';\nimport { Subscription } from '../Subscription';\nimport { AsyncScheduler } from './AsyncScheduler';\nimport { intervalProvider } from './intervalProvider';\nimport { arrRemove } from '../util/arrRemove';\nimport { TimerHandle } from './timerHandle';\n\nexport class AsyncAction extends Action {\n public id: TimerHandle | undefined;\n public state?: T;\n // @ts-ignore: Property has no initializer and is not definitely assigned\n public delay: number;\n protected pending: boolean = false;\n\n constructor(protected scheduler: AsyncScheduler, protected work: (this: SchedulerAction, state?: T) => void) {\n super(scheduler, work);\n }\n\n public schedule(state?: T, delay: number = 0): Subscription {\n if (this.closed) {\n return this;\n }\n\n // Always replace the current state with the new state.\n this.state = state;\n\n const id = this.id;\n const scheduler = this.scheduler;\n\n //\n // Important implementation note:\n //\n // Actions only execute once by default, unless rescheduled from within the\n // scheduled callback. This allows us to implement single and repeat\n // actions via the same code path, without adding API surface area, as well\n // as mimic traditional recursion but across asynchronous boundaries.\n //\n // However, JS runtimes and timers distinguish between intervals achieved by\n // serial `setTimeout` calls vs. a single `setInterval` call. An interval of\n // serial `setTimeout` calls can be individually delayed, which delays\n // scheduling the next `setTimeout`, and so on. `setInterval` attempts to\n // guarantee the interval callback will be invoked more precisely to the\n // interval period, regardless of load.\n //\n // Therefore, we use `setInterval` to schedule single and repeat actions.\n // If the action reschedules itself with the same delay, the interval is not\n // canceled. If the action doesn't reschedule, or reschedules with a\n // different delay, the interval will be canceled after scheduled callback\n // execution.\n //\n if (id != null) {\n this.id = this.recycleAsyncId(scheduler, id, delay);\n }\n\n // Set the pending flag indicating that this action has been scheduled, or\n // has recursively rescheduled itself.\n this.pending = true;\n\n this.delay = delay;\n // If this action has already an async Id, don't request a new one.\n this.id = this.id ?? this.requestAsyncId(scheduler, this.id, delay);\n\n return this;\n }\n\n protected requestAsyncId(scheduler: AsyncScheduler, _id?: TimerHandle, delay: number = 0): TimerHandle {\n return intervalProvider.setInterval(scheduler.flush.bind(scheduler, this), delay);\n }\n\n protected recycleAsyncId(_scheduler: AsyncScheduler, id?: TimerHandle, delay: number | null = 0): TimerHandle | undefined {\n // If this action is rescheduled with the same delay time, don't clear the interval id.\n if (delay != null && this.delay === delay && this.pending === false) {\n return id;\n }\n // Otherwise, if the action's delay time is different from the current delay,\n // or the action has been rescheduled before it's executed, clear the interval id\n if (id != null) {\n intervalProvider.clearInterval(id);\n }\n\n return undefined;\n }\n\n /**\n * Immediately executes this action and the `work` it contains.\n * @return {any}\n */\n public execute(state: T, delay: number): any {\n if (this.closed) {\n return new Error('executing a cancelled action');\n }\n\n this.pending = false;\n const error = this._execute(state, delay);\n if (error) {\n return error;\n } else if (this.pending === false && this.id != null) {\n // Dequeue if the action didn't reschedule itself. Don't call\n // unsubscribe(), because the action could reschedule later.\n // For example:\n // ```\n // scheduler.schedule(function doWork(counter) {\n // /* ... I'm a busy worker bee ... */\n // var originalAction = this;\n // /* wait 100ms before rescheduling the action */\n // setTimeout(function () {\n // originalAction.schedule(counter + 1);\n // }, 100);\n // }, 1000);\n // ```\n this.id = this.recycleAsyncId(this.scheduler, this.id, null);\n }\n }\n\n protected _execute(state: T, _delay: number): any {\n let errored: boolean = false;\n let errorValue: any;\n try {\n this.work(state);\n } catch (e) {\n errored = true;\n // HACK: Since code elsewhere is relying on the \"truthiness\" of the\n // return here, we can't have it return \"\" or 0 or false.\n // TODO: Clean this up when we refactor schedulers mid-version-8 or so.\n errorValue = e ? e : new Error('Scheduled action threw falsy error');\n }\n if (errored) {\n this.unsubscribe();\n return errorValue;\n }\n }\n\n unsubscribe() {\n if (!this.closed) {\n const { id, scheduler } = this;\n const { actions } = scheduler;\n\n this.work = this.state = this.scheduler = null!;\n this.pending = false;\n\n arrRemove(actions, this);\n if (id != null) {\n this.id = this.recycleAsyncId(scheduler, id, null);\n }\n\n this.delay = null!;\n super.unsubscribe();\n }\n }\n}\n", "import { Action } from './scheduler/Action';\nimport { Subscription } from './Subscription';\nimport { SchedulerLike, SchedulerAction } from './types';\nimport { dateTimestampProvider } from './scheduler/dateTimestampProvider';\n\n/**\n * An execution context and a data structure to order tasks and schedule their\n * execution. Provides a notion of (potentially virtual) time, through the\n * `now()` getter method.\n *\n * Each unit of work in a Scheduler is called an `Action`.\n *\n * ```ts\n * class Scheduler {\n * now(): number;\n * schedule(work, delay?, state?): Subscription;\n * }\n * ```\n *\n * @class Scheduler\n * @deprecated Scheduler is an internal implementation detail of RxJS, and\n * should not be used directly. Rather, create your own class and implement\n * {@link SchedulerLike}. Will be made internal in v8.\n */\nexport class Scheduler implements SchedulerLike {\n public static now: () => number = dateTimestampProvider.now;\n\n constructor(private schedulerActionCtor: typeof Action, now: () => number = Scheduler.now) {\n this.now = now;\n }\n\n /**\n * A getter method that returns a number representing the current time\n * (at the time this function was called) according to the scheduler's own\n * internal clock.\n * @return {number} A number that represents the current time. May or may not\n * have a relation to wall-clock time. May or may not refer to a time unit\n * (e.g. milliseconds).\n */\n public now: () => number;\n\n /**\n * Schedules a function, `work`, for execution. May happen at some point in\n * the future, according to the `delay` parameter, if specified. May be passed\n * some context object, `state`, which will be passed to the `work` function.\n *\n * The given arguments will be processed an stored as an Action object in a\n * queue of actions.\n *\n * @param {function(state: ?T): ?Subscription} work A function representing a\n * task, or some unit of work to be executed by the Scheduler.\n * @param {number} [delay] Time to wait before executing the work, where the\n * time unit is implicit and defined by the Scheduler itself.\n * @param {T} [state] Some contextual data that the `work` function uses when\n * called by the Scheduler.\n * @return {Subscription} A subscription in order to be able to unsubscribe\n * the scheduled work.\n */\n public schedule(work: (this: SchedulerAction, state?: T) => void, delay: number = 0, state?: T): Subscription {\n return new this.schedulerActionCtor(this, work).schedule(state, delay);\n }\n}\n", "import { Scheduler } from '../Scheduler';\nimport { Action } from './Action';\nimport { AsyncAction } from './AsyncAction';\nimport { TimerHandle } from './timerHandle';\n\nexport class AsyncScheduler extends Scheduler {\n public actions: Array> = [];\n /**\n * A flag to indicate whether the Scheduler is currently executing a batch of\n * queued actions.\n * @type {boolean}\n * @internal\n */\n public _active: boolean = false;\n /**\n * An internal ID used to track the latest asynchronous task such as those\n * coming from `setTimeout`, `setInterval`, `requestAnimationFrame`, and\n * others.\n * @type {any}\n * @internal\n */\n public _scheduled: TimerHandle | undefined;\n\n constructor(SchedulerAction: typeof Action, now: () => number = Scheduler.now) {\n super(SchedulerAction, now);\n }\n\n public flush(action: AsyncAction): void {\n const { actions } = this;\n\n if (this._active) {\n actions.push(action);\n return;\n }\n\n let error: any;\n this._active = true;\n\n do {\n if ((error = action.execute(action.state, action.delay))) {\n break;\n }\n } while ((action = actions.shift()!)); // exhaust the scheduler queue\n\n this._active = false;\n\n if (error) {\n while ((action = actions.shift()!)) {\n action.unsubscribe();\n }\n throw error;\n }\n }\n}\n", "import { AsyncAction } from './AsyncAction';\nimport { AsyncScheduler } from './AsyncScheduler';\n\n/**\n *\n * Async Scheduler\n *\n * Schedule task as if you used setTimeout(task, duration)\n *\n * `async` scheduler schedules tasks asynchronously, by putting them on the JavaScript\n * event loop queue. It is best used to delay tasks in time or to schedule tasks repeating\n * in intervals.\n *\n * If you just want to \"defer\" task, that is to perform it right after currently\n * executing synchronous code ends (commonly achieved by `setTimeout(deferredTask, 0)`),\n * better choice will be the {@link asapScheduler} scheduler.\n *\n * ## Examples\n * Use async scheduler to delay task\n * ```ts\n * import { asyncScheduler } from 'rxjs';\n *\n * const task = () => console.log('it works!');\n *\n * asyncScheduler.schedule(task, 2000);\n *\n * // After 2 seconds logs:\n * // \"it works!\"\n * ```\n *\n * Use async scheduler to repeat task in intervals\n * ```ts\n * import { asyncScheduler } from 'rxjs';\n *\n * function task(state) {\n * console.log(state);\n * this.schedule(state + 1, 1000); // `this` references currently executing Action,\n * // which we reschedule with new state and delay\n * }\n *\n * asyncScheduler.schedule(task, 3000, 0);\n *\n * // Logs:\n * // 0 after 3s\n * // 1 after 4s\n * // 2 after 5s\n * // 3 after 6s\n * ```\n */\n\nexport const asyncScheduler = new AsyncScheduler(AsyncAction);\n\n/**\n * @deprecated Renamed to {@link asyncScheduler}. Will be removed in v8.\n */\nexport const async = asyncScheduler;\n", "import { AsyncAction } from './AsyncAction';\nimport { Subscription } from '../Subscription';\nimport { QueueScheduler } from './QueueScheduler';\nimport { SchedulerAction } from '../types';\nimport { TimerHandle } from './timerHandle';\n\nexport class QueueAction extends AsyncAction {\n constructor(protected scheduler: QueueScheduler, protected work: (this: SchedulerAction, state?: T) => void) {\n super(scheduler, work);\n }\n\n public schedule(state?: T, delay: number = 0): Subscription {\n if (delay > 0) {\n return super.schedule(state, delay);\n }\n this.delay = delay;\n this.state = state;\n this.scheduler.flush(this);\n return this;\n }\n\n public execute(state: T, delay: number): any {\n return delay > 0 || this.closed ? super.execute(state, delay) : this._execute(state, delay);\n }\n\n protected requestAsyncId(scheduler: QueueScheduler, id?: TimerHandle, delay: number = 0): TimerHandle {\n // If delay exists and is greater than 0, or if the delay is null (the\n // action wasn't rescheduled) but was originally scheduled as an async\n // action, then recycle as an async action.\n\n if ((delay != null && delay > 0) || (delay == null && this.delay > 0)) {\n return super.requestAsyncId(scheduler, id, delay);\n }\n\n // Otherwise flush the scheduler starting with this action.\n scheduler.flush(this);\n\n // HACK: In the past, this was returning `void`. However, `void` isn't a valid\n // `TimerHandle`, and generally the return value here isn't really used. So the\n // compromise is to return `0` which is both \"falsy\" and a valid `TimerHandle`,\n // as opposed to refactoring every other instanceo of `requestAsyncId`.\n return 0;\n }\n}\n", "import { AsyncScheduler } from './AsyncScheduler';\n\nexport class QueueScheduler extends AsyncScheduler {\n}\n", "import { QueueAction } from './QueueAction';\nimport { QueueScheduler } from './QueueScheduler';\n\n/**\n *\n * Queue Scheduler\n *\n * Put every next task on a queue, instead of executing it immediately\n *\n * `queue` scheduler, when used with delay, behaves the same as {@link asyncScheduler} scheduler.\n *\n * When used without delay, it schedules given task synchronously - executes it right when\n * it is scheduled. However when called recursively, that is when inside the scheduled task,\n * another task is scheduled with queue scheduler, instead of executing immediately as well,\n * that task will be put on a queue and wait for current one to finish.\n *\n * This means that when you execute task with `queue` scheduler, you are sure it will end\n * before any other task scheduled with that scheduler will start.\n *\n * ## Examples\n * Schedule recursively first, then do something\n * ```ts\n * import { queueScheduler } from 'rxjs';\n *\n * queueScheduler.schedule(() => {\n * queueScheduler.schedule(() => console.log('second')); // will not happen now, but will be put on a queue\n *\n * console.log('first');\n * });\n *\n * // Logs:\n * // \"first\"\n * // \"second\"\n * ```\n *\n * Reschedule itself recursively\n * ```ts\n * import { queueScheduler } from 'rxjs';\n *\n * queueScheduler.schedule(function(state) {\n * if (state !== 0) {\n * console.log('before', state);\n * this.schedule(state - 1); // `this` references currently executing Action,\n * // which we reschedule with new state\n * console.log('after', state);\n * }\n * }, 0, 3);\n *\n * // In scheduler that runs recursively, you would expect:\n * // \"before\", 3\n * // \"before\", 2\n * // \"before\", 1\n * // \"after\", 1\n * // \"after\", 2\n * // \"after\", 3\n *\n * // But with queue it logs:\n * // \"before\", 3\n * // \"after\", 3\n * // \"before\", 2\n * // \"after\", 2\n * // \"before\", 1\n * // \"after\", 1\n * ```\n */\n\nexport const queueScheduler = new QueueScheduler(QueueAction);\n\n/**\n * @deprecated Renamed to {@link queueScheduler}. Will be removed in v8.\n */\nexport const queue = queueScheduler;\n", "import { AsyncAction } from './AsyncAction';\nimport { AnimationFrameScheduler } from './AnimationFrameScheduler';\nimport { SchedulerAction } from '../types';\nimport { animationFrameProvider } from './animationFrameProvider';\nimport { TimerHandle } from './timerHandle';\n\nexport class AnimationFrameAction extends AsyncAction {\n constructor(protected scheduler: AnimationFrameScheduler, protected work: (this: SchedulerAction, state?: T) => void) {\n super(scheduler, work);\n }\n\n protected requestAsyncId(scheduler: AnimationFrameScheduler, id?: TimerHandle, delay: number = 0): TimerHandle {\n // If delay is greater than 0, request as an async action.\n if (delay !== null && delay > 0) {\n return super.requestAsyncId(scheduler, id, delay);\n }\n // Push the action to the end of the scheduler queue.\n scheduler.actions.push(this);\n // If an animation frame has already been requested, don't request another\n // one. If an animation frame hasn't been requested yet, request one. Return\n // the current animation frame request id.\n return scheduler._scheduled || (scheduler._scheduled = animationFrameProvider.requestAnimationFrame(() => scheduler.flush(undefined)));\n }\n\n protected recycleAsyncId(scheduler: AnimationFrameScheduler, id?: TimerHandle, delay: number = 0): TimerHandle | undefined {\n // If delay exists and is greater than 0, or if the delay is null (the\n // action wasn't rescheduled) but was originally scheduled as an async\n // action, then recycle as an async action.\n if (delay != null ? delay > 0 : this.delay > 0) {\n return super.recycleAsyncId(scheduler, id, delay);\n }\n // If the scheduler queue has no remaining actions with the same async id,\n // cancel the requested animation frame and set the scheduled flag to\n // undefined so the next AnimationFrameAction will request its own.\n const { actions } = scheduler;\n if (id != null && actions[actions.length - 1]?.id !== id) {\n animationFrameProvider.cancelAnimationFrame(id as number);\n scheduler._scheduled = undefined;\n }\n // Return undefined so the action knows to request a new async id if it's rescheduled.\n return undefined;\n }\n}\n", "import { AsyncAction } from './AsyncAction';\nimport { AsyncScheduler } from './AsyncScheduler';\n\nexport class AnimationFrameScheduler extends AsyncScheduler {\n public flush(action?: AsyncAction): void {\n this._active = true;\n // The async id that effects a call to flush is stored in _scheduled.\n // Before executing an action, it's necessary to check the action's async\n // id to determine whether it's supposed to be executed in the current\n // flush.\n // Previous implementations of this method used a count to determine this,\n // but that was unsound, as actions that are unsubscribed - i.e. cancelled -\n // are removed from the actions array and that can shift actions that are\n // scheduled to be executed in a subsequent flush into positions at which\n // they are executed within the current flush.\n const flushId = this._scheduled;\n this._scheduled = undefined;\n\n const { actions } = this;\n let error: any;\n action = action || actions.shift()!;\n\n do {\n if ((error = action.execute(action.state, action.delay))) {\n break;\n }\n } while ((action = actions[0]) && action.id === flushId && actions.shift());\n\n this._active = false;\n\n if (error) {\n while ((action = actions[0]) && action.id === flushId && actions.shift()) {\n action.unsubscribe();\n }\n throw error;\n }\n }\n}\n", "import { AnimationFrameAction } from './AnimationFrameAction';\nimport { AnimationFrameScheduler } from './AnimationFrameScheduler';\n\n/**\n *\n * Animation Frame Scheduler\n *\n * Perform task when `window.requestAnimationFrame` would fire\n *\n * When `animationFrame` scheduler is used with delay, it will fall back to {@link asyncScheduler} scheduler\n * behaviour.\n *\n * Without delay, `animationFrame` scheduler can be used to create smooth browser animations.\n * It makes sure scheduled task will happen just before next browser content repaint,\n * thus performing animations as efficiently as possible.\n *\n * ## Example\n * Schedule div height animation\n * ```ts\n * // html:

\n * import { animationFrameScheduler } from 'rxjs';\n *\n * const div = document.querySelector('div');\n *\n * animationFrameScheduler.schedule(function(height) {\n * div.style.height = height + \"px\";\n *\n * this.schedule(height + 1); // `this` references currently executing Action,\n * // which we reschedule with new state\n * }, 0, 0);\n *\n * // You will see a div element growing in height\n * ```\n */\n\nexport const animationFrameScheduler = new AnimationFrameScheduler(AnimationFrameAction);\n\n/**\n * @deprecated Renamed to {@link animationFrameScheduler}. Will be removed in v8.\n */\nexport const animationFrame = animationFrameScheduler;\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\n\n/**\n * A simple Observable that emits no items to the Observer and immediately\n * emits a complete notification.\n *\n * Just emits 'complete', and nothing else.\n *\n * ![](empty.png)\n *\n * A simple Observable that only emits the complete notification. It can be used\n * for composing with other Observables, such as in a {@link mergeMap}.\n *\n * ## Examples\n *\n * Log complete notification\n *\n * ```ts\n * import { EMPTY } from 'rxjs';\n *\n * EMPTY.subscribe({\n * next: () => console.log('Next'),\n * complete: () => console.log('Complete!')\n * });\n *\n * // Outputs\n * // Complete!\n * ```\n *\n * Emit the number 7, then complete\n *\n * ```ts\n * import { EMPTY, startWith } from 'rxjs';\n *\n * const result = EMPTY.pipe(startWith(7));\n * result.subscribe(x => console.log(x));\n *\n * // Outputs\n * // 7\n * ```\n *\n * Map and flatten only odd numbers to the sequence `'a'`, `'b'`, `'c'`\n *\n * ```ts\n * import { interval, mergeMap, of, EMPTY } from 'rxjs';\n *\n * const interval$ = interval(1000);\n * const result = interval$.pipe(\n * mergeMap(x => x % 2 === 1 ? of('a', 'b', 'c') : EMPTY),\n * );\n * result.subscribe(x => console.log(x));\n *\n * // Results in the following to the console:\n * // x is equal to the count on the interval, e.g. (0, 1, 2, 3, ...)\n * // x will occur every 1000ms\n * // if x % 2 is equal to 1, print a, b, c (each on its own)\n * // if x % 2 is not equal to 1, nothing will be output\n * ```\n *\n * @see {@link Observable}\n * @see {@link NEVER}\n * @see {@link of}\n * @see {@link throwError}\n */\nexport const EMPTY = new Observable((subscriber) => subscriber.complete());\n\n/**\n * @param scheduler A {@link SchedulerLike} to use for scheduling\n * the emission of the complete notification.\n * @deprecated Replaced with the {@link EMPTY} constant or {@link scheduled} (e.g. `scheduled([], scheduler)`). Will be removed in v8.\n */\nexport function empty(scheduler?: SchedulerLike) {\n return scheduler ? emptyScheduled(scheduler) : EMPTY;\n}\n\nfunction emptyScheduled(scheduler: SchedulerLike) {\n return new Observable((subscriber) => scheduler.schedule(() => subscriber.complete()));\n}\n", "import { SchedulerLike } from '../types';\nimport { isFunction } from './isFunction';\n\nexport function isScheduler(value: any): value is SchedulerLike {\n return value && isFunction(value.schedule);\n}\n", "import { SchedulerLike } from '../types';\nimport { isFunction } from './isFunction';\nimport { isScheduler } from './isScheduler';\n\nfunction last(arr: T[]): T | undefined {\n return arr[arr.length - 1];\n}\n\nexport function popResultSelector(args: any[]): ((...args: unknown[]) => unknown) | undefined {\n return isFunction(last(args)) ? args.pop() : undefined;\n}\n\nexport function popScheduler(args: any[]): SchedulerLike | undefined {\n return isScheduler(last(args)) ? args.pop() : undefined;\n}\n\nexport function popNumber(args: any[], defaultValue: number): number {\n return typeof last(args) === 'number' ? args.pop()! : defaultValue;\n}\n", "export const isArrayLike = ((x: any): x is ArrayLike => x && typeof x.length === 'number' && typeof x !== 'function');", "import { isFunction } from \"./isFunction\";\n\n/**\n * Tests to see if the object is \"thennable\".\n * @param value the object to test\n */\nexport function isPromise(value: any): value is PromiseLike {\n return isFunction(value?.then);\n}\n", "import { InteropObservable } from '../types';\nimport { observable as Symbol_observable } from '../symbol/observable';\nimport { isFunction } from './isFunction';\n\n/** Identifies an input as being Observable (but not necessary an Rx Observable) */\nexport function isInteropObservable(input: any): input is InteropObservable {\n return isFunction(input[Symbol_observable]);\n}\n", "import { isFunction } from './isFunction';\n\nexport function isAsyncIterable(obj: any): obj is AsyncIterable {\n return Symbol.asyncIterator && isFunction(obj?.[Symbol.asyncIterator]);\n}\n", "/**\n * Creates the TypeError to throw if an invalid object is passed to `from` or `scheduled`.\n * @param input The object that was passed.\n */\nexport function createInvalidObservableTypeError(input: any) {\n // TODO: We should create error codes that can be looked up, so this can be less verbose.\n return new TypeError(\n `You provided ${\n input !== null && typeof input === 'object' ? 'an invalid object' : `'${input}'`\n } where a stream was expected. You can provide an Observable, Promise, ReadableStream, Array, AsyncIterable, or Iterable.`\n );\n}\n", "export function getSymbolIterator(): symbol {\n if (typeof Symbol !== 'function' || !Symbol.iterator) {\n return '@@iterator' as any;\n }\n\n return Symbol.iterator;\n}\n\nexport const iterator = getSymbolIterator();\n", "import { iterator as Symbol_iterator } from '../symbol/iterator';\nimport { isFunction } from './isFunction';\n\n/** Identifies an input as being an Iterable */\nexport function isIterable(input: any): input is Iterable {\n return isFunction(input?.[Symbol_iterator]);\n}\n", "import { ReadableStreamLike } from '../types';\nimport { isFunction } from './isFunction';\n\nexport async function* readableStreamLikeToAsyncGenerator(readableStream: ReadableStreamLike): AsyncGenerator {\n const reader = readableStream.getReader();\n try {\n while (true) {\n const { value, done } = await reader.read();\n if (done) {\n return;\n }\n yield value!;\n }\n } finally {\n reader.releaseLock();\n }\n}\n\nexport function isReadableStreamLike(obj: any): obj is ReadableStreamLike {\n // We don't want to use instanceof checks because they would return\n // false for instances from another Realm, like an + + +

What methods exist for the 3x3x3x3?

+

Many 3D methods can just be scaled up and used on the 4D cube. Some notable methods are:

+ +

As of 2024, 3-Block is the most popular speedsolving method and what’s used in the world record, but CFOP is also competitive.

+

How do I start learning to solve 4D puzzles?

+

First, download Hyperspeedcube or MC4D and start experimenting with the 34! Try to solve one-move scrambles and keep practicing that until you’re comfortable. Once you can solve one-move scrambles with ease, pick a method to learn.

+

Why not start with the physical 2x2x2x2?

+

You will never understand physical puzzles before understanding virtual puzzles. Computer simulations are the most direct way to experience 4D puzzles, and it’s what all physical puzzles are emulating. It’s effectively impossible to develop new physical puzzles without first understanding the equivalent virtual puzzles.

+
+

Grant Staten

+

I think that approaching hypercubing by starting with only physical puzzles does not lead to an understanding of how the puzzles actually work. Even if you can solve them.

+

Before I touched a virtual puzzle, I had:

+
    +
  • almost gotten sub-2 on a physical 24 I’d built
    • +
  • started building physical 2x2x2x3
    • +
  • rendered physical 2x2x2x3, 2x2x3x3, 2x3x3x3 (piggybacking off of Luna’s work and just following patterns in place)
    • +
+

However, looking back, I 100% honestly did not yet understand how any of those puzzles actually worked at the time.

+

I have absolutely nothing against someone only focusing on physical puzzles. But in my experience you don’t gain an actual understanding of how the puzzles work through solely using the physical puzzles.

+
+

Why not start with the virtual 2x2x2x2?

+

The 24 is particularly disorienting for beginners because half of the puzzle turns at once. As a result, while the 24 strategy is technically simpler, it’s actually more challenging to wrap your head around, especially when you’re new to 4D puzzles. Just like how the 33 is a better starting puzzle in 3D, you can learn lots of important concepts from the 34 that will help you with other 4D puzzles.

+
+

Sergej Volkov

+

I regret not starting with a 34. I first solved virtual 24 using Rowan’s physical method and it was extremely painful and did not really help to develop any 4D intuition. I ended up just drawing the physical representation of the puzzle on a piece of paper.

+
+

What is God’s number for [puzzle]?

+

See God’s Number.

+

Physical puzzles

+

What is a physical 4D puzzle?

+

The physical 4D puzzles are puzzles that are perfectly analogous to the virtual 4D puzzles, but implemented in the physical world. See these links:

+ +

How can I buy a physical 2x2x2x2?

+

See the Ordering Melinda’s 2x2x2x2 on the Superliminal website. Also see Melinda’s 2x2x2x2 on the Superliminal site, which includes the history, statistics, and Hall of Fame.

+

How can I buy other physical puzzles?

+

Melinda’s 24 is the only physical puzzle for sale. The physical 34 and hypercuboids are currently one-of-a-kind. If you want one, you need to design and 3D print it yourself.

+

Can I download 3D files for the physical 2x2x2x2?

+

No. Melinda has put a lot of work into her physical 24 designs, and invested quite a bit of her own capital into selling prototypes at a loss and getting them mass-produced, so we respect her wishes to not make those files public. If you want to create your own files modeling Melinda’s physical 24 and 3D print them, that is fine, but we ask that you do not make the files avaliable for download unless Melinda is OK with it.

+

What physical 4D puzzles have been built?

+

See Physical Puzzles for a comprehensive list.

+

What physical 4D puzzles are possible?

+

While it’s always possible to just arrange the stickers on a table, the real challenge is in finding a design that is piece-based instead of sticker-based and fits in a compact shape that isn’t too horrendous to turn. This requires some out-of-the-box thinking and, in extreme cases, application of group theory. We currently have several renderings for physical puzzles that haven’t been built in real life yet; see the Physical Puzzles page for an incomplete list.

+

Hyperspeedcube

+

Does Hyperspeedcube run on my OS?

+

Hyperspeedcube runs natively on Windows, macOS, and Linux. There is also a web version, which runs on Chromebooks or other devices where the downloaded version does not work. For mobile phones, it is possible to run Hyperspeedcube in a browser but not recommended. See Software for alternatives.

+

I get an error when I try to run Hyperspeedcube

+

See Hyperspeedcube - Troubleshooting.

+

How do I use keybinds in Hyperspeedcube?

+ + +

How do I use piece filters in Hyperspeedcube?

+ + +

Hyperspeedcube 2

+

When will Hyperspeedcube 2 be ready?

+

When it’s done. Hopefully before summer 2025.

+

What features are planned for Hyperspeedcube 2?

+

See Hactar’s website.

+

Does Hyperspeedcube 2 have [feature] yet?

+

See Hactar’s website.

+

Can I download the latest development build?

+

You can access the latest development builds either by donating to Hactar on Ko-fi (any amount one time should work) or building it yourself from the source code, which takes some time to set up but generally isn’t too hard.

+

I’m having trouble with Hyperspeedcube 2?

+

Please do not ask Hactar for help building or using development builds of Hyperspeedcube 2 unless the build on Ko-fi is broken. Everything in the program is subject to change, and any time spent helping an individual user is time that could instead be spent writing documentation.

+

Where can I follow the latest development updates?

+

See the Hyperspeedcube 2.0 Development Updates thread in #hyper-forum on the Hypercubers Discord Server. Once Hyperspeedcube 2.0 is ready for general use, there will be an announcement that pings the @Hyperspeedcube Update role (and possibly @everyone).

+

Speedsolving

+

What are the speedsolving records for 4D puzzles?

+

See the leaderboards. To get on the leaderboard, read the submission guidelines and submit a video of your solve to this form.

+

Why not use speedrun.com?

+

Speedrun.com does not allow “generic Rubik’s Cube simulators.” We applied and were rejected.

+

I don’t know full OLL/PLL/ZBLL/etc. Can I still get fast at 4D?

+

Absolutely! Most 4D speed methods are highly intuitive, and world-record times often use just 2-look OLL and PLL. Executing algorithms is a very negligible part of the solve compared to the massive amounts of pair or block building.

+
+

Hactar (mid-2024)

+

Ok I actually ran the numbers:

+ +

Consider that the second solve is basically the best case for a 2-look last layer (the goal of full OLL+PLL), using algorithms that have really nice RKT cancels and are easy to execute with my keybinds and I’ve practiced them a ton, but it was still only 4 seconds faster, which is ~2% of the total solve. Additionally, my experience is that it takes much more practice to execute 4D algorithms at max speed with a keyboard compared to executing 3D algorithms at max speed on a 3^3.

+

Based on all that, I can confidently say I don’t think full OLL+PLL will ever be meaningfully better than RKT-canceled 2-look OLL + 2-look PLL, with one exception: There’s a handful of full OLL algorithms (namely the ones composed of fruruf and sune) that I think are worth using if you know them from 3D, but aren’t at all crucial. I do use these during 4D solves when I recognize them, and with some effort we may be able to find a few more cases with easy cancels, but I don’t think it’s worth learning these just for 4D solves.

+
+

What 4D algorithms are there?

+

There’s so many cases for each step of the solve that creating a complete algorithm set is basically impossible, and there’s so many options for moves that algorithm explorers are infeasible. Almost every algorithm we have is based on an algorithm from 3D, and the only search program we have is a sort of optimizer for one very specific kind of algorithm derived from 3D.

+

How can I get faster at solving 3x3x3x3?

+

First, learn 3-Block! Pairs are more popular since the recognition is easier, but solutions using triplets tend to be shorter; as far as we know, both are viable. If you’re getting times in the 8-15 minute range, do slowsolves where you focus on efficient solutions to F2L pairs/triplets.

+
+

Hactar

+

From most to least significant:

+
    +
  • 1-key-per-move keybinds that prioritize R and I cells. This is more than 2x faster than default keybinds, because you do not need to time the release of keys.
    • +
  • 1-key-per-move RKTbinds with RKT cancels. These two optimizations complement each other so well, because it spreads out the work among all your fingers.
    • +
  • Single keys for some 180-degree turns, because repeated keys are slow! By adding keys for x2 and y2, you can reorient a cell into any orientation with just two keypresses.
    • +
  • RKT-canceled triggers during F2L-3. I leave debt on U and have muscle memory for the common RKT-canceled trigger R (flip) U' R' (flip) (where (flip) = {1-2}Ozx2), its inverse (flip) R U (flip) R'*, and their back and left-handed forms. This + U moves accounts for basically everything you do in F2L other than R U2 R', which you can usually avoid or in worst case just (flip) R (flip) U2 (flip) R' (flip).
    • +
+
+

How could the 3x3x3x3 record be improved?

+
+

Hactar (mid-2024)

+

Here’s what I think would make a big difference, from most to least significant:

+
    +
  • Lookahead during F2L-4. This is essentially unexplored.
    • +
  • Optimal solutions for F2L-b cases. My gut says this could save an average of ~3 moves per pair over my solutions during left block, and ~5 moves per pair during right block, which adds up to ~30 STM saved. I think there’d be some value in a comprehensive study of 3-block F2L cases and what techniques are effective in solving them, and research into the psychology of how to recognize them. For some examples of what I mean: I remember reading somewhere that on 3^3 it’s better to find an F2L corner and then search for the matching edge, rather than the other way around. I also know that once you find the edge, you can recognize its orientation to know whether you can solve the pair using just or whether you need less-ergonomic moves. Figuring out and documenting these sort of tricks for 4D would help.
    • +
+

I think both of those improvements could save 15-30 seconds, bringing it from ~1:15 to ~0:45. I also feel like general lookahead/efficiency improvements in PLC might save 10-15 seconds but I can’t put my finger on exactly what they would be. I know my OLC feels very variable during solves, but in practice I can brute-force pretty much anything into a valid final case by spending a second or two flipping one edge/corner.

+
+

Does this puzzle exist?

+

2D Rubik’s Cube

+

Depending on how you define “2D Rubik’s cube,” it might or might not exist.

+

A 3D Rubik’s cube has 6 square faces and each face twists within its 2D plane (with 4 possible rotations). A 4D Rubik’s cube has 8 cubic cells and each cell twists within its 3D plane (with 24 possible rotations). By analogy, a 2D Rubik’s cube has 4 edges and each edge can be rotated within its 1D line … but there’s no way to do a rotation in 1D. So if twists have to be rotations, then a 2D Rubik’s cube doesn’t have any turns and so isn’t a puzzle (or is a trivial one).

+

Using different definitions, we can create a few different puzzles that could reasonably be called a 2D Rubik’s cube:

+
    +
  • If we allow reflections instead of rotations, we get the Reflesquare, which generalizes to the Reflecube.
    • +
  • If we allow translations instead of rotations, we get Loopover.
    • +
  • If we allow circular cuts instead of flat cuts, we get various MagicTile puzzles, including some that are actually equivalent to an ordinary 3D Rubik’s cube!
    • +
+

4D Square-1

+

Square-1 is fundamentally a bandaged dodecagonal prism. There are so many ways to extend that into 4D that there isn’t really a canonical “4D square-1”

+

4D Skewb

+

There’s so many ways to generalize a skewb to 4D that we have a whole page full of them!

+

8-dimensional and higher

+

Above 5 dimensions, cube puzzles aren’t more difficult or interesting, just more tedious and computationally expensive. But there might be some wild hyperpuzzles yet to be discovered up there, say one based on the very special geometry of the E8 Lattice!

+

3D Rubik’s Clock

+

Instead of rotating circles in 2D, you can rotate spheres in 3D. This is a more interesting puzzle than the traditional Rubik’s Clock because moves don’t commute. No one’s written a program yet to simulate it but you totally could!

+

How do I make a 4D [thing]?

+

✨ Generalising Things to 4D: A Handy Guide ✨

+
    +
  1. Understand and define the thing you’re generalising
    2. +
  2. Find where your definitions reference or assume something dimension-specific
    3. +
  3. Rewrite your definitions to avoid dimension-specific references or assumptions
    4. +
  4. Find what 4D object fits your new definitions (there may be one, several, or none)
    5. +
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Glossary

+

This is a glossary of terms used in the hypercubing community. We take a mostly prescriptivist1 approach: terminology has a great effect on how we think about puzzles, and we try to be mindful when naming concepts and inventing notation to ensure that they encourage better understanding of puzzles and are useful in as many contexts as possible.

+
+

Before you propose new terminology …

+

We’ve wrestled in the past with poor terminology that actively hurt understanding. First, gain hands-on experience and intuition for the thing you want to describe, and then see what terms are actually needed. There’s no value in making up words for the pieces on a 7-dimensional puzzle, for example, if there’s no need to communicate about them.

+
+

Puzzle elements

+

A 1-dimensional turning axis is not always well-defined for higher-dimensional puzzles, because rotations generally happen in a plane, not around an axis.

+

Polytope elements

+

For an \(N\)-dimensional polytope: +(Some of these terms are from Polytope - Wikipedia)

+
    +
  • vertex = rank 0, single point
    • +
  • edge = rank 1, line connecting two vertices
    • +
  • face = rank 2, polygon constructed from edges
    • +
  • cell = rank 3, polyhedron constructed from faces
    • +
  • …
    • +
  • \(N\)-face = rank \(N\), polytope constructed from rank \(N-1\) elements
    • +
  • peak = \(N-3\) face
    • +
  • ridge = \(N-2\) face
    • +
  • facet = rank \(N-1\), polytope constructed from rank \(N-2\) elements
    • +
+

In 4D, we prefer facet rather than cell. In simple terms: on most puzzles, a facet is the thing with a single color.

+

Pieces

+

Basic definitions for an \(N\)-dimensional hypercubic puzzle:

+
    +
  • corner = piece with \(N\) colors (4 colors in 4D)
    • +
  • edge = piece with \(N-1\) colors (3 colors in 4D)
    • +
  • peak or 3c = piece with 3 colors (5D+)
    • +
  • ridge or 2c = piece with 2 colors (4D+)
    • +
  • center or 1c = piece with 1 color
    • +
+

We prefer words rather than 1c, 2c, etc. because the words generalize better to other, non-facet-turning puzzles and describe how a piece behaves rather than how it looks.

+

Moves

+
    +
  • axis or turning axis = ray start from the center of the puzzle, around which puzzle elements rotate during twists
    • +
  • twist or move or turn = movement of pieces that changes the puzzle state
    • +
  • rotation or full-puzzle rotation = rotation of the whole puzzle that does not change its state
    • +
+

Solving

+

Actions

+

An action is sequence of moves that preserves invariants of the stage. Usually, an action keeps certain pieces solved. For example, when a 4n has been reduced to a 3n using big cube reduction, outer layer moves are the only actions. In this case, the actions are reduced moves. Another common set of actions is RKT.

+

Parity

+

There is no community consensus on the definitions of parity. Below are some proposed definitions:

+
    +
  • group theory parity = a case where the puzzle is in an unexpected coset of a subgroup of index 2
      +
    • It is often more broady applied to a case where the puzzle is in an unexpected coset of a subgroup of any index.
      • +
    +
    • +
  • cuber parity = a case that is difficult to solve that the solver didn’t expect
      +
    • Melinda’s definition: a local maximum, where the puzzle is largely solved but requires many moves to fix
      • +
    • Hactar’s definition: a case which cannot be solved using the actions expected at this stage in the solve
      • +
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None of these definitions are satisfactory. According to most of these definitions, RKT parity is not parity at all, but is more accurately called “RKT error.” According to Melinda’s definition, most PLL cases are parity. The first definition given for “cuber parity” is highly subjective, but is the only one that captures its current use.

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Open questions

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  • Is there a definition for “cuber parity” that captures the way it’s naturally used?
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  • Is there a catchy term we can use instead of “RKT parity”? Melinda proposes “RKT error.”
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F2L

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F2L is a very general solving strategy that works by building a small block of pieces and then inserting the block into its solved position. F2L stands for “first two layers” because it was originally developed to solve the first two layers of 33, but in hypercubing we use it for many other puzzles.

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F2L axes

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  • free axes = axes which affect only unsolved pieces; can be turned freely
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  • top axis = the free axis currently being worked on
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Examples

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  • In F2L on a 33, D is the only base axis, U is the top axis (the only free axis), and R, L, F, & B are all side axes.
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  • When beginning F2L on a megaminx there are, 6 free axes, 5 side axes, and 1 base axis.
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  • Near the end of F2L on a megaminx there are, 1 free axis, 5 side axes, and 5 base axes.
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We use the letter T to represent the top axis, R & F to represent intersecting side axes, and R & L to represent non-intersecting side axes.

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F2L blocks

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An F2L block or pair is a group of pieces that is paired and solved as one unit. There’s usually a head and body, where the head intersects with more twisting axes than the body.

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The base sticker of a head is the sticker which will be facing the base axis when it is solved. The facing direction of the head of a block is whatever direction its base sticker is facing. The facing direction of the body of a block is the same as the head, when they are paired. This notion of which direction a head or body faces gives a way to describe edge orientation before the pieces have been paired, which is helpful especially in 4D+ where edge orientation is otherwise difficult to define.

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Examples

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  • On the 33 an F2L pair consists of a corner (the head) and an edge (the body).
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  • On the 34, an F2L-a pair consists of an edge (the head) and a ridge (the body).
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  • paired = fully assembled
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  • split pair = one move away from paired, or can be paired as part of inserting the block
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F2L action terminology

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  • breaking the base = unsolving some pieces that were solved
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  • restoring the base = re-solving some pieces
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  • push = a twist of a side axis that breaks the base and puts new pieces on top
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  • pull = a twist of a side axis that restores the base and puts new pieces on top
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  • overpush = push again after pushing (e.g., R U R U R2’)
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  • overpull = push as a continuation of a pull (e.g., R U R2’ U’ R)
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  • push pair = formation of a pair via a push
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  • pull pair = formation of a pair via a pull
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  • hide = to remove a piece from the top (using a push or pull)
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  • reveal = to bring a piece to the top (using a push or pull)
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  • rebase or reorient = to reorient a piece to face a different direction (i.e., change where its base sticker is facing)
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  • cap = to twist T to form a pair (where the head is on top and the body is not on the top)
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  • uncap = to separate the head and body of a pair by twisting T
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Open question

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What should we call a move like RT on 34, which doesn’t change the set of pieces on T and might or might not unsolve some pieces?

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Methods in higher dimensions

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In higher and higher dimensions, it gets annoying to have to say stuff like “permuting the last cell of the last cell of the…” etc. To avoid this, we simply add a hyphen and the rank of the thing you’re solving at the end. Examples:

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  • PLL-4 is the PLL step on a rank-4 object, which permutes a rank-3 object. With CFOP on 34, it consists of permuting the 2c’s, then permuting the rest like a 33.
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  • For F2L, you put the number before the letter at the end e.g. F2L-5a, F2L-6d, etc.
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  • If you were solving a 36 with pure CFOP and you were solving the F2L of the final cube with triple RKT, that would be F2L-3 of PLL-4 of PLL-5 of PLL-6.
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Puzzle descriptors

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This section is a work-in-progress.

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  • Solid vs. tiling vs. soup
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  • Doctrinaire
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  • Reduced
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  • Bandaged
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  • Unbandaged
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  • Shapeshifting
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  • Sliding vs. twisting
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  • Circle
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  • Super
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  • Real
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  • Complex
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  • Stickermod
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  • Shapemod
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  • Cuboid terms (tower, brick, floppy, domino, pancake)
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  • Other common puzzle families: weirdling, bubbloid, rotate-gap, sliding-gap (15-puzzle), loopover
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Cut depth

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This section is a work-in-progress.

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  • Shallow cut
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  • Half cut
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  • Deep cut
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    • Deeper than adjacent
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    • Deeper than origin
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  • Semideep cut?
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    Go ahead, run us over with the descriptivist bus. ↩

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History of Hypercubing

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1988-2002: The Beginning of Hypercubing

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MagicCube4D in 2001

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Old MagicCube4D screenshot

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Shortly after the Rubik’s Cube was invented in 1974, people began thinking about different versions of the puzzle. The idea of higher dimensional analogues was intriguing to many people. This article from 1982 is one of the earliest known documents theorizing properties of the 4D Rubik’s Cube. In 1988 Melinda Green and Don Hatch wrote MagicCube4D, and the first solutions to the 34 puzzle came shortly thereafter. The Superliminal website was created some time around 2001, and Charles Doan has an excellent video about an old version of MC4D. Communication was very difficult because of how extremely niche hypercubing was.

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2002-Present: Mailing List Years

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In August 2003, the Hypercubing Yahoo Groups mailing list was created. Anyone could subscribe to the list and join the discussion via email. This greatly improved the speed of sharing knowledge. Many new people joined and discussed methods, puzzles, and even had some speedsolving competitions! It was moderately active, up until Yahoo groups began removing past content in 2019. Click here to view an archive of all the past messages. After that, the Hypercubing Google Groups mailing list was created, with pretty much the same members and type of discussions as before. It wasn’t until 2021 that a Discord server was created to speed up communication even more.

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2010-2017: Andrey’s Influence

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Andrey Astrelin was a Russian programmer that became known to the hypercubing community when he claimed to have programmed and solved a 34 before MC4D was created in 1988. He had no proof for this, but was able to program a new simulator in only 129 lines of code, which certified his legitimacy. Andrey’s main legacy came in the form of creating many programs for the community, including Magic Puzzle Ultimate, Magic Cube 7D, Magic Simplex 5D, and Magic Hyperbolic Tile. These programs unlocked a massive amount of higher dimensional puzzles that were probably never seen or solved before, enriching the hypercubing community. Unfortunately Andrey passed away in 20171, but his contributions to the hypercubing community will never be forgotten.

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2017-Present: Making Physical 4D Puzzles

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Throughout 2013 and 2014, Melinda Green got in contact with Oskar van Deventer to try and think of any possible mechanism for a physical 24. After failing to find a reasonable physical mechanism, they ended up going with only using magnets in order to stick the hypercubies together. In 2017 the first design was finalized, and Melinda began selling the puzzles by 3D printing the parts from Shapeways. The quality of the puzzle improved in 2022 when Melinda secured a deal with a company to mass produce the pieces via injection moulding.

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In late 2021, some speculation began as to if you could expand the physical 24 into a physical 2x2x2x3 hypercuboid. After Rowan learned that Grant had 3D printed his own physical 24, he challenged him to build a physical 2x2x2x3. Grant completed the challenge, and began designing and printing many more physical puzzles, all the way up to the physical 3x3x3x3. Many physical puzzle designs have still yet to be built…

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Physical puzzles page

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  • 2013-2015: Brainstorming ideas to make the 2x2x2x2 in 3D space
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  • 2017 Feb 08: First puzzle made with magnetic dice mechanism
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  • 2017 Apr 26: 2nd 24 made using 3D printing from Shapeways
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  • 2022 Feb 03: Grant finishes the 2x2x2x3
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  • 2023 May: First brainstorming of floppyhypercubes
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  • 2022 May 14: Grant finishes the 2x2x3x3
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  • 2022 Jul 06: Grant finishes the 2x3x3x3
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  • 2022 Jul 22: Grant finishes the 3x3x3x3
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  • 2022 Aug: Markceluna designs a physical simplex
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  • 2024 Jan: Tymon finishes making 1x2x2x2 through 1x3x3x3 hypercuboids
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  • 2024 Jan: Dietl builds a physical simplex based on a completely different design than Markceluna’s
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2022-Present: The Hyperspeedsolving Revolution

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On 2021 Jun 10, Rowan Fortier created the Hypercubers Discord server, and in late 2021 Hactar began working on the Hyperspeedcube program, which was a massive upgrade over older programs like MC4D. Features like piece filters and customizable keybinds were instrumental to lowering solving times. November 2022 saw the beginning of the race for the 34 world record. The record was traded almost exclusively by Grant and Hactar, smashing all the barriers from sub-10 minutes all the way down to sub-2! There were also a few smaller record races for bigger layered n4s.

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In early 2024, the first speedsolves of the 35 were done. A lot of hypercubers were waiting for HSC 2 to come out so they could use keybinds and better piece filters to solve in 5D, but using MC7D or MPU isn’t that bad at all.

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  • 2022 Jan: HSC v0.1 includes 33 and 34, filtering by piece type (not color), and customizable keybinds.
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  • 2022 Aug: HSC v0.8 adds modern piece filters. HSC v0.9 adds multiple keybind sets.
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  • 2022 Nov 06: First 34 sub-10:00 by Hactar
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  • 2022 Nov 18: First 34 sub-9:00 by Rowan
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  • 2022 Nov 19: First 34 sub-8:00 by Hactar
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  • 2022 Nov 22: First 34 sub-7:00 by Grant
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  • 2022 Nov 23: First 34 sub-6:00 by Hactar
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  • 2022 Nov 26: First 34 sub-5:00 by Hactar
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  • 2022 Dec 07: First 34 sub-4:00 by Grant
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  • 2022 Dec 22: First 34 sub-3:00 by Hactar
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  • 2023 May 13: First 34 sub-2:00 by Hactar
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    Read Andrey’s biography here ↩

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function (arg, view) { + return cachedTextEncoder.encodeInto(arg, view); +} + : function (arg, view) { + const buf = cachedTextEncoder.encode(arg); + view.set(buf); + return { + read: arg.length, + written: buf.length + }; +}); + +function passStringToWasm0(arg, malloc, realloc) { + + if (realloc === undefined) { + const buf = cachedTextEncoder.encode(arg); + const ptr = malloc(buf.length); + getUint8Memory0().subarray(ptr, ptr + buf.length).set(buf); + WASM_VECTOR_LEN = buf.length; + return ptr; + } + + let len = arg.length; + let ptr = malloc(len); + + const mem = getUint8Memory0(); + + let offset = 0; + + for (; offset < len; offset++) { + const code = arg.charCodeAt(offset); + if (code > 0x7F) break; + mem[ptr + offset] = code; + } + + if (offset !== len) { + if (offset !== 0) { + arg = arg.slice(offset); + } + ptr = realloc(ptr, len, len = offset + arg.length * 3); + const view = getUint8Memory0().subarray(ptr + offset, ptr + len); + const ret = encodeString(arg, view); + + offset += ret.written; + } + + WASM_VECTOR_LEN = offset; + return ptr; +} + +function isLikeNone(x) { + return x === undefined || x === null; +} + +let cachedInt32Memory0 = new Int32Array(); + +function getInt32Memory0() { + if (cachedInt32Memory0.byteLength === 0) { + cachedInt32Memory0 = new Int32Array(wasm.memory.buffer); + } + return cachedInt32Memory0; +} + +let cachedFloat64Memory0 = new Float64Array(); + +function getFloat64Memory0() { + if (cachedFloat64Memory0.byteLength === 0) { + cachedFloat64Memory0 = new Float64Array(wasm.memory.buffer); + } + return cachedFloat64Memory0; +} + +function debugString(val) { + // primitive types + const type = typeof val; + if (type == 'number' || type == 'boolean' || val == null) { + return `${val}`; + } + if (type == 'string') { + return `"${val}"`; + } + if (type == 'symbol') { + const description = val.description; + if (description == null) { + return 'Symbol'; + } else { + return `Symbol(${description})`; + } + } + if (type == 'function') { + const name = val.name; + if (typeof name == 'string' && name.length > 0) { + return `Function(${name})`; + } else { + return 'Function'; + } + } + // objects + if (Array.isArray(val)) { + const length = val.length; + let debug = '['; + if (length > 0) { + debug += debugString(val[0]); + } + for(let i = 1; i < length; i++) { + debug += ', ' + debugString(val[i]); + } + debug += ']'; + return debug; + } + // Test for built-in + const builtInMatches = /\[object ([^\]]+)\]/.exec(toString.call(val)); + let className; + if (builtInMatches.length > 1) { + className = builtInMatches[1]; + } else { + // Failed to match the standard '[object ClassName]' + return toString.call(val); + } + if (className == 'Object') { + // we're a user defined class or Object + // JSON.stringify avoids problems with cycles, and is generally much + // easier than looping through ownProperties of `val`. + try { + return 'Object(' + JSON.stringify(val) + ')'; + } catch (_) { + return 'Object'; + } + } + // errors + if (val instanceof Error) { + return `${val.name}: ${val.message}\n${val.stack}`; + } + // TODO we could test for more things here, like `Set`s and `Map`s. + return className; +} + +function makeMutClosure(arg0, arg1, dtor, f) { + const state = { a: arg0, b: arg1, cnt: 1, dtor }; + const real = (...args) => { + // First up with a closure we increment the internal reference + // count. This ensures that the Rust closure environment won't + // be deallocated while we're invoking it. + state.cnt++; + const a = state.a; + state.a = 0; + try { + return f(a, state.b, ...args); + } finally { + if (--state.cnt === 0) { + wasm.__wbindgen_export_2.get(state.dtor)(a, state.b); + + } else { + state.a = a; + } + } + }; + real.original = state; + + return real; +} +function __wbg_adapter_32(arg0, arg1, arg2) { + wasm._dyn_core__ops__function__FnMut__A____Output___R_as_wasm_bindgen__closure__WasmClosure___describe__invoke__h21a8a87d0ff6cf7d(arg0, arg1, addHeapObject(arg2)); +} + +function __wbg_adapter_51(arg0, arg1) { + wasm._dyn_core__ops__function__FnMut_____Output___R_as_wasm_bindgen__closure__WasmClosure___describe__invoke__hcacf8a7eac086784(arg0, arg1); +} + +function __wbg_adapter_54(arg0, arg1, arg2) { + wasm._dyn_core__ops__function__FnMut__A____Output___R_as_wasm_bindgen__closure__WasmClosure___describe__invoke__hf454c416e0e2cf0f(arg0, arg1, addHeapObject(arg2)); +} + +function handleError(f, args) { + try { + return f.apply(this, args); + } catch (e) { + wasm.__wbindgen_exn_store(addHeapObject(e)); + } +} + +function getArrayU8FromWasm0(ptr, len) { + return getUint8Memory0().subarray(ptr / 1, ptr / 1 + len); +} + +let cachedFloat32Memory0 = new Float32Array(); + +function getFloat32Memory0() { + if (cachedFloat32Memory0.byteLength === 0) { + cachedFloat32Memory0 = new Float32Array(wasm.memory.buffer); + } + return cachedFloat32Memory0; +} + +function getArrayF32FromWasm0(ptr, len) { + return getFloat32Memory0().subarray(ptr / 4, ptr / 4 + len); +} + +function getArrayI32FromWasm0(ptr, len) { + return getInt32Memory0().subarray(ptr / 4, ptr / 4 + len); +} + +let cachedUint32Memory0 = new Uint32Array(); + +function getUint32Memory0() { + if (cachedUint32Memory0.byteLength === 0) { + cachedUint32Memory0 = new Uint32Array(wasm.memory.buffer); + } + return cachedUint32Memory0; +} + +function getArrayU32FromWasm0(ptr, len) { + return getUint32Memory0().subarray(ptr / 4, ptr / 4 + len); +} + +async function load(module, imports) { + if (typeof Response === 'function' && module instanceof Response) { + if (typeof WebAssembly.instantiateStreaming === 'function') { + try { + return await WebAssembly.instantiateStreaming(module, imports); + + } catch (e) { + if (module.headers.get('Content-Type') != 'application/wasm') { + console.warn("`WebAssembly.instantiateStreaming` failed because your server does not serve wasm with `application/wasm` MIME type. Falling back to `WebAssembly.instantiate` which is slower. Original error:\n", e); + + } else { + throw e; + } + } + } + + const bytes = await module.arrayBuffer(); + return await WebAssembly.instantiate(bytes, imports); + + } else { + const instance = await WebAssembly.instantiate(module, imports); + + if (instance instanceof WebAssembly.Instance) { + return { instance, module }; + + } else { + return instance; + } + } +} + +function getImports() { + const imports = {}; + imports.wbg = {}; + imports.wbg.__wbindgen_object_drop_ref = function(arg0) { + takeObject(arg0); + }; + imports.wbg.__wbindgen_object_clone_ref = function(arg0) { + const ret = getObject(arg0); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_cb_drop = function(arg0) { + const obj = takeObject(arg0).original; + if (obj.cnt-- == 1) { + obj.a = 0; + return true; + } + const ret = false; + return ret; + }; + imports.wbg.__wbindgen_string_new = function(arg0, arg1) { + const ret = getStringFromWasm0(arg0, arg1); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_string_get = function(arg0, arg1) { + const obj = getObject(arg1); + const ret = typeof(obj) === 'string' ? obj : undefined; + var ptr0 = isLikeNone(ret) ? 0 : passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + var len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbindgen_number_get = function(arg0, arg1) { + const obj = getObject(arg1); + const ret = typeof(obj) === 'number' ? obj : undefined; + getFloat64Memory0()[arg0 / 8 + 1] = isLikeNone(ret) ? 0 : ret; + getInt32Memory0()[arg0 / 4 + 0] = !isLikeNone(ret); + }; + imports.wbg.__wbindgen_is_undefined = function(arg0) { + const ret = getObject(arg0) === undefined; + return ret; + }; + imports.wbg.__wbindgen_is_function = function(arg0) { + const ret = typeof(getObject(arg0)) === 'function'; + return ret; + }; + imports.wbg.__wbg_mark_40e050a77cc39fea = function(arg0, arg1) { + performance.mark(getStringFromWasm0(arg0, arg1)); + }; + imports.wbg.__wbg_log_c9486ca5d8e2cbe8 = function(arg0, arg1) { + try { + console.log(getStringFromWasm0(arg0, arg1)); + } finally { + wasm.__wbindgen_free(arg0, arg1); + } + }; + imports.wbg.__wbg_log_aba5996d9bde071f = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7) { + try { + console.log(getStringFromWasm0(arg0, arg1), getStringFromWasm0(arg2, arg3), getStringFromWasm0(arg4, arg5), getStringFromWasm0(arg6, arg7)); + } finally { + wasm.__wbindgen_free(arg0, arg1); + } + }; + imports.wbg.__wbg_measure_aa7a73f17813f708 = function() { return handleError(function (arg0, arg1, arg2, arg3) { + try { + performance.measure(getStringFromWasm0(arg0, arg1), getStringFromWasm0(arg2, arg3)); + } finally { + wasm.__wbindgen_free(arg0, arg1); + wasm.__wbindgen_free(arg2, arg3); + } + }, arguments) }; + imports.wbg.__wbg_new_abda76e883ba8a5f = function() { + const ret = new Error(); + return addHeapObject(ret); + }; + imports.wbg.__wbg_stack_658279fe44541cf6 = function(arg0, arg1) { + const ret = getObject(arg1).stack; + const ptr0 = passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + const len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbg_error_f851667af71bcfc6 = function(arg0, arg1) { + try { + console.error(getStringFromWasm0(arg0, arg1)); + } finally { + wasm.__wbindgen_free(arg0, arg1); + } + }; + imports.wbg.__wbg_alert_1e8e774a1999e4b9 = function(arg0, arg1) { + alert(getStringFromWasm0(arg0, arg1)); + }; + imports.wbg.__wbg_confirm_6743dafc3f7534ae = function(arg0, arg1) { + const ret = confirm(getStringFromWasm0(arg0, arg1)); + return ret; + }; + imports.wbg.__wbindgen_boolean_get = function(arg0) { + const v = getObject(arg0); + const ret = typeof(v) === 'boolean' ? (v ? 1 : 0) : 2; + return ret; + }; + imports.wbg.__wbindgen_number_new = function(arg0) { + const ret = arg0; + return addHeapObject(ret); + }; + imports.wbg.__wbg_crypto_e1d53a1d73fb10b8 = function(arg0) { + const ret = getObject(arg0).crypto; + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_is_object = function(arg0) { + const val = getObject(arg0); + const ret = typeof(val) === 'object' && val !== null; + return ret; + }; + imports.wbg.__wbg_process_038c26bf42b093f8 = function(arg0) { + const ret = getObject(arg0).process; + return addHeapObject(ret); + }; + imports.wbg.__wbg_versions_ab37218d2f0b24a8 = function(arg0) { + const ret = getObject(arg0).versions; + return addHeapObject(ret); + }; + imports.wbg.__wbg_node_080f4b19d15bc1fe = function(arg0) { + const ret = getObject(arg0).node; + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_is_string = function(arg0) { + const ret = typeof(getObject(arg0)) === 'string'; + return ret; + }; + imports.wbg.__wbg_require_78a3dcfbdba9cbce = function() { return handleError(function () { + const ret = module.require; + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_msCrypto_6e7d3e1f92610cbb = function(arg0) { + const ret = getObject(arg0).msCrypto; + return addHeapObject(ret); + }; + imports.wbg.__wbg_randomFillSync_6894564c2c334c42 = function() { return handleError(function (arg0, arg1, arg2) { + getObject(arg0).randomFillSync(getArrayU8FromWasm0(arg1, arg2)); + }, arguments) }; + imports.wbg.__wbg_getRandomValues_805f1c3d65988a5a = function() { return handleError(function (arg0, arg1) { + getObject(arg0).getRandomValues(getObject(arg1)); + }, arguments) }; + imports.wbg.__wbg_instanceof_WebGl2RenderingContext_fcfa91cd777063f3 = function(arg0) { + let result; + try { + result = getObject(arg0) instanceof WebGL2RenderingContext; + } catch { + result = false; + } + const ret = result; + return ret; + }; + imports.wbg.__wbg_beginQuery_909ec673d606f873 = function(arg0, arg1, arg2) { + getObject(arg0).beginQuery(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_bindBufferRange_b8f6dc19661d5cf7 = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).bindBufferRange(arg1 >>> 0, arg2 >>> 0, getObject(arg3), arg4, arg5); + }; + imports.wbg.__wbg_bindSampler_4b0e0e598e2cae44 = function(arg0, arg1, arg2) { + getObject(arg0).bindSampler(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_bindVertexArray_9d12800e272184b0 = function(arg0, arg1) { + getObject(arg0).bindVertexArray(getObject(arg1)); + }; + imports.wbg.__wbg_blitFramebuffer_cdc1ebf043046b70 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10) { + getObject(arg0).blitFramebuffer(arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9 >>> 0, arg10 >>> 0); + }; + imports.wbg.__wbg_bufferData_6ce28904b25c8be9 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).bufferData(arg1 >>> 0, arg2, arg3 >>> 0); + }; + imports.wbg.__wbg_bufferData_8d206d7adf6751c0 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).bufferData(arg1 >>> 0, getObject(arg2), arg3 >>> 0); + }; + imports.wbg.__wbg_bufferSubData_0e04c6c7fec3c949 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).bufferSubData(arg1 >>> 0, arg2, getObject(arg3)); + }; + imports.wbg.__wbg_clearBufferfi_92173f77d7147a2f = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).clearBufferfi(arg1 >>> 0, arg2, arg3, arg4); + }; + imports.wbg.__wbg_clearBufferfv_5cc4edeacbcf72e8 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).clearBufferfv(arg1 >>> 0, arg2, getArrayF32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_clearBufferiv_8bb0c2b97eedc22b = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).clearBufferiv(arg1 >>> 0, arg2, getArrayI32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_clearBufferuiv_1f5c5e9baa9a3d9b = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).clearBufferuiv(arg1 >>> 0, arg2, getArrayU32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_clientWaitSync_ad323ab9e423d0cf = function(arg0, arg1, arg2, arg3) { + const ret = getObject(arg0).clientWaitSync(getObject(arg1), arg2 >>> 0, arg3 >>> 0); + return ret; + }; + imports.wbg.__wbg_compressedTexSubImage2D_5b2a7dc8dc7b3e73 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9) { + getObject(arg0).compressedTexSubImage2D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7 >>> 0, arg8, arg9); + }; + imports.wbg.__wbg_compressedTexSubImage2D_fd1cef4f6a5da5c3 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8) { + getObject(arg0).compressedTexSubImage2D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7 >>> 0, getObject(arg8)); + }; + imports.wbg.__wbg_compressedTexSubImage3D_0df5a8ddb9ebafc2 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10, arg11) { + getObject(arg0).compressedTexSubImage3D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9 >>> 0, arg10, arg11); + }; + imports.wbg.__wbg_compressedTexSubImage3D_9c916feb243112db = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10) { + getObject(arg0).compressedTexSubImage3D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9 >>> 0, getObject(arg10)); + }; + imports.wbg.__wbg_copyBufferSubData_11187dccce72b79b = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).copyBufferSubData(arg1 >>> 0, arg2 >>> 0, arg3, arg4, arg5); + }; + imports.wbg.__wbg_copyTexSubImage3D_7a262558a6a33f2e = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9) { + getObject(arg0).copyTexSubImage3D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9); + }; + imports.wbg.__wbg_createSampler_288fd761eabe283d = function(arg0) { + const ret = getObject(arg0).createSampler(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createVertexArray_8467a75e68fec199 = function(arg0) { + const ret = getObject(arg0).createVertexArray(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_deleteQuery_77a7ae09eda297e1 = function(arg0, arg1) { + getObject(arg0).deleteQuery(getObject(arg1)); + }; + imports.wbg.__wbg_deleteSampler_ec3ca2243d8cfcad = function(arg0, arg1) { + getObject(arg0).deleteSampler(getObject(arg1)); + }; + imports.wbg.__wbg_deleteSync_48aed3df05f4f497 = function(arg0, arg1) { + getObject(arg0).deleteSync(getObject(arg1)); + }; + imports.wbg.__wbg_deleteVertexArray_00194a31d79df7e5 = function(arg0, arg1) { + getObject(arg0).deleteVertexArray(getObject(arg1)); + }; + imports.wbg.__wbg_drawArraysInstanced_951a1d7e32c4f855 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).drawArraysInstanced(arg1 >>> 0, arg2, arg3, arg4); + }; + imports.wbg.__wbg_drawBuffers_23c1572f12f90db2 = function(arg0, arg1) { + getObject(arg0).drawBuffers(getObject(arg1)); + }; + imports.wbg.__wbg_drawElementsInstanced_2e05a96af17fe284 = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).drawElementsInstanced(arg1 >>> 0, arg2, arg3 >>> 0, arg4, arg5); + }; + imports.wbg.__wbg_endQuery_05baee8fc782e5f0 = function(arg0, arg1) { + getObject(arg0).endQuery(arg1 >>> 0); + }; + imports.wbg.__wbg_fenceSync_91d72c970c880844 = function(arg0, arg1, arg2) { + const ret = getObject(arg0).fenceSync(arg1 >>> 0, arg2 >>> 0); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_framebufferTextureLayer_d5e78fc74b8261e3 = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).framebufferTextureLayer(arg1 >>> 0, arg2 >>> 0, getObject(arg3), arg4, arg5); + }; + imports.wbg.__wbg_getBufferSubData_6b00169c609c16f7 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).getBufferSubData(arg1 >>> 0, arg2, getObject(arg3)); + }; + imports.wbg.__wbg_getIndexedParameter_d4a2b68e14a022a1 = function() { return handleError(function (arg0, arg1, arg2) { + const ret = getObject(arg0).getIndexedParameter(arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_getQueryParameter_358ea490fb85e05c = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getQueryParameter(getObject(arg1), arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_getSyncParameter_ab2f9499a91faae0 = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getSyncParameter(getObject(arg1), arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_getUniformBlockIndex_a6f3a994dcc7399d = function(arg0, arg1, arg2, arg3) { + const ret = getObject(arg0).getUniformBlockIndex(getObject(arg1), getStringFromWasm0(arg2, arg3)); + return ret; + }; + imports.wbg.__wbg_invalidateFramebuffer_802e38619851791e = function() { return handleError(function (arg0, arg1, arg2) { + getObject(arg0).invalidateFramebuffer(arg1 >>> 0, getObject(arg2)); + }, arguments) }; + imports.wbg.__wbg_readBuffer_f20d42ed12643534 = function(arg0, arg1) { + getObject(arg0).readBuffer(arg1 >>> 0); + }; + imports.wbg.__wbg_readPixels_e855be1f94815442 = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7) { + getObject(arg0).readPixels(arg1, arg2, arg3, arg4, arg5 >>> 0, arg6 >>> 0, getObject(arg7)); + }, arguments) }; + imports.wbg.__wbg_readPixels_5d4e6205291096f0 = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7) { + getObject(arg0).readPixels(arg1, arg2, arg3, arg4, arg5 >>> 0, arg6 >>> 0, arg7); + }, arguments) }; + imports.wbg.__wbg_renderbufferStorageMultisample_3e76453eed60554b = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).renderbufferStorageMultisample(arg1 >>> 0, arg2, arg3 >>> 0, arg4, arg5); + }; + imports.wbg.__wbg_samplerParameterf_6eda655d7213cb18 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).samplerParameterf(getObject(arg1), arg2 >>> 0, arg3); + }; + imports.wbg.__wbg_samplerParameteri_390f1debfe40f83b = function(arg0, arg1, arg2, arg3) { + getObject(arg0).samplerParameteri(getObject(arg1), arg2 >>> 0, arg3); + }; + imports.wbg.__wbg_texStorage2D_d25a76ad1b1ea98f = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).texStorage2D(arg1 >>> 0, arg2, arg3 >>> 0, arg4, arg5); + }; + imports.wbg.__wbg_texStorage3D_19979792a7a67f59 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6) { + getObject(arg0).texStorage3D(arg1 >>> 0, arg2, arg3 >>> 0, arg4, arg5, arg6); + }; + imports.wbg.__wbg_texSubImage2D_421e29fed0db07ab = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9) { + getObject(arg0).texSubImage2D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7 >>> 0, arg8 >>> 0, getObject(arg9)); + }, arguments) }; + imports.wbg.__wbg_texSubImage2D_f06e46b3b25ee691 = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9) { + getObject(arg0).texSubImage2D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7 >>> 0, arg8 >>> 0, arg9); + }, arguments) }; + imports.wbg.__wbg_texSubImage3D_ebb9e6f80d19a411 = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10, arg11) { + getObject(arg0).texSubImage3D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9 >>> 0, arg10 >>> 0, arg11); + }, arguments) }; + imports.wbg.__wbg_texSubImage3D_591b8511a3c7593a = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10, arg11) { + getObject(arg0).texSubImage3D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9 >>> 0, arg10 >>> 0, getObject(arg11)); + }, arguments) }; + imports.wbg.__wbg_uniform2fv_a611afaf4a045f7e = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform2fv(getObject(arg1), getArrayF32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform2iv_b1b33c9425d5791b = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform2iv(getObject(arg1), getArrayI32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform3fv_740a7286bf6328ee = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform3fv(getObject(arg1), getArrayF32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform3iv_df752fa54b2b8b7b = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform3iv(getObject(arg1), getArrayI32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform4fv_737873ef0bcd5e6c = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform4fv(getObject(arg1), getArrayF32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform4iv_67eed4073c7e55c5 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform4iv(getObject(arg1), getArrayI32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniformBlockBinding_50ced0c985f91a02 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniformBlockBinding(getObject(arg1), arg2 >>> 0, arg3 >>> 0); + }; + imports.wbg.__wbg_uniformMatrix2fv_f4fc5e6214cc5549 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).uniformMatrix2fv(getObject(arg1), arg2 !== 0, getArrayF32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_uniformMatrix3fv_a02aa02ecb8e5f99 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).uniformMatrix3fv(getObject(arg1), arg2 !== 0, getArrayF32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_uniformMatrix4fv_68d11b378757596e = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).uniformMatrix4fv(getObject(arg1), arg2 !== 0, getArrayF32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_vertexAttribDivisor_2dc16945a591d4c6 = function(arg0, arg1, arg2) { + getObject(arg0).vertexAttribDivisor(arg1 >>> 0, arg2 >>> 0); + }; + imports.wbg.__wbg_vertexAttribIPointer_167c7ed4319992e7 = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).vertexAttribIPointer(arg1 >>> 0, arg2, arg3 >>> 0, arg4, arg5); + }; + imports.wbg.__wbg_activeTexture_6a9afd67cc0ade73 = function(arg0, arg1) { + getObject(arg0).activeTexture(arg1 >>> 0); + }; + imports.wbg.__wbg_attachShader_90ad543fb1bccb18 = function(arg0, arg1, arg2) { + getObject(arg0).attachShader(getObject(arg1), getObject(arg2)); + }; + imports.wbg.__wbg_bindBuffer_66e359418f5c82d7 = function(arg0, arg1, arg2) { + getObject(arg0).bindBuffer(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_bindFramebuffer_5c01742edd5d843a = function(arg0, arg1, arg2) { + getObject(arg0).bindFramebuffer(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_bindRenderbuffer_f66dee160b94e5ef = function(arg0, arg1, arg2) { + getObject(arg0).bindRenderbuffer(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_bindTexture_ae9620ea4a6ffb97 = function(arg0, arg1, arg2) { + getObject(arg0).bindTexture(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_blendColor_50e203e2f58784cb = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).blendColor(arg1, arg2, arg3, arg4); + }; + imports.wbg.__wbg_blendEquation_72746aedc87e3f72 = function(arg0, arg1) { + getObject(arg0).blendEquation(arg1 >>> 0); + }; + imports.wbg.__wbg_blendEquationSeparate_f0abe930082fff02 = function(arg0, arg1, arg2) { + getObject(arg0).blendEquationSeparate(arg1 >>> 0, arg2 >>> 0); + }; + imports.wbg.__wbg_blendFunc_99b48b64bde98c6f = function(arg0, arg1, arg2) { + getObject(arg0).blendFunc(arg1 >>> 0, arg2 >>> 0); + }; + imports.wbg.__wbg_blendFuncSeparate_cecb7dfda39dc38d = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).blendFuncSeparate(arg1 >>> 0, arg2 >>> 0, arg3 >>> 0, arg4 >>> 0); + }; + imports.wbg.__wbg_colorMask_12687df5490e9bc9 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).colorMask(arg1 !== 0, arg2 !== 0, arg3 !== 0, arg4 !== 0); + }; + imports.wbg.__wbg_compileShader_822f38928f6f2a08 = function(arg0, arg1) { + getObject(arg0).compileShader(getObject(arg1)); + }; + imports.wbg.__wbg_copyTexSubImage2D_4c72e3ef713b65e6 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8) { + getObject(arg0).copyTexSubImage2D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7, arg8); + }; + imports.wbg.__wbg_createBuffer_a6cffb7f7d5b92a3 = function(arg0) { + const ret = getObject(arg0).createBuffer(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createFramebuffer_d5f3985ce3652661 = function(arg0) { + const ret = getObject(arg0).createFramebuffer(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createProgram_dc6b23d3caa1d86e = function(arg0) { + const ret = getObject(arg0).createProgram(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createRenderbuffer_531167a301a60e27 = function(arg0) { + const ret = getObject(arg0).createRenderbuffer(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createShader_46a66dce5a9e22d0 = function(arg0, arg1) { + const ret = getObject(arg0).createShader(arg1 >>> 0); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createTexture_269f67d411bdc4dc = function(arg0) { + const ret = getObject(arg0).createTexture(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_cullFace_d6b862a4ad70b414 = function(arg0, arg1) { + getObject(arg0).cullFace(arg1 >>> 0); + }; + imports.wbg.__wbg_deleteBuffer_12fd7d93834069ef = function(arg0, arg1) { + getObject(arg0).deleteBuffer(getObject(arg1)); + }; + imports.wbg.__wbg_deleteFramebuffer_d7551444a28f508e = function(arg0, arg1) { + getObject(arg0).deleteFramebuffer(getObject(arg1)); + }; + imports.wbg.__wbg_deleteProgram_ce56000628d7f1ce = function(arg0, arg1) { + getObject(arg0).deleteProgram(getObject(arg1)); + }; + imports.wbg.__wbg_deleteRenderbuffer_58c540348fb8606d = function(arg0, arg1) { + getObject(arg0).deleteRenderbuffer(getObject(arg1)); + }; + imports.wbg.__wbg_deleteShader_246e6e678f3eb957 = function(arg0, arg1) { + getObject(arg0).deleteShader(getObject(arg1)); + }; + imports.wbg.__wbg_deleteTexture_68a539339fd87792 = function(arg0, arg1) { + getObject(arg0).deleteTexture(getObject(arg1)); + }; + imports.wbg.__wbg_depthFunc_1015c3364a49cd2f = function(arg0, arg1) { + getObject(arg0).depthFunc(arg1 >>> 0); + }; + imports.wbg.__wbg_depthMask_55f538b7411e5023 = function(arg0, arg1) { + getObject(arg0).depthMask(arg1 !== 0); + }; + imports.wbg.__wbg_depthRange_c6ed3371d3b601f8 = function(arg0, arg1, arg2) { + getObject(arg0).depthRange(arg1, arg2); + }; + imports.wbg.__wbg_disable_1659dc1efb5fb934 = function(arg0, arg1) { + getObject(arg0).disable(arg1 >>> 0); + }; + imports.wbg.__wbg_disableVertexAttribArray_6f3d27dd0ad6aabf = function(arg0, arg1) { + getObject(arg0).disableVertexAttribArray(arg1 >>> 0); + }; + imports.wbg.__wbg_drawArrays_d587302f7a868d91 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).drawArrays(arg1 >>> 0, arg2, arg3); + }; + imports.wbg.__wbg_drawElements_241caa588795bcb1 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).drawElements(arg1 >>> 0, arg2, arg3 >>> 0, arg4); + }; + imports.wbg.__wbg_enable_4791414dce6f602a = function(arg0, arg1) { + getObject(arg0).enable(arg1 >>> 0); + }; + imports.wbg.__wbg_enableVertexAttribArray_a1ffc091f3999354 = function(arg0, arg1) { + getObject(arg0).enableVertexAttribArray(arg1 >>> 0); + }; + imports.wbg.__wbg_framebufferRenderbuffer_963b305ac8cb6fd6 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).framebufferRenderbuffer(arg1 >>> 0, arg2 >>> 0, arg3 >>> 0, getObject(arg4)); + }; + imports.wbg.__wbg_framebufferTexture2D_4b810902dffa1ef3 = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).framebufferTexture2D(arg1 >>> 0, arg2 >>> 0, arg3 >>> 0, getObject(arg4), arg5); + }; + imports.wbg.__wbg_frontFace_97d7f9493791771d = function(arg0, arg1) { + getObject(arg0).frontFace(arg1 >>> 0); + }; + imports.wbg.__wbg_getActiveUniform_97472b76b9daa461 = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getActiveUniform(getObject(arg1), arg2 >>> 0); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_getExtension_e7912bce04869d40 = function() { return handleError(function (arg0, arg1, arg2) { + const ret = getObject(arg0).getExtension(getStringFromWasm0(arg1, arg2)); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_getParameter_4e2ccc745690476a = function() { return handleError(function (arg0, arg1) { + const ret = getObject(arg0).getParameter(arg1 >>> 0); + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_getProgramInfoLog_1e37a3d1d090ec1c = function(arg0, arg1, arg2) { + const ret = getObject(arg1).getProgramInfoLog(getObject(arg2)); + var ptr0 = isLikeNone(ret) ? 0 : passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + var len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbg_getProgramParameter_acf4ae158143e2b2 = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getProgramParameter(getObject(arg1), arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_getShaderInfoLog_451545b963646762 = function(arg0, arg1, arg2) { + const ret = getObject(arg1).getShaderInfoLog(getObject(arg2)); + var ptr0 = isLikeNone(ret) ? 0 : passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + var len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbg_getShaderParameter_6cd8c36fded266ea = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getShaderParameter(getObject(arg1), arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_getSupportedExtensions_b84494641d686623 = function(arg0) { + const ret = getObject(arg0).getSupportedExtensions(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_getUniformLocation_0da0c93f626244a2 = function(arg0, arg1, arg2, arg3) { + const ret = getObject(arg0).getUniformLocation(getObject(arg1), getStringFromWasm0(arg2, arg3)); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_linkProgram_c33885d9ea798810 = function(arg0, arg1) { + getObject(arg0).linkProgram(getObject(arg1)); + }; + imports.wbg.__wbg_pixelStorei_51c83dc5117bea35 = function(arg0, arg1, arg2) { + getObject(arg0).pixelStorei(arg1 >>> 0, arg2); + }; + imports.wbg.__wbg_polygonOffset_7af170d91752512c = function(arg0, arg1, arg2) { + getObject(arg0).polygonOffset(arg1, arg2); + }; + imports.wbg.__wbg_renderbufferStorage_0b6269243d09a9f7 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).renderbufferStorage(arg1 >>> 0, arg2 >>> 0, arg3, arg4); + }; + imports.wbg.__wbg_scissor_b1b9e314ab6aac29 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).scissor(arg1, arg2, arg3, arg4); + }; + imports.wbg.__wbg_shaderSource_5111981e7afb61fb = function(arg0, arg1, arg2, arg3) { + getObject(arg0).shaderSource(getObject(arg1), getStringFromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_stencilFuncSeparate_2939e543fa4caa77 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).stencilFuncSeparate(arg1 >>> 0, arg2 >>> 0, arg3, arg4 >>> 0); + }; + imports.wbg.__wbg_stencilMask_4eb0f989e4108b15 = function(arg0, arg1) { + getObject(arg0).stencilMask(arg1 >>> 0); + }; + imports.wbg.__wbg_stencilMaskSeparate_69e9937a9533f4ab = function(arg0, arg1, arg2) { + getObject(arg0).stencilMaskSeparate(arg1 >>> 0, arg2 >>> 0); + }; + imports.wbg.__wbg_stencilOpSeparate_c57c8bbe863e9f57 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).stencilOpSeparate(arg1 >>> 0, arg2 >>> 0, arg3 >>> 0, arg4 >>> 0); + }; + imports.wbg.__wbg_texParameteri_21fd6b6b394882c9 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).texParameteri(arg1 >>> 0, arg2 >>> 0, arg3); + }; + imports.wbg.__wbg_uniform1f_ade6c204580582c8 = function(arg0, arg1, arg2) { + getObject(arg0).uniform1f(getObject(arg1), arg2); + }; + imports.wbg.__wbg_uniform1i_49986febd844f2c4 = function(arg0, arg1, arg2) { + getObject(arg0).uniform1i(getObject(arg1), arg2); + }; + imports.wbg.__wbg_uniform4f_d564461a6e4fdfe0 = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).uniform4f(getObject(arg1), arg2, arg3, arg4, arg5); + }; + imports.wbg.__wbg_useProgram_35a58ac1e0d9577b = function(arg0, arg1) { + getObject(arg0).useProgram(getObject(arg1)); + }; + imports.wbg.__wbg_vertexAttribPointer_3b06d737566f0745 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6) { + getObject(arg0).vertexAttribPointer(arg1 >>> 0, arg2, arg3 >>> 0, arg4 !== 0, arg5, arg6); + }; + imports.wbg.__wbg_viewport_319ab5302767fcc9 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).viewport(arg1, arg2, arg3, arg4); + }; + imports.wbg.__wbg_instanceof_Window_acc97ff9f5d2c7b4 = function(arg0) { + let result; + try { + result = getObject(arg0) instanceof Window; + } catch { + result = false; + } + const ret = result; + return ret; + }; + imports.wbg.__wbg_document_3ead31dbcad65886 = function(arg0) { + const ret = getObject(arg0).document; + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_navigator_d1dcf282b97e2495 = function(arg0) { + const ret = getObject(arg0).navigator; + return addHeapObject(ret); + }; + imports.wbg.__wbg_innerWidth_ffa584f74d721fce = function() { return handleError(function (arg0) { + const ret = getObject(arg0).innerWidth; + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_innerHeight_f4804c803fcf02b0 = function() { return handleError(function (arg0) { + const ret = getObject(arg0).innerHeight; + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_devicePixelRatio_476ddb014eb2520a = function(arg0) { + const ret = getObject(arg0).devicePixelRatio; + return ret; + }; + imports.wbg.__wbg_localStorage_753b6d15a844c3dc = function() { return handleError(function (arg0) { + const ret = getObject(arg0).localStorage; + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_cancelAnimationFrame_679ac3913d7f9b34 = function() { return handleError(function (arg0, arg1) { + getObject(arg0).cancelAnimationFrame(arg1); + }, arguments) }; + imports.wbg.__wbg_matchMedia_0b5dc8aaf445df72 = function() { return handleError(function (arg0, arg1, arg2) { + const ret = getObject(arg0).matchMedia(getStringFromWasm0(arg1, arg2)); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_open_8b947bc096f5bb21 = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4) { + const ret = getObject(arg0).open(getStringFromWasm0(arg1, arg2), getStringFromWasm0(arg3, arg4)); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_requestAnimationFrame_4181656476a7d86c = function() { return handleError(function (arg0, arg1) { + const ret = getObject(arg0).requestAnimationFrame(getObject(arg1)); + return ret; + }, arguments) }; + imports.wbg.__wbg_get_55f248d76a5aa3d1 = function(arg0, arg1, arg2) { + const ret = getObject(arg0)[getStringFromWasm0(arg1, arg2)]; + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_clearTimeout_7d6f7bfeed34b348 = function(arg0, arg1) { + getObject(arg0).clearTimeout(arg1); + }; + imports.wbg.__wbg_setTimeout_d6fcf0d9067b8e64 = function() { return handleError(function (arg0, arg1, arg2) { + const ret = getObject(arg0).setTimeout(getObject(arg1), arg2); + return ret; + }, arguments) }; + imports.wbg.__wbg_getBoundingClientRect_06acb6ac1c23e409 = function(arg0) { + const ret = getObject(arg0).getBoundingClientRect(); + return addHeapObject(ret); + }; + imports.wbg.__wbg_requestFullscreen_7d41309612540445 = function() { return handleError(function (arg0) { + getObject(arg0).requestFullscreen(); + }, arguments) }; + imports.wbg.__wbg_setAttribute_d8436c14a59ab1af = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).setAttribute(getStringFromWasm0(arg1, arg2), getStringFromWasm0(arg3, arg4)); + }, arguments) }; + imports.wbg.__wbg_setPointerCapture_7cc6c6e831d5dae0 = function() { return handleError(function (arg0, arg1) { + getObject(arg0).setPointerCapture(arg1); + }, arguments) }; + imports.wbg.__wbg_fullscreenElement_de98779ddf556e06 = function(arg0) { + const ret = getObject(arg0).fullscreenElement; + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createElement_976dbb84fe1661b5 = function() { return handleError(function (arg0, arg1, arg2) { + const ret = getObject(arg0).createElement(getStringFromWasm0(arg1, arg2)); + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_getElementById_3a708b83e4f034d7 = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getElementById(getStringFromWasm0(arg1, arg2)); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_querySelector_3628dc2c3319e7e0 = function() { return handleError(function (arg0, arg1, arg2) { + const ret = getObject(arg0).querySelector(getStringFromWasm0(arg1, arg2)); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_style_e9380748cee29f13 = function(arg0) { + const ret = getObject(arg0).style; + return addHeapObject(ret); + }; + imports.wbg.__wbg_instanceof_HtmlCanvasElement_97761617af6ea089 = function(arg0) { + let result; + try { + result = getObject(arg0) instanceof HTMLCanvasElement; + } catch { + result = false; + } + const ret = result; + return ret; + }; + imports.wbg.__wbg_width_2f4b0cbbf1c850d9 = function(arg0) { + const ret = getObject(arg0).width; + return ret; + }; + imports.wbg.__wbg_setwidth_afb418d3fbf71ba7 = function(arg0, arg1) { + getObject(arg0).width = arg1 >>> 0; + }; + imports.wbg.__wbg_height_a81d308a000d91d0 = function(arg0) { + const ret = getObject(arg0).height; + return ret; + }; + imports.wbg.__wbg_setheight_3eb8729b59493242 = function(arg0, arg1) { + getObject(arg0).height = arg1 >>> 0; + }; + imports.wbg.__wbg_getContext_a6ea7a8e317f182a = function() { return handleError(function (arg0, arg1, arg2, arg3) { + const ret = getObject(arg0).getContext(getStringFromWasm0(arg1, arg2), getObject(arg3)); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_clipboard_7f9a0c2ee555ca31 = function(arg0) { + const ret = getObject(arg0).clipboard; + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_debug_64711eb2fc6980ef = function(arg0, arg1, arg2, arg3) { + console.debug(getObject(arg0), getObject(arg1), getObject(arg2), getObject(arg3)); + }; + imports.wbg.__wbg_error_ef9a0be47931175f = function(arg0) { + console.error(getObject(arg0)); + }; + imports.wbg.__wbg_error_02ffd4185a83fe18 = function(arg0, arg1) { + console.error(getObject(arg0), getObject(arg1)); + }; + imports.wbg.__wbg_error_00c5d571f754f629 = function(arg0, arg1, arg2, arg3) { + console.error(getObject(arg0), getObject(arg1), getObject(arg2), getObject(arg3)); + }; + imports.wbg.__wbg_info_d60a960a4e955dc2 = function(arg0, arg1, arg2, arg3) { + console.info(getObject(arg0), getObject(arg1), getObject(arg2), getObject(arg3)); + }; + imports.wbg.__wbg_log_de258f66ad9eb784 = function(arg0, arg1, arg2, arg3) { + console.log(getObject(arg0), getObject(arg1), getObject(arg2), getObject(arg3)); + }; + imports.wbg.__wbg_warn_be542501a57387a5 = function(arg0, arg1, arg2, arg3) { + console.warn(getObject(arg0), getObject(arg1), getObject(arg2), getObject(arg3)); + }; + imports.wbg.__wbg_x_419967b8271dcf59 = function(arg0) { + const ret = getObject(arg0).x; + return ret; + }; + imports.wbg.__wbg_y_0f67486e0f88b265 = function(arg0) { + const ret = getObject(arg0).y; + return ret; + }; + imports.wbg.__wbg_drawArraysInstancedANGLE_89a45d6f51cd0483 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).drawArraysInstancedANGLE(arg1 >>> 0, arg2, arg3, arg4); + }; + imports.wbg.__wbg_drawElementsInstancedANGLE_6ac21f9a1ebe5f6b = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).drawElementsInstancedANGLE(arg1 >>> 0, arg2, arg3 >>> 0, arg4, arg5); + }; + imports.wbg.__wbg_vertexAttribDivisorANGLE_d5931335aaf0c735 = function(arg0, arg1, arg2) { + getObject(arg0).vertexAttribDivisorANGLE(arg1 >>> 0, arg2 >>> 0); + }; + imports.wbg.__wbg_addEventListener_cbe4c6f619b032f3 = function() { return handleError(function (arg0, arg1, arg2, arg3) { + getObject(arg0).addEventListener(getStringFromWasm0(arg1, arg2), getObject(arg3)); + }, arguments) }; + imports.wbg.__wbg_addEventListener_1fc744729ac6dc27 = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).addEventListener(getStringFromWasm0(arg1, arg2), getObject(arg3), getObject(arg4)); + }, arguments) }; + imports.wbg.__wbg_removeEventListener_dd20475efce70084 = function() { return handleError(function (arg0, arg1, arg2, arg3) { + getObject(arg0).removeEventListener(getStringFromWasm0(arg1, arg2), getObject(arg3)); + }, arguments) }; + imports.wbg.__wbg_charCode_b0f31612a52c2bff = function(arg0) { + const ret = getObject(arg0).charCode; + return ret; + }; + imports.wbg.__wbg_keyCode_72faed4278f77f2c = function(arg0) { + const ret = getObject(arg0).keyCode; + return ret; + }; + imports.wbg.__wbg_altKey_6dbe46bf3ae42d67 = function(arg0) { + const ret = getObject(arg0).altKey; + return ret; + }; + imports.wbg.__wbg_ctrlKey_fd79f035994d9387 = function(arg0) { + const ret = getObject(arg0).ctrlKey; + return ret; + }; + imports.wbg.__wbg_shiftKey_908ae224b8722a41 = function(arg0) { + const ret = getObject(arg0).shiftKey; + return ret; + }; + imports.wbg.__wbg_metaKey_cdd15bf44efb510e = function(arg0) { + const ret = getObject(arg0).metaKey; + return ret; + }; + imports.wbg.__wbg_key_ad4fc49423a94efa = function(arg0, arg1) { + const ret = getObject(arg1).key; + const ptr0 = passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + const len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbg_code_06787cd3c7a60600 = function(arg0, arg1) { + const ret = getObject(arg1).code; + const ptr0 = passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + const len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbg_getModifierState_135305ae40997dc7 = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getModifierState(getStringFromWasm0(arg1, arg2)); + return ret; + }; + imports.wbg.__wbg_bindVertexArrayOES_84540c072ea96b75 = function(arg0, arg1) { + getObject(arg0).bindVertexArrayOES(getObject(arg1)); + }; + imports.wbg.__wbg_createVertexArrayOES_00a5c523e5b17eff = function(arg0) { + const ret = getObject(arg0).createVertexArrayOES(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_deleteVertexArrayOES_98b83132b3d85825 = function(arg0, arg1) { + getObject(arg0).deleteVertexArrayOES(getObject(arg1)); + }; + imports.wbg.__wbg_readText_e285bdd5dcfea7ab = function(arg0) { + const ret = getObject(arg0).readText(); + return addHeapObject(ret); + }; + imports.wbg.__wbg_writeText_14e311eab7a9f719 = function(arg0, arg1, arg2) { + const ret = getObject(arg0).writeText(getStringFromWasm0(arg1, arg2)); + return addHeapObject(ret); + }; + imports.wbg.__wbg_setProperty_e489dfd8c0a6bffc = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).setProperty(getStringFromWasm0(arg1, arg2), getStringFromWasm0(arg3, arg4)); + }, arguments) }; + imports.wbg.__wbg_size_878ba1bf0c2ec606 = function(arg0) { + const ret = getObject(arg0).size; + return ret; + }; + imports.wbg.__wbg_type_ca7819eaadc2049f = function(arg0) { + const ret = getObject(arg0).type; + return ret; + }; + imports.wbg.__wbg_name_2473476082bed625 = function(arg0, arg1) { + const ret = getObject(arg1).name; + const ptr0 = passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + const len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbg_deltaX_6b627fd6f4c19e51 = function(arg0) { + const ret = getObject(arg0).deltaX; + return ret; + }; + imports.wbg.__wbg_deltaY_a5393ec7ac0f7bb4 = function(arg0) { + const ret = getObject(arg0).deltaY; + return ret; + }; + imports.wbg.__wbg_deltaMode_a90be314f5c676f1 = function(arg0) { + const ret = getObject(arg0).deltaMode; + return ret; + }; + imports.wbg.__wbg_matches_0ffc2232d99a6034 = function(arg0) { + const ret = getObject(arg0).matches; + return ret; + }; + imports.wbg.__wbg_addListener_19238ce0935173e6 = function() { return handleError(function (arg0, arg1) { + getObject(arg0).addListener(getObject(arg1)); + }, arguments) }; + imports.wbg.__wbg_removeListener_c08dac8493263a47 = function() { return handleError(function (arg0, arg1) { + getObject(arg0).removeListener(getObject(arg1)); + }, arguments) }; + imports.wbg.__wbg_matches_206d50bc7cb1f89e = function(arg0) { + const ret = getObject(arg0).matches; + return ret; + }; + imports.wbg.__wbg_pointerId_18be034781db46f3 = function(arg0) { + const ret = getObject(arg0).pointerId; + return ret; + }; + imports.wbg.__wbg_target_bf704b7db7ad1387 = function(arg0) { + const ret = getObject(arg0).target; + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_cancelBubble_8c0bdf21c08f1717 = function(arg0) { + const ret = getObject(arg0).cancelBubble; + return ret; + }; + imports.wbg.__wbg_preventDefault_3209279b490de583 = function(arg0) { + getObject(arg0).preventDefault(); + }; + imports.wbg.__wbg_stopPropagation_eca3af16f2d02a91 = function(arg0) { + getObject(arg0).stopPropagation(); + }; + imports.wbg.__wbg_clientX_e39206f946859108 = function(arg0) { + const ret = getObject(arg0).clientX; + return ret; + }; + imports.wbg.__wbg_clientY_e376bb2d8f470c88 = function(arg0) { + const ret = getObject(arg0).clientY; + return ret; + }; + imports.wbg.__wbg_offsetX_8891849b36542d53 = function(arg0) { + const ret = getObject(arg0).offsetX; + return ret; + }; + imports.wbg.__wbg_offsetY_1f52082687af467b = function(arg0) { + const ret = getObject(arg0).offsetY; + return ret; + }; + imports.wbg.__wbg_ctrlKey_4795fb55a59f026c = function(arg0) { + const ret = getObject(arg0).ctrlKey; + return ret; + }; + imports.wbg.__wbg_shiftKey_81014521a7612e6a = function(arg0) { + const ret = getObject(arg0).shiftKey; + return ret; + }; + imports.wbg.__wbg_altKey_2b8d6d80ead4bad7 = function(arg0) { + const ret = getObject(arg0).altKey; + return ret; + }; + imports.wbg.__wbg_metaKey_49e49046d8402fb7 = function(arg0) { + const ret = getObject(arg0).metaKey; + return ret; + }; + imports.wbg.__wbg_button_2bb5dc0116d6b89b = function(arg0) { + const ret = getObject(arg0).button; + return ret; + }; + imports.wbg.__wbg_buttons_047716c1296e3d1c = function(arg0) { + const ret = getObject(arg0).buttons; + return ret; + }; + imports.wbg.__wbg_movementX_f5947c282009d740 = function(arg0) { + const ret = getObject(arg0).movementX; + return ret; + }; + imports.wbg.__wbg_movementY_2c81eed268321a0a = function(arg0) { + const ret = getObject(arg0).movementY; + return ret; + }; + imports.wbg.__wbg_now_8172cd917e5eda6b = function(arg0) { + const ret = getObject(arg0).now(); + return ret; + }; + imports.wbg.__wbg_getItem_845e475f85f593e4 = function() { return handleError(function (arg0, arg1, arg2, arg3) { + const ret = getObject(arg1).getItem(getStringFromWasm0(arg2, arg3)); + var ptr0 = isLikeNone(ret) ? 0 : passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + var len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }, arguments) }; + imports.wbg.__wbg_setItem_9c469d634d0c321c = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).setItem(getStringFromWasm0(arg1, arg2), getStringFromWasm0(arg3, arg4)); + }, arguments) }; + imports.wbg.__wbg_bufferData_d6fac0d761e08fec = function(arg0, arg1, arg2, arg3) { + getObject(arg0).bufferData(arg1 >>> 0, arg2, arg3 >>> 0); + }; + imports.wbg.__wbg_bufferData_a33528a74dd300f4 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).bufferData(arg1 >>> 0, getObject(arg2), arg3 >>> 0); + }; + imports.wbg.__wbg_bufferSubData_a116fea11850b38f = function(arg0, arg1, arg2, arg3) { + getObject(arg0).bufferSubData(arg1 >>> 0, arg2, getObject(arg3)); + }; + imports.wbg.__wbg_compressedTexSubImage2D_30943b654d04ee44 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8) { + getObject(arg0).compressedTexSubImage2D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7 >>> 0, getObject(arg8)); + }; + imports.wbg.__wbg_readPixels_db685489e1779d63 = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7) { + getObject(arg0).readPixels(arg1, arg2, arg3, arg4, arg5 >>> 0, arg6 >>> 0, getObject(arg7)); + }, arguments) }; + imports.wbg.__wbg_texSubImage2D_cb339dd200dd1179 = function() { return handleError(function (arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9) { + getObject(arg0).texSubImage2D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7 >>> 0, arg8 >>> 0, getObject(arg9)); + }, arguments) }; + imports.wbg.__wbg_uniform2fv_3aad4d306a1cb8af = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform2fv(getObject(arg1), getArrayF32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform2iv_8c390eac30cb1de3 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform2iv(getObject(arg1), getArrayI32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform3fv_d1ef35c158c348e7 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform3fv(getObject(arg1), getArrayF32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform3iv_76acc51e8e6fe1a4 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform3iv(getObject(arg1), getArrayI32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform4fv_a513dc4d02f192d3 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform4fv(getObject(arg1), getArrayF32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniform4iv_19aa13960dc767c2 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).uniform4iv(getObject(arg1), getArrayI32FromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_uniformMatrix2fv_4173a282fcaa5508 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).uniformMatrix2fv(getObject(arg1), arg2 !== 0, getArrayF32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_uniformMatrix3fv_2b7de3010c8ed627 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).uniformMatrix3fv(getObject(arg1), arg2 !== 0, getArrayF32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_uniformMatrix4fv_f16e4a5553357886 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).uniformMatrix4fv(getObject(arg1), arg2 !== 0, getArrayF32FromWasm0(arg3, arg4)); + }; + imports.wbg.__wbg_activeTexture_02b7c73c76c2c06b = function(arg0, arg1) { + getObject(arg0).activeTexture(arg1 >>> 0); + }; + imports.wbg.__wbg_attachShader_f4d51147351a1906 = function(arg0, arg1, arg2) { + getObject(arg0).attachShader(getObject(arg1), getObject(arg2)); + }; + imports.wbg.__wbg_bindBuffer_8b5135aa633680f5 = function(arg0, arg1, arg2) { + getObject(arg0).bindBuffer(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_bindFramebuffer_080d0b0cf22e1645 = function(arg0, arg1, arg2) { + getObject(arg0).bindFramebuffer(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_bindRenderbuffer_6da549f066c1b8a5 = function(arg0, arg1, arg2) { + getObject(arg0).bindRenderbuffer(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_bindTexture_6f1dec563e82e818 = function(arg0, arg1, arg2) { + getObject(arg0).bindTexture(arg1 >>> 0, getObject(arg2)); + }; + imports.wbg.__wbg_blendColor_3bea829c60b1f6f2 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).blendColor(arg1, arg2, arg3, arg4); + }; + imports.wbg.__wbg_blendEquation_5d5abe2ee10109a9 = function(arg0, arg1) { + getObject(arg0).blendEquation(arg1 >>> 0); + }; + imports.wbg.__wbg_blendEquationSeparate_fa6aebc5cd0c5285 = function(arg0, arg1, arg2) { + getObject(arg0).blendEquationSeparate(arg1 >>> 0, arg2 >>> 0); + }; + imports.wbg.__wbg_blendFunc_49ea28240d4c1084 = function(arg0, arg1, arg2) { + getObject(arg0).blendFunc(arg1 >>> 0, arg2 >>> 0); + }; + imports.wbg.__wbg_blendFuncSeparate_9fef8acb74d50df5 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).blendFuncSeparate(arg1 >>> 0, arg2 >>> 0, arg3 >>> 0, arg4 >>> 0); + }; + imports.wbg.__wbg_colorMask_bc13c97d0db65962 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).colorMask(arg1 !== 0, arg2 !== 0, arg3 !== 0, arg4 !== 0); + }; + imports.wbg.__wbg_compileShader_22b038faa1f49857 = function(arg0, arg1) { + getObject(arg0).compileShader(getObject(arg1)); + }; + imports.wbg.__wbg_copyTexSubImage2D_e815f93a9ef52dd2 = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8) { + getObject(arg0).copyTexSubImage2D(arg1 >>> 0, arg2, arg3, arg4, arg5, arg6, arg7, arg8); + }; + imports.wbg.__wbg_createBuffer_6e747d928c9ba46d = function(arg0) { + const ret = getObject(arg0).createBuffer(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createFramebuffer_9b5b0507480146cd = function(arg0) { + const ret = getObject(arg0).createFramebuffer(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createProgram_1c5f8dffd1066e71 = function(arg0) { + const ret = getObject(arg0).createProgram(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createRenderbuffer_69c2f0554298bf89 = function(arg0) { + const ret = getObject(arg0).createRenderbuffer(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createShader_4017d9fbc36659af = function(arg0, arg1) { + const ret = getObject(arg0).createShader(arg1 >>> 0); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_createTexture_4ce49e8a8c655124 = function(arg0) { + const ret = getObject(arg0).createTexture(); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_cullFace_aa9f8eea262690c0 = function(arg0, arg1) { + getObject(arg0).cullFace(arg1 >>> 0); + }; + imports.wbg.__wbg_deleteBuffer_6fd9bca7f8a6d9de = function(arg0, arg1) { + getObject(arg0).deleteBuffer(getObject(arg1)); + }; + imports.wbg.__wbg_deleteFramebuffer_2617e39d2c39b4da = function(arg0, arg1) { + getObject(arg0).deleteFramebuffer(getObject(arg1)); + }; + imports.wbg.__wbg_deleteProgram_e8636e3cb5a18a59 = function(arg0, arg1) { + getObject(arg0).deleteProgram(getObject(arg1)); + }; + imports.wbg.__wbg_deleteRenderbuffer_e5b3450b8b57b395 = function(arg0, arg1) { + getObject(arg0).deleteRenderbuffer(getObject(arg1)); + }; + imports.wbg.__wbg_deleteShader_89369612f61ec145 = function(arg0, arg1) { + getObject(arg0).deleteShader(getObject(arg1)); + }; + imports.wbg.__wbg_deleteTexture_5c40169772519141 = function(arg0, arg1) { + getObject(arg0).deleteTexture(getObject(arg1)); + }; + imports.wbg.__wbg_depthFunc_2ac2c797a8220f09 = function(arg0, arg1) { + getObject(arg0).depthFunc(arg1 >>> 0); + }; + imports.wbg.__wbg_depthMask_88ab181c23c32dcd = function(arg0, arg1) { + getObject(arg0).depthMask(arg1 !== 0); + }; + imports.wbg.__wbg_depthRange_5dccc27b5cdd74b3 = function(arg0, arg1, arg2) { + getObject(arg0).depthRange(arg1, arg2); + }; + imports.wbg.__wbg_disable_6835d16c2cd3fa26 = function(arg0, arg1) { + getObject(arg0).disable(arg1 >>> 0); + }; + imports.wbg.__wbg_disableVertexAttribArray_ab474d273ff59265 = function(arg0, arg1) { + getObject(arg0).disableVertexAttribArray(arg1 >>> 0); + }; + imports.wbg.__wbg_drawArrays_c0dcb4151e0bf007 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).drawArrays(arg1 >>> 0, arg2, arg3); + }; + imports.wbg.__wbg_drawElements_e09dbef58c8f099a = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).drawElements(arg1 >>> 0, arg2, arg3 >>> 0, arg4); + }; + imports.wbg.__wbg_enable_fc393941ac400f72 = function(arg0, arg1) { + getObject(arg0).enable(arg1 >>> 0); + }; + imports.wbg.__wbg_enableVertexAttribArray_3d21f4936ad4a378 = function(arg0, arg1) { + getObject(arg0).enableVertexAttribArray(arg1 >>> 0); + }; + imports.wbg.__wbg_framebufferRenderbuffer_6b8dd5a111d341e6 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).framebufferRenderbuffer(arg1 >>> 0, arg2 >>> 0, arg3 >>> 0, getObject(arg4)); + }; + imports.wbg.__wbg_framebufferTexture2D_499d1c21458d0113 = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).framebufferTexture2D(arg1 >>> 0, arg2 >>> 0, arg3 >>> 0, getObject(arg4), arg5); + }; + imports.wbg.__wbg_frontFace_5fd354be6327d46b = function(arg0, arg1) { + getObject(arg0).frontFace(arg1 >>> 0); + }; + imports.wbg.__wbg_getActiveUniform_fd021da851153e8c = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getActiveUniform(getObject(arg1), arg2 >>> 0); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_getParameter_585a5b83c595ada8 = function() { return handleError(function (arg0, arg1) { + const ret = getObject(arg0).getParameter(arg1 >>> 0); + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_getProgramInfoLog_e47d5073d57fb18d = function(arg0, arg1, arg2) { + const ret = getObject(arg1).getProgramInfoLog(getObject(arg2)); + var ptr0 = isLikeNone(ret) ? 0 : passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + var len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbg_getProgramParameter_eaf768a9b399b7cf = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getProgramParameter(getObject(arg1), arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_getShaderInfoLog_ec7e5b959e47645b = function(arg0, arg1, arg2) { + const ret = getObject(arg1).getShaderInfoLog(getObject(arg2)); + var ptr0 = isLikeNone(ret) ? 0 : passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + var len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbg_getShaderParameter_42a35b974329561c = function(arg0, arg1, arg2) { + const ret = getObject(arg0).getShaderParameter(getObject(arg1), arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_getUniformLocation_8e9cc276a231ddcd = function(arg0, arg1, arg2, arg3) { + const ret = getObject(arg0).getUniformLocation(getObject(arg1), getStringFromWasm0(arg2, arg3)); + return isLikeNone(ret) ? 0 : addHeapObject(ret); + }; + imports.wbg.__wbg_linkProgram_25cda5f9318ea316 = function(arg0, arg1) { + getObject(arg0).linkProgram(getObject(arg1)); + }; + imports.wbg.__wbg_pixelStorei_bee1e2da4cb1115b = function(arg0, arg1, arg2) { + getObject(arg0).pixelStorei(arg1 >>> 0, arg2); + }; + imports.wbg.__wbg_polygonOffset_4cba459d8eacb66d = function(arg0, arg1, arg2) { + getObject(arg0).polygonOffset(arg1, arg2); + }; + imports.wbg.__wbg_renderbufferStorage_4ceec9b17dbd1e76 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).renderbufferStorage(arg1 >>> 0, arg2 >>> 0, arg3, arg4); + }; + imports.wbg.__wbg_scissor_4b89b60091ee8f0e = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).scissor(arg1, arg2, arg3, arg4); + }; + imports.wbg.__wbg_shaderSource_a0001b8eab5d44f4 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).shaderSource(getObject(arg1), getStringFromWasm0(arg2, arg3)); + }; + imports.wbg.__wbg_stencilFuncSeparate_1f0226d5d3acaf47 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).stencilFuncSeparate(arg1 >>> 0, arg2 >>> 0, arg3, arg4 >>> 0); + }; + imports.wbg.__wbg_stencilMask_00541859199befd2 = function(arg0, arg1) { + getObject(arg0).stencilMask(arg1 >>> 0); + }; + imports.wbg.__wbg_stencilMaskSeparate_5e7b9b536eac0c5d = function(arg0, arg1, arg2) { + getObject(arg0).stencilMaskSeparate(arg1 >>> 0, arg2 >>> 0); + }; + imports.wbg.__wbg_stencilOpSeparate_153523493abc8ec8 = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).stencilOpSeparate(arg1 >>> 0, arg2 >>> 0, arg3 >>> 0, arg4 >>> 0); + }; + imports.wbg.__wbg_texParameteri_1b210b807f1ea723 = function(arg0, arg1, arg2, arg3) { + getObject(arg0).texParameteri(arg1 >>> 0, arg2 >>> 0, arg3); + }; + imports.wbg.__wbg_uniform1f_f60e1072e28b8c49 = function(arg0, arg1, arg2) { + getObject(arg0).uniform1f(getObject(arg1), arg2); + }; + imports.wbg.__wbg_uniform1i_50124a48de1da66b = function(arg0, arg1, arg2) { + getObject(arg0).uniform1i(getObject(arg1), arg2); + }; + imports.wbg.__wbg_uniform4f_5b62a0acebac4494 = function(arg0, arg1, arg2, arg3, arg4, arg5) { + getObject(arg0).uniform4f(getObject(arg1), arg2, arg3, arg4, arg5); + }; + imports.wbg.__wbg_useProgram_156511a425feb519 = function(arg0, arg1) { + getObject(arg0).useProgram(getObject(arg1)); + }; + imports.wbg.__wbg_vertexAttribPointer_63d2aef49627302b = function(arg0, arg1, arg2, arg3, arg4, arg5, arg6) { + getObject(arg0).vertexAttribPointer(arg1 >>> 0, arg2, arg3 >>> 0, arg4 !== 0, arg5, arg6); + }; + imports.wbg.__wbg_viewport_a93f3881c4202d5e = function(arg0, arg1, arg2, arg3, arg4) { + getObject(arg0).viewport(arg1, arg2, arg3, arg4); + }; + imports.wbg.__wbg_drawBuffersWEBGL_482a093ae5a4ad55 = function(arg0, arg1) { + getObject(arg0).drawBuffersWEBGL(getObject(arg1)); + }; + imports.wbg.__wbg_get_57245cc7d7c7619d = function(arg0, arg1) { + const ret = getObject(arg0)[arg1 >>> 0]; + return addHeapObject(ret); + }; + imports.wbg.__wbg_length_6e3bbe7c8bd4dbd8 = function(arg0) { + const ret = getObject(arg0).length; + return ret; + }; + imports.wbg.__wbg_new_1d9a920c6bfc44a8 = function() { + const ret = new Array(); + return addHeapObject(ret); + }; + imports.wbg.__wbg_newnoargs_b5b063fc6c2f0376 = function(arg0, arg1) { + const ret = new Function(getStringFromWasm0(arg0, arg1)); + return addHeapObject(ret); + }; + imports.wbg.__wbg_get_765201544a2b6869 = function() { return handleError(function (arg0, arg1) { + const ret = Reflect.get(getObject(arg0), getObject(arg1)); + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_call_97ae9d8645dc388b = function() { return handleError(function (arg0, arg1) { + const ret = getObject(arg0).call(getObject(arg1)); + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_new_0b9bfdd97583284e = function() { + const ret = new Object(); + return addHeapObject(ret); + }; + imports.wbg.__wbg_self_6d479506f72c6a71 = function() { return handleError(function () { + const ret = self.self; + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_window_f2557cc78490aceb = function() { return handleError(function () { + const ret = window.window; + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_globalThis_7f206bda628d5286 = function() { return handleError(function () { + const ret = globalThis.globalThis; + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_global_ba75c50d1cf384f4 = function() { return handleError(function () { + const ret = global.global; + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_of_d79bf3cec607f7a4 = function(arg0) { + const ret = Array.of(getObject(arg0)); + return addHeapObject(ret); + }; + imports.wbg.__wbg_push_740e4b286702d964 = function(arg0, arg1) { + const ret = getObject(arg0).push(getObject(arg1)); + return ret; + }; + imports.wbg.__wbg_call_168da88779e35f61 = function() { return handleError(function (arg0, arg1, arg2) { + const ret = getObject(arg0).call(getObject(arg1), getObject(arg2)); + return addHeapObject(ret); + }, arguments) }; + imports.wbg.__wbg_is_40a66842732708e7 = function(arg0, arg1) { + const ret = Object.is(getObject(arg0), getObject(arg1)); + return ret; + }; + imports.wbg.__wbg_resolve_99fe17964f31ffc0 = function(arg0) { + const ret = Promise.resolve(getObject(arg0)); + return addHeapObject(ret); + }; + imports.wbg.__wbg_then_11f7a54d67b4bfad = function(arg0, arg1) { + const ret = getObject(arg0).then(getObject(arg1)); + return addHeapObject(ret); + }; + imports.wbg.__wbg_then_cedad20fbbd9418a = function(arg0, arg1, arg2) { + const ret = getObject(arg0).then(getObject(arg1), getObject(arg2)); + return addHeapObject(ret); + }; + imports.wbg.__wbg_buffer_3f3d764d4747d564 = function(arg0) { + const ret = getObject(arg0).buffer; + return addHeapObject(ret); + }; + imports.wbg.__wbg_newwithbyteoffsetandlength_890b478c8d7226ff = function(arg0, arg1, arg2) { + const ret = new Int8Array(getObject(arg0), arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_newwithbyteoffsetandlength_698c5100ae9c3365 = function(arg0, arg1, arg2) { + const ret = new Int16Array(getObject(arg0), arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_newwithbyteoffsetandlength_7be13f49af2b2012 = function(arg0, arg1, arg2) { + const ret = new Int32Array(getObject(arg0), arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_newwithbyteoffsetandlength_d9aa266703cb98be = function(arg0, arg1, arg2) { + const ret = new Uint8Array(getObject(arg0), arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_new_8c3f0052272a457a = function(arg0) { + const ret = new Uint8Array(getObject(arg0)); + return addHeapObject(ret); + }; + imports.wbg.__wbg_set_83db9690f9353e79 = function(arg0, arg1, arg2) { + getObject(arg0).set(getObject(arg1), arg2 >>> 0); + }; + imports.wbg.__wbg_length_9e1ae1900cb0fbd5 = function(arg0) { + const ret = getObject(arg0).length; + return ret; + }; + imports.wbg.__wbg_newwithbyteoffsetandlength_5540e144e9b8b907 = function(arg0, arg1, arg2) { + const ret = new Uint16Array(getObject(arg0), arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_newwithbyteoffsetandlength_9cc9adccd861aa26 = function(arg0, arg1, arg2) { + const ret = new Uint32Array(getObject(arg0), arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_newwithbyteoffsetandlength_be22e5fcf4f69ab4 = function(arg0, arg1, arg2) { + const ret = new Float32Array(getObject(arg0), arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_newwithlength_f5933855e4f48a19 = function(arg0) { + const ret = new Uint8Array(arg0 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_subarray_58ad4efbb5bcb886 = function(arg0, arg1, arg2) { + const ret = getObject(arg0).subarray(arg1 >>> 0, arg2 >>> 0); + return addHeapObject(ret); + }; + imports.wbg.__wbg_set_bf3f89b92d5a34bf = function() { return handleError(function (arg0, arg1, arg2) { + const ret = Reflect.set(getObject(arg0), getObject(arg1), getObject(arg2)); + return ret; + }, arguments) }; + imports.wbg.__wbindgen_debug_string = function(arg0, arg1) { + const ret = debugString(getObject(arg1)); + const ptr0 = passStringToWasm0(ret, wasm.__wbindgen_malloc, wasm.__wbindgen_realloc); + const len0 = WASM_VECTOR_LEN; + getInt32Memory0()[arg0 / 4 + 1] = len0; + getInt32Memory0()[arg0 / 4 + 0] = ptr0; + }; + imports.wbg.__wbindgen_throw = function(arg0, arg1) { + throw new Error(getStringFromWasm0(arg0, arg1)); + }; + imports.wbg.__wbindgen_memory = function() { + const ret = wasm.memory; + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper723 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper725 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper727 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper729 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper731 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper733 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper735 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper737 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper739 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_32); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper741 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 90, __wbg_adapter_51); + return addHeapObject(ret); + }; + imports.wbg.__wbindgen_closure_wrapper3687 = function(arg0, arg1, arg2) { + const ret = makeMutClosure(arg0, arg1, 1128, __wbg_adapter_54); + return addHeapObject(ret); + }; + + return imports; +} + +function initMemory(imports, maybe_memory) { + +} + +function finalizeInit(instance, module) { + wasm = instance.exports; + init.__wbindgen_wasm_module = module; + cachedFloat32Memory0 = new Float32Array(); + cachedFloat64Memory0 = new Float64Array(); + cachedInt32Memory0 = new Int32Array(); + cachedUint32Memory0 = new Uint32Array(); + cachedUint8Memory0 = new Uint8Array(); + + wasm.__wbindgen_start(); + return wasm; +} + +function initSync(module) { + const imports = getImports(); + + initMemory(imports); + + if (!(module instanceof WebAssembly.Module)) { + module = new WebAssembly.Module(module); + } + + const instance = new WebAssembly.Instance(module, imports); + + return finalizeInit(instance, module); +} + +async function init(input) { + if (typeof input === 'undefined') { + input = new URL('hyperspeedcube_bg.wasm', import.meta.url); + } + const imports = getImports(); + + if (typeof input === 'string' || (typeof Request === 'function' && input instanceof Request) || (typeof URL === 'function' && input instanceof URL)) { + input = fetch(input); + } + + initMemory(imports); + + const { instance, module } = await load(await input, imports); + + return finalizeInit(instance, module); +} + +export { initSync } +export default init; diff --git a/hyperspeedcube/hyperspeedcube_bg.wasm b/hyperspeedcube/hyperspeedcube_bg.wasm new file mode 100644 index 0000000000000000000000000000000000000000..d26638f151730368c5df9a8fe180b8b7051a67ac 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Hypercubing

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3×3×3×3 in Magic Cube 4D

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Hypercubing is the hobby of solving twisty puzzles (such as the Rubik’s cube) in higher dimensions. The ways that twisty puzzles move are mathematically well defined, and can be generalized to higher spatial dimensions. These puzzles can then be visualized and simulated using computer software.

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The most well known 4D shape is the hypercube (also called the tesseract, 8-cell, octachoron, or 4-cube). It has 8 cubic sides that are called cells. Turning any of the cells involves rotating it like a cube to any of 24 orientations.

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The short article Abstracting Rubik’s Cube introduces a number of the hypercubing puzzles.

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Introduction to 4D

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This page aims to teach you the fundamentals of understanding higher dimensional space, some of the shapes that exist in them, and how we can turn them into twisty puzzles to play with.

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What is the 4th dimension?

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The dimensionality of a space/object is the number of coordinates needed to describe its position. First, think of a 1-dimensional number line. To describe a position on the number line, all you need is one set of numbers for the coordinate. In 2-dimensional space like a graph, you need 2 sets of numbers to represent a point, such as \((3,4)\). This continues on to higher dimensions, where you need more and more numbers in order to locate a point.

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The 4th dimension is a space where you need 4 sets of numbers to describe the coordinates of something. As 3D beings in a 3D universe, we can’t interact with or see the 4th dimension. But computers can easily simulate higher dimensions, and project it down to 2D so we can see it on a computer screen.

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Dimension Levels (by NerdBoy1392 - Own work, CC BY-SA 3.0, link)

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Dimensions

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Understanding 4D shapes

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Hypercube rotating

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tesseract

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Puzzles are (usually) based on physical geometric shapes, so you have to understand what 4D shapes there are. 4D shapes have sides, or cells, that take the form of 3D shapes, and enclose some amount of “hypervolume”. As the most popular 4D puzzles are the hypercubes, so it’s important to at least understand its geometry.

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A hypercube (also known as a tesseract, 8-cell, octachoron, or 4-cube) is a shape with 8 cubic sides. 3 sides meet at each edge, giving it the Schläfli symbol {\(4,3,3\)}. Watch the animation below, as the tesseract unfolds itself into its 3-dimensional net.

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4D tesseract unfolding into its 3D net of 8 cubes

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But of course, there are many other shapes besides the tesseract. Check out these resources for a deeper understanding:

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Making puzzles out of shapes

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To make twisty puzzles out of higher dimensional shapes, you have to slice it up! Adding 2 evenly spaced planar cuts along each dimension, we get the 3x3x3x3, or 34.

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3×3×3×3 in Magic Cube 4D

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Congratulations, you have completed the tutorial!

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Hypercubing community in-jokes

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Like any niche internet community, we’ve accumulated some in-jokes over the years. In an effort to be welcoming to newcomers, here’s a list of them!

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This page is specifically for in-jokes that have been used by multiple people over a relatively long time, and wouldn’t be obvious to newcomers.

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My carrerr

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On 2022-10-31, Grant did a one-handed solve of the physical 24 on a whim and immediately set a world record of 3:55.62, beating Rowan’s prior record by two and a half minutes and breaking three minute barriers. In response:

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Rowan: I worked so hard for my sub-7:00
+Rowan: wth is this
+Grant: first try too
+Grant: lol
+Grant: it’s uploading rn
+Rowan: wow
+Grant: first sub 6, sub 5, and sub 4
+Luna: rowan in shambles
+Luna: what will he do
+Rowan: my carrerr
+Rowan: it’s over

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A graph of Rowan's carrerr, which starts by sloping upward and then curves downward. It ends in an arrow, implying that Rowan's carrerr will continue to decline.

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Sticker soup

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Hactar: You can always just make a soup of flat stickers with rules about how to move them around. It just becomes less and less usable. Physical 24 is special because the rules are fairly straightforward.

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In the wake of the Rowan’s video The Quest to Build a 4D Rubik’s Cube, many people joined the server proposing “3D physical” puzzles, which are flat 2D constructions of 3D puzzles. Generally, hypercubers are only interested in these as a way to explain how 4D physical puzzles work, since “3D physical” puzzles aren’t mathematically interesting in their own right and they’re more difficult to use than real 3D puzzles. They tend to become a “soup” of stickers that move according to arbitrary rules, a generalization of twisty puzzles that throws away all geometric relations.

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Luna: how many times do I have to say this
+Luna: dimensional 👏 analogies 👏 don’t 👏 work 👏 for 👏 physical 👏 puzzle 👏 design 👏

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Designing physical versions of 4D+ puzzles is generally very difficult and is usually mathematically impossible to do well. In the end, most end up becoming sticker soup.

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A soup of Rubik's cube stickers, generated by Rowan using some image-generation AI

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Dunning-Kruger graph

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From Dunning-Kruger effect - Wikipedia

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The Dunning–Kruger effect is a cognitive bias whereby people with low ability, expertise, or experience regarding a type of task or area of knowledge tend to overestimate their ability or knowledge.

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The Dunning-Kruger effect is especially common in hypercubing, which lies at the intersection of puzzle theory and higher-dimensional geometry, two niche mathematical fields. Of course, you don’t need to be an expert in math to solve 4D puzzles, but if you have no 4D mathematical foundations then your understanding will probably be wrong at first.

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The Dunning-Kruger effect is sometimes described using this graph, where “competence” increases to the right and “confidence” increases upward:

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An unlabeled graph starting with a steep slope up from zero to the top, followed by a bowl shape that covers the majority of the graph

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Of course, in reality the effect is not quite so extreme, and varies from person to person.

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Jambling

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On 2023-01-03, in Hypercubers voice chat, a very sleepy Hactar mispronounced the word “jumbling” as “jambling.” Since then, some hypercubers have been using the phrases “jambling” or “jamble” instead of “jumbling”.

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Fake Anderson

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On 2023-01-12, in Hypercubers voice chat, Hactar and Luna were conversing with Anderson Taurence. During that time, a user named Jedi Cabanna Boy (Andreas Thompson) joined and continued the conversation, and Anderson left. But, their voices were similar enough that Hactar, who was distracted speedsolving 54, didn’t notice the switch, and repeatedly addressed Jedi as Anderson. Luna noticed and did not correct him.

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Once Hactar finally realized what was happening, Jedi changed their nickname on the server to Anderson (Beta), and later 🅱a n d e r s o n.

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Cactus

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On 2023-02-19, several members were brainstorming possible shapes for physical 5D puzzle pieces and named one of them “Cactus.”

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Banderson: then yeah I think (compound of four cubes)-generated tetragonal icositetrahedron is good; C4CTi or something
+Banderson: C4QTi ?
+troller: 🌵
+Banderson: oh wait no C4Ct.I
+Banderson: or better yet I’ll let akkei make a shorthand/pick a better name to describe it lol
+troller: Cacti
+Rowan: squeez cheezahedron
+Grant: Your shape, your name akkei
+Akkei: i still like the
+Akkei: we Do Not Have A Scheme for the cactus
+Grant: Cactus LOL
+…
+Grant: I don’t care what actual name we end up giving it, I’m calling it cactus in this channel

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Silence, derivative

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In early 2018, at the 13th Gathering for Gardner, Melinda’s physical 2x2x2x2 was shown to Ernő Rubik himself. It was reported that his only comment was that none of the derivative puzzles matter and that only his original invention is important.

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On the night of 2023-06-25, Kitcen lay awake, unable to sleep, their mind plagued with the thought of Ernő Rubik with lasers coming out of his eyes to blow up a physical 24 while denouncing it and calling it “derivative.”

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30 minutes later, Akkei responded wordlessly with this image:

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Exactly the exact image described above

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Grant Standingslice

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In earlier versions of hypercubing.xyz, individual letters such as R, F, etc. showed tooltips indicating their use in common notation. On 2023-01-06, Hactar posted a screenshot of the leaderboards page in which the “S” in “Grant S.” had a tooltip that said “Standing slice (parallel to F).”

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Hactar: Fun fact: the “S” in Grant’s name stands for “Standing slice”

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Mimcaf and Celse pign

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Many hypercubers play Minecraft and Celeste, and have adopted common misspellings of those games. Akkei was the first to use “mimcaf” on 2022-12-03, and Rowan was the first to use the phrase “celse pign” on 2023-09-30 as a misspelling of “Celeste ping.”

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That may I can

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On 2024-11-09, OmnipotentEntity sent the following message in #offtopic-or-memes:

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I don’t often remember my dreams. Last night I dreamt that xkcd started including puns of the phrase “that may I can.” And it was supposed to be a reference to something. And everyone but me got it, and no one would explain it to me.

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Several members then responded with various images punning on the phrase.

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A soup can says "That May I" +There is a stick figure for every month except for May, which has an open can +A stick figure wearing a black hat is standing on a platform holding a closed can and saying "You may not" to a stick figure who is not on the platform and is reaching for the can +An arrow labeled "THAT" pointing toward a can labeled "may I"

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Adaptability

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Hactar & Luna, 2024-04-15

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Neophyte Grigor had been learning under Master Yuwi for several months, strengthening his skills with the 3x3x3. Over time, he grew frustrated; he had come to the Temple of Intuition to learn to solve more exciting puzzles.

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“Master Yuwi, thank you for your mentorship. You have taught me many new methods of solving a 3x3x3, but are these useful? I had already learned the method CFOP from the Temple of the Swift Solve, and could solve the cube in just 20 seconds. These new methods are slower and more confusing; what is their use?”

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Master Yuwi nodded. “The techniques I have taught you are slower. Certainly your Method of the Swift Solve is superior.” She then handed him an Astroid Attack. “I trust you can solve this using CFOP.”

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The neophyte stared at the puzzle for a moment. It was like nothing he had ever seen before. Certainly the Orientation and Permutation algorithms of CFOP would be of no use here. Nonetheless, when he applied the slower methods of Commutators and Blockbuilding, he was able to completely solve the puzzle piece by piece. By the end, he had understood exactly the purpose of each and every move in his solution.

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At that moment, Grigor was enlightened.

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Applications of Algorithms

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Hactar, 2024-04-15

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Neophyte Cherto toiled into the late hours of the night, eager to complete their latest assignment: producing a written guide to their own solution to Rubik’s Puzzle of Four Dimensions. They had solved the puzzle once before, but Master Nazir had suggested that they solve it again before beginning their writing, for there was a case they had not yet encountered.

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As Cherto approached the end of the Orientation of the Last Cell of their 3x3x3x3, they found that only a single piece was twisted in place. Cherto, being wise in the ways of Isolation and the construction of Commutators, was easily able to solve this case. They carefully noted the sequence of moves they had performed, added it to their guide, and then continued on.

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Some days later, another disciple came upon Cherto and addressed them. “I am Neophyte Nima. I have nearly solved my puzzle, save for a single 3-color piece twisted in place. Its stickers are permuted in a cycle, and I cannot seem to rotate it without rotating another piece. Master Nazir said that you had recently solved this case and suggested that I learn from the solution from you.”

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Cherto excitedly withdrew the latest draft of their solution guide and began copying their algorithm onto a slip of parchment for Nima to read. They began: U R2 ...

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Later that day, Master Nazir found Nima still struggling on the same monotwist case, and inquired whether she had found help from Cherto. She replied: “Cherto was kind enough to provide their method, but I believe they copied it wrong, for my puzzle becomes scrambled when I apply their algorithm!”

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“Cherto has given you an algorithm?” Master Nazir peered over her shoulder. “I see you struggle with the application of Cherto’s algorithm. Did they explain the derivation of this algorithm to you?” Nima responded that they had not. Master Nazir left the room, leaving Nima to her struggles.

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Shortly after, Master Nazir returned with Neophyte Cherto. Cherto watched Nima diligently applying their algorithm to the hemimegaminx in front of her, unaware of its origins.

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At that moment, Cherto was enlightened.

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Behold!

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Hactar, 2024-04-15

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Neophyte Horatio had been studying hyperbolic tilings under Master Nikolai for nearly a month, and was learning well the ways of the petal, the fourfold corner, and orientability. He woke up to find an envelope outside his door containing a message from his mentor, Master Nikolai:

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Hyperbolic {8,6} 6-color tiling, face-turning

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Neophyte Horatio,
+I have fallen ill and cannot attend your instruction today. Enclosed is a hyperbolic tiling which I am sure you have solved before. I expect you will have it solved by tomorrow.
+- Master Nikolai

+
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The neophyte unfolded the puzzle and gasped in horror. The polygons of this tiling were eight-sided, and six met at each vertex! He had learned by now that each additional polygon at a vertex immensely increased the complexity of the cut interactions, so surely this puzzle would require some extensive sequences to isolate pieces. But he had no room! The puzzle had only six colors, and often the same face met multiple times at one vertex! Nonetheless, he began building an edge, a petal, another edge, another petal, … but as he inserted the second petal, the first petal popped out. Try as he might, he could not remove the second petal without replacing the first as well.

+

Horatio sat and pondered. He had heard fragments from Neophyte Atenea, who had been studying The Theory of Grips, and had an idea. He counted the colors, the cuts, the interactions between pieces, …

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At that moment, Horatio was enlightened.

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He solved the rest of the puzzle trivially, using not a single hyperbolic technique.

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Index

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Welcome, neophyte!

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Here you will find a selection of short stories of other Learners such as yourself, who have become enlightened on the path of Hypercubing. To start, you might want to read about Neophyte Cherto’s Application of Algorithms.

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This is a work of fiction. Resemblence to any persons, real or fictitious, is entirely coincidental. Resemblence to any puzzles, real or complex, is probably intentional.

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Projection

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Rowan, 2024-06-12

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Neophyte Reyhan had recently seen Neophyte Horatio ascend the dimensional barrier by conquering the legendary 3x3x3x3x3 puzzle, and wanted to join him in this accomplishment. But Reyhan was confused by the spell Horatio used for the construction of his puzzle. It seemed to show stickers and pieces in strange locations that she couldn’t make sense of.

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Later that night, Reyhan went to the altar and crafted her dimensional spell. Finally after several days of solving, she had solved the puzzle! Reyhan excitedly rushed to find Horatio, to tell him about her tremendous achievement, but Horatio was also confused about Reyhan’s puzzle.

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“Why do you have so many duplicate stickers?” he asked. “Furthermore, even with those duplicate stickers, you still cannot see every piece at the same time.”

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That’s when Master Nilu strolled past, casually solving a 3x3x3x3x3 (formed using yet another different spell). Both Reyhan and Horatio ran up to Nilu and started arguing about whose puzzle was correct.

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“My neoyphytes”, Nilu sighed as she guided them towards the Temple of the Sacred Hyperstone.

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“Reyhan, you will stand to the left of the Hyperstone. Horatio, stand on the right. I will stand in front.” After a brief pause to allow a moment of reflection, Master Nilu inquired “now, what are we looking at?”

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Reyhan and Horatio each described how the sacred hyperstone looked from their viewpoints…

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“But it is the same stone!” Nilu exclaimed.

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At that moment, Reyhan and Horatio were enlightened.

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Soup

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Hactar, Luna, Banderson, et al., 2024-04-15

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One autumn day, Neophyte Cherto approached their mentor, Master Nazir, with a new design for a puzzle. Cherto had heard of the Shrine of Manifestation, where the Priests of High Symmetry imbue 4-dimensional or even 5-dimensional symmetry unto ordinary 3-dimensional shapes, and decided to try their hand at designing such a spell. They had counted the stickers and the pieces of a 4D puzzle, and had constructed a 3D shape with the same stickers.

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Master Nazir, having been presented with the neophyte’s design, sighed deeply and gently shook his head as they walked through the garden together.

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“Young neophyte, how will the pieces twist? Have you accounted for the orientations of each piece? Do they change according to the moves?”

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Neophyte Cherto scrambled to fix the issues. “Well we can adjust the orientation of the pieces after each twist!”

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“And how, young neophyte, is that any better a puzzle than the soup we eat for lunch?”

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Cherto’s face fell. They knew that when incantations failed at the Shrine of Manifestation, the polytopes on the altar melted into a formless blob that the elders called “soup.”

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“But surely there must be a way to make this work! I have even constructed an analogous 2D representation of a similar 3D puzzle–“

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Master Nazir hushed them and gestured toward a nearby tree. “Surely there must be a way for you to climb this tree! I have even routed a path through a similar tree.”

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“But that tree is dead.” Cherto paused, then continued more confidently. “The branches may be weak; it cannot be compared to another tree.”

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At that moment, Cherto was enlightened.

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Square Prism

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Hactar & Luna, 2024-04-15

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Master Nazir was working on a new puzzle, a novel cut of a square-pentagon duoprism, when he noticed Neophyte Sama standing awkwardly, as though she were waiting for something.

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“Do you have a question for me?” Master Nazir asked, not bothering to look up from his dihedral angle calculations.

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“Y- yes. Master, what is this shape? I did not know a 4-dimensional shape could be constructed only from cubes and pentagonal prisms.”

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“You recognize cubes in this shape?”

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“Are these cells not cubes? They appear to be cubes.”

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Master Nazir stood up from his work and led the neophyte around the room. The puzzle workshop extended quite far in all directions, and several monks from other temples were working on other designs throughout.

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He asked, “Is this room a cube?”

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“It certainly appears to be, if the ceiling is as tall as the length of the wall.”

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“Are you able to measure it?”

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“No.”

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“Why not?”

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“I cannot walk on the wall.”

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“And yet you walk on the floor.”

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At that moment, Sama was enlightened.

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Twice Spun

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Hactar, 2024-04-15

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One day Acolyte Quenta found herself wandering the gardens, as she often did, and upon crossing a small bridge she came upon a curious puzzle. It appeared in the form of an ordinary 3x3x3, but with an odd coating on the stickers that presented a gradient over each piece that tilted strangely as she observed it from different angles. She took the cube and scrambled it, intending to return it once she had solved it.

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As she twisted the puzzle, she became sure that it was playing tricks on her. Even whole-puzzle rotations seemed to behave oddly. She had placed a blue piece next to an orange piece, but upon spinning the puzzle in her hands, the blue piece became orange and the orange piece had become blue! After struggling for a half-hour, managing only to pair a handful of pieces, she resolved to find Master Eiji, who was an expert in variations of the 3x3x3.

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Upon showing the mysterious puzzle to Master Eiji, he led her inside to a bare room with a column in the center. “Walk around the column and return to me. Then you shall learn how to solve this puzzle,” he instructed. Quenta gave him a quizzical look, but then walked around the column– and found that Master Eiji had disappeared! Furthermore, the room was slightly different; she could not see the entrance they had come in from, and instead there sat a bench against the wall. She walked another full circle around the column, and then found Master Eiji.

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“Master Eiji!” she exclaimed. “What strange geometry is this room constructed from? It is neither Spherical nor Hyperbolic, as the light does not seem to bend. But it cannot be Euclidean, because I circled the room twice before returning to you–” she stopped.

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At that moment, Quenta was enlightened.

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World Record History

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NameTimeDateProgram
Andrew Farkas (Hactar)1m 56s 420ms2023-05-13HSC
Grant Staten2m 03s 830ms2023-04-21HSC
Andrew Farkas (Hactar)2m 07s 300ms2023-04-08HSC
Grant Staten2m 11s 330ms2023-03-16HSC
Andrew Farkas (Hactar)2m 17s 970ms2023-03-11HSC
Grant Staten2m 21s 010ms2023-03-08HSC
Andrew Farkas (Hactar)2m 26s 460ms2023-02-18HSC
Andrew Farkas (Hactar)2m 29s 730ms2023-01-11HSC
Andrew Farkas (Hactar)2m 39s 470ms2023-01-08HSC
Andrew Farkas (Hactar)2m 41s 220ms2023-01-03HSC
Andrew Farkas (Hactar)2m 42s 810ms2022-12-30HSC
Grant Staten2m 59s 340ms2022-12-23HSC
Andrew Farkas (Hactar)2m 59s 970ms2022-12-22HSC
Grant Staten3m 06s 060ms2022-12-21HSC
Grant Staten3m 13s 970ms2022-12-21HSC
Grant Staten3m 16s 440ms2022-12-20HSC
Grant Staten3m 21s 520ms2022-12-15HSC
Grant Staten3m 23s 260ms2022-12-14HSC
Grant Staten3m 34s 430ms2022-12-14HSC
Grant Staten3m 45s 690ms2022-12-13HSC
Grant Staten3m 55s 320ms2022-12-10HSC
Grant Staten3m 58s 290ms2022-12-07HSC
Grant Staten4m 02s 600ms2022-12-03HSC
Grant Staten4m 09s 190ms2022-12-03HSC
Grant Staten4m 10s 840ms2022-12-02HSC
Andrew Farkas (Hactar)4m 30s 880ms2022-12-01HSC
Grant Staten4m 43s 860ms2022-11-30HSC
Andrew Farkas (Hactar)4m 50s 730ms2022-11-29HSC
Grant Staten4m 52s 940ms2022-11-27HSC
Andrew Farkas (Hactar)4m 58s 140ms2022-11-26HSC
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World record database for higher dimensional twisty puzzle speedsolving!

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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/leaderboards/other-leaderboards/index.html b/leaderboards/other-leaderboards/index.html new file mode 100644 index 00000000..61bb904e --- /dev/null +++ b/leaderboards/other-leaderboards/index.html @@ -0,0 +1,3940 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Other Leaderboards - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Other Leaderboards

+

Magic Cube 4D Hall of Fame +- The original Hall of Fame. New submissions are no longer accepted due to reaching 500 solvers.

+

Magic Cube 4D Extended Records +- Records for many other puzzles found in MC4D.

+

Magic Cube 5D Hall of Insanity +- List of everyone who has solved 25-75, and also FMC solutions.

+

Magic Cube 7D Solvers +- List of 6D and 7D solvers in MC7D.

+

Magic 120 Cell Solvers +- List of 120 cell solvers in M120C or MPU.

+

MagicTile Klein Bottle challenge +- List of the first 100 people to solve the Klein Bottle Rubik’s Cube from a competition video made by Mathologer.

+

Speedsolving Wiki List of Unofficial World Records +- List of a few speedsolving and FMC records.

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+ + + +
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+
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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/leaderboards/records/index.html b/leaderboards/records/index.html new file mode 100644 index 00000000..891d747b --- /dev/null +++ b/leaderboards/records/index.html @@ -0,0 +1,4245 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Records - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + +

Current World Records + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + +

+ + +
+View Options + + + + + + + + + + + + + + +
Time Format
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EventNameTimeDateProgram
3x3x3x3 SingleAndrew Farkas (Hactar)1m 56s 420ms2023-05-13HSC
3x3x3x3 Ao5Andrew Farkas (Hactar)2m 03s 180ms2024-05-23HSC
3x3x3x3 BlindfoldedSebastian Yuste1h 21m 17s 680ms2022-03-17MC4D
3x3x3x3 No FiltersPKEarth4m 48s 970ms2024-08-10HSC
2x2x2x2 SingleBilal Mourad16s 310ms2024-12-13HSC
2x2x2x2 Ao5Bilal Mourad22s 390ms2024-12-26HSC
2x2x2x2 BlindfoldedKevin Liu10m 12s 770ms2024-10-12MC4D
4x4x4x4 SingleAndrew Farkas (Hactar)7m 52s 610ms2024-04-24HSC
5x5x5x5 SingleAndrew Farkas (Hactar)19m 35s 750ms2024-05-16HSC
6x6x6x6 SingleGrant Staten56m 03s 940ms2023-06-02HSC
7x7x7x7 SingleGrant Staten1h 58m 21s 230ms2023-07-18HSC
1x3x3x3 SingleTymon Fro57s 460ms2023-11-11MPU
2x2x2x3 SingleRowan Fortier2m 58s 900ms2024-06-20MPU
2x2x3x3 SingleEmanuele Battistin22m 12s 690ms2024-06-12MPU
Physical 2x2x2x2 SingleThe Cube Dude31s 580ms2024-12-24-
Physical 2x2x2x2 Ao5The Cube Dude59s 890ms2024-12-27-
Physical 2x2x2x2 One-HandedTymon Fro2m 56s 980ms2023-10-08-
Physical 2x2x2x2 BlindfoldedAsa Kaplan47m 14s 000ms2022-08-08-
Physical 3x3x3x3 SingleThe Cube Dude56m 22s 110ms2024-07-28-
Virtual Physical 3x3x3x3 SingleGrant Staten9m 58s 950ms2023-05-03AKKEI-SIM
3-Layer Simplex SingleGrant Staten1m 17s 700ms2023-03-15MC4D
3x3x3x3x3 SingleRowan Fortier1h 52m 29s 730ms2024-07-01MC7D
2x2x2x2x2 SingleTymon Fro48m 22s 310ms2024-07-02MPU
4x4x4x4x4 SingleLuna Harran15h 20m 24s 490ms2024-04-26MC7D
Hemimegaminx SingleMilo Jacquet1m 31s 670ms2024-03-31MT
Canon-Cut Klein Quartic SingleEthan Davis22m 32s 810ms2024-09-23MT
3x3x3 in 2D projection with 1D Vision SingleGrant Staten3m 22s 200ms2023-07-05MC3D
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Hypercubing Leaderboard Submission Rules

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These rules are not written in legalese. Please don’t try to exploit them. Ask in the Hypercubers Discord if you have any questions, comments, or concerns about these rules.

+

These rules may be flexible, especially for new categories. For example, if you’re speedsolving a really big puzzle that no one’s done before, it might be reasonable to allow macros for it.

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General rules

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    +
  • These programs are recommended for speedsolving. If using a program not listed here, you should ask on the Discord first.
    • +
  • The puzzle must be solved from a full scramble.
      +
    • No twists may be applied between the puzzle being scrambled and the timer starting, or else the solve counts as a DNF. Full-puzzle rotations, including gyros, are allowed.
      • +
    • When the timer is stopped the puzzle must be solved, or else the solve counts as a DNF.
      • +
    • Twists may be applied after the timer is stopped, but these moves are discounted.
      • +
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    • +
  • Spectators may commentate, but must not help the solver.
    • +
  • Algorithm reference sheets are strongly discouraged.1
    • +
  • Reversing the scramble is not allowed; the solution must be novel.
    • +
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Speedsolving rules

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These rules apply generally to speedsolving submissions unless the category has specific rules.

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  • Macros are not allowed.
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    • Exception: macros consisting of a single move STM are allowed, provided that they are created during the solve or demonstrated in the video before the solve starts.
      • +
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    • +
  • Piece filters are allowed.
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  • All speedsolves require video evidence, such as a screen recording.
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    • The video must include scrambling the puzzle. When using Hyperspeedcube, the statusbar must be visible with the text “Scrambled fully.”
      • +
    • A timer with precision of 0.01 seconds or better must be visible on the screen during the solve. Timers built into the puzzle software are preferred.
      • +
    • Inspection time must not exceed 60 seconds. In other words, the solve timer must be started within 60 seconds of the puzzle being completely scrambled.
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    • +
  • If using keyboard controls:
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    • If possible, a keybinds reference must be visible during the solve.2
      • +
    • The keybinds should not be hyperoptimized for specific algorithms.3
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    • +
  • Log file is not required, but you should still save your PBs!
    • +
  • Using software to compute a solution to the puzzle state is not allowed.
    • +
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Blindsolving rules

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  • Macros are allowed during solving.
    • +
  • Piece filters/highlightings are allowed during memorization but not during solving.
    • +
  • All blindsolves require video evidence of the solver that clearly shows the screen and that they are not cheating by using some external reference.
    • +
  • The solver does not need to be blindfolded, but the colors on the puzzle must be invisible during the solve.
    • +
  • Using software to compute a solution to the puzzle state is not allowed.
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  • Besides those rules, typical blindsolving rules apply:
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    • Final time = memorization time + solving time.
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    • Writing anything down isn’t allowed.
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    • +
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Real-life puzzle rules

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  • All real-life puzzle solves must have video evidence that shows the puzzle fully in frame throughout the solve and shows the entire scrambling and solving process as well as the timer.
    • +
  • Scrambles must be generated by a computer program that produces a random state or at least this many random moves:
      +
    • Physical 24: see below
      • +
    • Hemimegaminx: 30 moves
      • +
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    • +
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Physical 24 rules

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See Canonical Moves for more details.

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  • All rules for real-life puzzles apply.
    • +
  • Allowed moves for physical 2x2x2x2 events are limited to 2 categories, the canonical moveset originally made by the hypercubing mailing list as well as a few additional moves:
      +
    • Canonical moves:
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      • All R and L cell rotations
        • +
      • U/D y2
        • +
      • F/B z2
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      • I/O x* (a.k.a. axial twist)
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      • Any of many equivalent gyro algorithms
        • +
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      • +
    • Additional moves:
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      • U/D x2/z2
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      • F/B x2/y2
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      • I y2/z2
        • +
      +
      • +
    +
    • +
  • For 24, scrambles must be generated by this page or any equivalent scramble generator.
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    • Hand scrambles are no longer allowed, as they have an increased chance to have an easier solution than random state scrambles.
      • +
    • It’s ok if you mess up the computer generated scramble a little bit. It should still be way more random than any hand scramble.
      • +
    +
    • +
  • Any full puzzle rotation that doesn’t change the state (including the Gyro) may be applied to the puzzle during inspection; other moves may not.
    • +
+

Fewest-moves solving rules

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    +
  • Log files are required for submission.
    • +
  • Video evidence is not required.
    • +
  • Collaboration is allowed; each collaborator must be given the option to be credited for the solve, and should be credited if their contributions were significant.
    • +
  • Moves are counted using STM.
    • +
  • Use of computer software to generate a partial or full solution is allowed, but must be noted in the submission.
    • +
+

Principles when writing new speedsolving software

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If you’re thinking of writing a new program for speedsolving or suggesting a new feature for Hyperspeedcube, here are some principles we try to follow:

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    +
  • No more than one move per keypress.
    • +
  • If your program has keybinds (especially customizable ones), add some reference that shows what keys are being pressed and what they’re doing.
    • +
  • New features should hopefully make solving strategy more interesting. Some examples:
      +
    • Macros on the 34 encourage commutator spam, which is very boring compared to layer-by-layer or F2L methods, so they are not allowed for speedsolves.
      • +
    • Piece filters reduce time spent looking for pieces, which is boring to do and boring to watch. There’s a trade-off there: piece filters encourage a stricter solving order, which is less interesting in a way, but this also enables 3-Block, a very interesting method.
      • +
    +
    • +
  • New features should ideally have some justification based on 3D speedsolving. For example, multiple keybind sets is analogous to different grips on a 3D puzzle.4
    • +
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    +
  1. +

    We can’t stop you from using an algorithm reference, but it’s not in the spirit of the competition and it’ll probably slow you down. ↩

    +
    2. +
  2. +

    If the program doesn’t have a built-in keybinds reference (such as akkei’s physical 34 simulator) then you probably don’t need one. ↩

    +
    3. +
  3. +

    In particular, you should not be able to execute a meaningful algorithm by pressing a sequence of keys in a line on the keyboard. Keybinds should be laid out in a way that makes sense moreso than a way that is useful for an algorithm. A keybind set optimized for general RKT is fine. If you have a question about this, ask on the Discord. ↩

    +
    4. +
  4. +

    You might hold a 33 differently to execute <M,U> moves more efficiently compared to <R,U,F> moves. Similarly, a 4D creature might have a specialized grip on the 34 for RKT compared to general moves. ↩

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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/leaderboards/solvers/adam/index.html b/leaderboards/solvers/adam/index.html new file mode 100644 index 00000000..15cdda10 --- /dev/null +++ b/leaderboards/solvers/adam/index.html @@ -0,0 +1,4646 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Adam Marcellus Kelly - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Rankings

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EventRankTimeDateProgram
3x3x3x3 Single93m 28s 220ms2023-08-15HSC
3x3x3x3 Ao543m 58s 960ms2023-08-15HSC
2x2x2x2 Single + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 323s 680ms2024-08-07HSC
2x2x2x2 Ao5 + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 328s 080ms2024-08-28HSC
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History

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93m 28s 220ms2023-08-15HSC
3m 49s 300ms2023-01-12HSC
4m 34s 620ms2023-01-10HSC
5m 44s 860ms2023-01-09HSC
6m 58s 670ms2023-01-09HSC
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43m 58s 960ms2023-08-15HSC
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RankTimeDateProgram
+ + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 323s 680ms2024-08-07HSC
25s 790ms2024-06-29HSC
26s 370ms2024-06-29HSC
26s 490ms2023-09-23HSC
39s 400ms2023-01-16HSC
43s 540ms2023-01-16HSC
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RankTimeDateProgram
+ + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 328s 080ms2024-08-28HSC
31s 310ms2024-07-16HSC
33s 040ms2024-07-01HSC
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EventRankTimeDateProgram
Hemimegaminx Single518m 47s 120ms2024-03-21-
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History

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518m 47s 120ms2024-03-21-
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EventRankTimeDateProgram
3x3x3x3 Single2612m 13s 260ms2022-11-24HSC
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History

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2612m 13s 260ms2022-11-24HSC
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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/leaderboards/solvers/asa/index.html b/leaderboards/solvers/asa/index.html new file mode 100644 index 00000000..965cb92c --- /dev/null +++ b/leaderboards/solvers/asa/index.html @@ -0,0 +1,4281 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Asa Kaplan - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Asa Kaplan

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Physical 2x2x2x2 Blindfolded + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 147m 14s 000ms2022-08-08-
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+ + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 147m 14s 000ms2022-08-08
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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/leaderboards/solvers/battistin/index.html b/leaderboards/solvers/battistin/index.html new file mode 100644 index 00000000..e8dbcf64 --- /dev/null +++ b/leaderboards/solvers/battistin/index.html @@ -0,0 +1,4772 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Emanuele Battistin - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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EventRankTimeDateProgram
3x3x3x3 Single155m 29s 780ms2024-04-06HSC
3x3x3x3 Ao577m 42s 360ms2024-03-24HSC
2x2x2x2 Single121m 25s 170ms2024-06-01HSC
4x4x4x4 Single959m 55s 810ms2023-11-10HSC
2x2x2x3 Single + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 35m 52s 720ms2024-05-26MPU
2x2x3x3 Single + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 122m 12s 690ms2024-06-12MPU
Hemimegaminx Single45m 22s 970ms2024-03-26MT
Canon-Cut Klein Quartic Single41h 33m 54s 020ms2024-06-20MT
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155m 29s 780ms2024-04-06HSC
5m 50s 990ms2024-03-29HSC
6m 19s 690ms2023-11-06HSC
7m 24s 840ms2023-11-05HSC
7m 54s 790ms2023-11-04HSC
8m 18s 630ms2023-11-02HSC
8m 38s 990ms2023-10-31HSC
9m 54s 040ms2023-10-29HSC
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77m 42s 360ms2024-03-24HSC
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121m 25s 170ms2024-06-01HSC
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959m 55s 810ms2023-11-10HSC
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+ + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + 122m 12s 690ms2024-06-12MPU
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45m 22s 970ms2024-03-26MT
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41h 33m 54s 020ms2024-06-20MT
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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/methods/2x2x2x2-physical/grant/index.html b/methods/2x2x2x2-physical/grant/index.html new file mode 100644 index 00000000..0e7e6e4c --- /dev/null +++ b/methods/2x2x2x2-physical/grant/index.html @@ -0,0 +1,4084 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Grant's OBC Method - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Grant’s OBC Method for physical 2x2x2x2

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Grant’s OBC Method uses only 3 gyros, making it much faster than Rowan’s OBC Method. It works by secretly setting up to an H OCLL case while orienting both cells.

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Steps

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Inspection

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Before starting the solve, inspect the puzzle for an opposite colour pair that you can easily get exactly 8 stickers from that pair to be on the y-axis. They can be in any spot, as long as they are pointing up or down.

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Getting 8 Oriented to Y-axis, then gyro

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This is done in just a few simple moves, and should be fully planned during inspection for speedsolving. This step is very difficult to get used to, but gets easier over time. Once it looks like the image above, perform the gyro algorithm such that those 8 stickers end up on the x-axis (the “pipped” spots).

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Orient the remaining 8 to y-axis

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Now that 8 corners have been oriented, you can do any moves (besides a gyro), and not affect their orientation. Pair up the remaining 8 so that they are all on the same cell, and oriented to the y-axis.

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Finish orientation

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Rotate the cell with the pieces you just paired up to the w-axis (facing in/out). Now perform the gyro algorithm, and cancel into the 5 move H OCLL case, and then gyro again.

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GRCL Method

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The Grant Rowan Connor Luna method, or GRCL (pronounced gurkle) is a combination of the best parts of several methods, including Grant’s OBC method, Rowan’s OBC method, Luna’s cell blockbuilding technique, and Connor Lindsay’s P4L algorithms. The name GRCL was proposed by Hypercubers Discord member Jasper on July 18 2022.

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Steps

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  • Orient Both Cells using either Grant’s method or Rowan’s method (whatever looks better in that particular scramble)
    • +
  • Separate the oriented colours of both cells while orienting 2 layers on those cells by holding the puzzle vertically and using Luna’s method
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  • Permuate both layers of both cell using Rowan’s updated version of Connor’s P4L algorithms
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Luna’s Method for 2x2x2x2

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A method that Luna developed in 2017-18 making heavy use of an IU move sequence.

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Steps

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    +
  • Orient the D cell with opposite colours
    • +
  • Orient the U cell making use of the IU move
    • +
  • Permute a single cell via blockbuilding
    • +
  • Permute the last cell using RKT
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Resources

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P4L

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History

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PBLBC/P4L/PAL is an advanced algorithm set for speedsolving Melinda’s Physical 2x2x2x2. It is used after orientation and separation of 2 opposite cells, followed by (or simultaneously) orienting both layers on both of those cells. This leaves you with two 2x2x2 PBL cases, one on each cell. The original idea for this came from Connor Lindsay, who created most of the algorithms in his algorithm sheet.

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2 years later, Rowan tried optimizing the algorithms, and compiling them in a new document, which can be found here.

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Algorithms

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Each layer can be in 3 different permutations: solved, adjacent swap, and opposite swap. This gives us 81 cases, but some of these are impossible. Cases can also be rearranged via Case Manipulation, further reducing the number of cases to 29. Practically however, only a few algorithms are used because all the bad cases are really slow to execute.

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Big 3D Cases

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Big 3D cases are the cases where you have 2 pairs of the same layers, such as OOAA. In such a case, you can just use the normal 23 PBL algorithm for that case (but it must be a version with the moveset <R2,L2,F2,B2,U*,D*> only, due to the restrictions in turning the puzzle).

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Rowan’s OBC Method for 2x2x2x2

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Rowan developed this method for Orienting Both Cells while trying to solve the physical 2x2x2x2 for the first time in a YouTube video.

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Steps

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Inspection

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Before starting the solve, inspect the puzzle for an opposite colour pair that has 4 or fewer stickers oriented to the Left/Right axis (x-axis). They can be made of any colour in the colour pair, and can be in any spot on the puzzle.

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4 or less on 2×2×2×2

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4 red/orange corners oriented to L/R

+

Getting 8 Oriented to Y-axis

+

Hold the puzzle vertically, and use block building to orient 8 pieces on a cell to the y-axis using the same colour pair that has 4 or less from inspection. This step is intuitive and does not require any RKT.

+

8 to y on 2×2×2×2

+

8 red/orange corners oriented to U/D

+

Getting 12 Oriented to Y-axis

+

Now use RKT to orient a layer of 4 stickers on the other cell, leaving you with a “last layer” of 4 unoriented pieces on one of the cells.

+

12 to y on 2×2×2×2

+

12 red/orange corners oriented to U/D

+

Fixing the Last Layer

+

Gyro the puzzle such that the 12 pieces that were oriented end up on the x-axis. Now you should have 4 (or fewer) stickers from that colour pair oriented in other directions besides the x-axis. All you need to do is use RKT to make it into a layer that looks like a possible OCLL case, not worrying about messing up the other 12 corners.

+

H case on 2×2×2×2

+

an H OCLL case set up on LU

+

From here, gyro back. Now you will have a normal OCLL case that you can solve using RKT. After doing that algorithm, you should have all 16 corners oriented to the y-axis. Finally, you just gyro to put them all on the x-axis, and OBC is solved.

+

Analysis

+

This technique takes more moves than other known OBC methods, and it uses up to 4 gyros. Besides the skip case where everything is already oriented, the best possible case for this technique uses 3 gyros, making it slower than other techniques. It also requires knowing the OCLL algorithms, which other techniques don’t.

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4tega

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4tega is the 4-dimensional equivalent of the Ortega Method on 23. There are 2 main variants, as there is no perfect way to generalize this method to 4d.

+

Variant 1

+

Orient Both Cells

+

Start by orienting a single side, using either colour from that side or its opposite side. This can easily be done without RKT by using blockbuilding. Next, use 24 OLC strategies to orient the other cell. Because the first cell isn’t solved, you don’t have to worry about messing it up, allowing you to save moves.

+

Separate colors

+

Separate the colors of the cells that are oriented, leaving you with 2 23 solves.

+

Permute Both Cells

+

Solve the first cell like a 23 without RKT, then solve the other cell like a 23, but using RKT.

+

Variant 2

+

Orient Both Cells

+

Same as in Variant 1.

+

Separate colors

+

Same as in Variant 1.

+

Orient Both Layers of Both Cells

+

Use RKT to orient both layers of both cells, just like OBL in 3D Ortega. This can be done using the OCLL algorithms.

+

P4L/PBLBC

+

Permtue all 4 layers of both cells at once using algorithms. Some algorithms can be found here.

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Cell by Cell

+

Cell by Cell is the 4D equivalent of Layer by Layer.

+

1. Solve the first cell

+

Completely solve one of the sides of the puzzle. You can do this by blockbuilding, or by getting all the same color one on side and then using RKT to permute the pieces around until they are solved.

+

2. Solve the last cell

+

a. Orient the last cell

+

Orient the last cell by using RKT to set up OCLL cases, and then using RKT to perform those algorithms.

+

b. Permute the last cell

+

Permute the last cell like a 23 using RKT. You may run into RKT parity in this step, or you can avoid it by using certain last layer algorithms.

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3-Block

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3-Block is a method designed for quickly speedsolving the 34. Many speedsolving world records have been set using this method. It was primarily invented by Luna and HactarCE, and has been described as “ZZ without EO”. 3-Block is most analogous to the 33 method FreeFOP (which omits a single cross piece in order to pair pieces easier). Compared to 4D CFOP, 3-Block uses ~20% fewer moves.

+
+

Prerequisites

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Summary

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  1. 4-Cross — Solve 4 out of the 6 cross pieces
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  2. Middle Block — Solve 4 2c3c pairs betwen the 4 cross pieces
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  3. Left Block — Blockbuild or pair pieces to solve the Left Cell
    4. +
  4. Right Block — Blockbuild or pair pieces to solve the Right Cell
    5. +
  5. OLL-4 - Orient LC 2c, 3c, and 4c pieces using 3D algorithms
    6. +
  6. PLL-4 - Permute LC 2c, 3c, and 4c pieces using 3D techniques
    7. +
+

Steps

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4-Cross

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Solve the four 2c cross pieces in a ring in the M slice. Leave the L and R cross pieces unsolved.

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4-cross

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Middle Block

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Tip

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For the entirety of F2L, the last layer is held on the I cell, as opposed to CFOP where it is held on the U cell. This allows us to see more information at once.

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Create and insert 4 F2L-a (2c3c) pairs into the 4/6 cross. This will solve 2/3 of the M slice. Because the Left and Right cells don’t have their cross pieces, you can use them to aid with building and inserting the pairs.

+

Middle block

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Left Block

+

Solve the Left cell. This is done in 3 blocks, hence the name of the method. The first block consists of the cross edge, followed by two 2c3c F2L-a pairs that are opposite of each other. This solves the middle column of the left cell. The final two blocks consist of a 2c3c pair, and two 3c4c pairs.

+

Left block

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Right Block

+

Solve the Right cell. This is also done by breaking it up into the 3 blocks, except now you don’t have an empty opposite cell to aid you in making pairs. You could just solve the cross edge, and then finish the whole solve using CFOP style F2L and Last Cell. You can also do it the 3 blocks way, except having less freedom means that it is slightly trickier to set up the correct cases.

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Right block

+

Last Layer

+

This is done in the exact same way as CFOP.

+

Big cubes

+

Pair up only the pieces you need during the step of 3-block you’re on. For example: after solving 4 centers, pair up 4 cross ridges. For last cell, you can either just orient everything, and then do an RKT 43 solve, or you can pair everything and end up with an RKT 33 solve.

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CFOP

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3D CFOP can be easily implemented on the 4D Rubik’s Cube. This method was also called Sheerin-Zhao Method (Hybrid) V1, named after the people who discovered its higher dimensional equivalence.

+
+

Prerequisites

+ +
+

Summary

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    +
  1. Cross — Build a cross by solving six 2c pieces on the O or D cell
    2. +
  2. F2L-a — Join 12 2c+3c pairs together and insert them into the first two layers
    3. +
  3. F2L-b — Join 8 3c+4c pairs together and insert them into the first two layers
    4. +
  4. OLL-4 — Orient LC 2c, 3c, and 4c pieces using 3D EOLL and OCLL
    5. +
  5. PLL-4 — Permute 2c pieces of the LL using EPLL, then solve the rest like a 33
    6. +
+

Steps

+

Cross

+

Using the same techniques from 3D, intuitively place the cross 2c pieces such that they lie between their centers, correctly oriented. After this step, the cross should be kept on the D layer.

+

F2L-a

+

Find a pair of 2C and 3C pieces. Find a way to bring them onto the P slice using easy intuitve setups. Now, you should be able to pair them up using “normal” looking 3x3x3 moves.

+

Repeat this until you have solved all 12 2c3c pairs of F2L-a.

+
+

Misoriented pairs

+

It is possible to insert a pair into its slot, but rotated in place. Make sure to line it up so that it looks like a normal 3x3x3 case before inserting the pair. Oftentimes, this just means doing a Ux2 or Uz2 move beforehand.

+
+

F2L-b

+

Find any 3C piece that doesn’t have a U cell colour. +Find its respective 4C piece.

+
    +
  • If they both have the same coloured sticker on the U cell, use RKT on the U cell to pair them up.
    • +
  • If the 4C is stuck in a slot in the D cell, bring the edge over the slot such that its colour on the I cell matches the 4C’s colour on the I cell. Then use RKT to pair them up.
    • +
  • If the 3C is stuck in a slot in the middle layer, bring the 4C on top of it until its colour on the I cell matches the 3C’s colour on the I cell. Then use RKT to pair them up.
    • +
  • If none of the above cases occured, then you kind of just have to fiddle around with it or pick a different pair to solve.
    • +
+

Repeat for all 8 3c4c pairs of F2L-b.

+

OLL-4

+

2c OLL-4

+

Use EOLL algorithms from 2-look OLL to orient the 2C pieces. This can always be done in 2 EOLL algorithms (or less).

+

3c OLL-4

+

Use RKT on the last cell to set up the slice layers of the last cell into configurations that look like possible OCLL cases. Then use the OCLL algorithms to solve that case. This can always be done in 3 OCLL algorithms (or less)

+
+

3c monotwist

+

It’s possible to have just 1 3c piece twisted in place. To avoid this, make sure that your last OCLL algorithm will solve all of the 3c pieces. For example if you have 5 left, you can’t set it up into an H OCLL case, because that will solve 4/5, leaving you with 1 left. Instead, you can set it up into a Sune case, which would then leave you with 2 (which you can solve using a T or U case OCLL algorithm).

+
+

4c OLL-4

+

Use RKT on the last cell to set up the 4c pieces into possible OCLL cases. Rotate the last cell to U, such that your OCLL case is in the IU plane, then execute that algorithm with RKT on I.

+
+

4c monoflip

+

It’s possible to have just 1 4c piece flipped in place. To avoid this, make sure that your last OCLL algorithm will solve all of the 4c pieces. For example if you have 5 left, you can’t set it up into an H OCLL case, because that will solve 4/5, leaving you with 1 left. Instead, you can set it up into a Sune case, which would then leave you with 2 (which you can solve using a T or U case OCLL algorithm).

+
+

PLL-4

+

2c PLL-4

+

It is always possible to solve 2c permutation using just U-perms, but many cases have faster methods. There are seven cases for 2c PLL-4:

+
    +
  • Solved
    • +
  • Adjacent swap - Do a 90-degree twist of I to turn this case into a 3-cycle, then use a 3D U-perm algorithm to solve it, such as M2 U M U2 M' U M2 (7 STM) to cycle IL → IF → IR.
    • +
  • Opposite swap - Do a 90-degree twist of I to turn this case into a pair of adjacent swaps, then use a 3D Z-perm algorithm to solve it, such as M2 U M2 U M' U2 M2 U2 M' (9 STM) to swap IF ↔ IR and IB ↔ IL.
    • +
  • Adjacent 3-cycle (clockwise) - Use [[RD, ID], {1-2}LO] = (RD ID RU IU) RO (IF RF IB RB) (RI) (9 STM + 1 move RKT debt) to cycle IF → IU → IR.
    • +
  • Adjacent 3-cycle (counterclockwise) - Use [[RF, IF], {1-2}LI] = (RF IF RB IB) RI (ID RD IU RU) (RO) (9 STM + 1 move RKT debt) to cycle IR → IU → IF.
    • +
  • Double adjacent swap (right-handed) - Use [[RU, IU] IR2 RU, IDR] = (RU IU RD IFR RU) IDR (RD IFR RU ID RD) (IUR) (11 STM + 1 move offset) to swap IF ↔ IU and IR ↔ IB.
    • +
  • Double adjacent swap (left-handed) - Use [[RU, IU] IR2 RU, IUR] = (RU IU RD IFR RU) IUR (RD IFR RU ID RD) (IDR) (11 STM + 1 move offset) to swap IF ↔ ID and IR ↔ IB.
    • +
+

This is currently no known reliable way to recognize which case you have.

+

PLL-3

+

From here, you use RKT to solve the rest of the puzzle like a whole 33. The CFOP method is recommended for this because you arrive at this step inspectionless, meaning that in a speedsolve, you don’t really have the time to count Edge Orientation, or plan a Roux First Block. Finding 4 cross pieces is pretty easy inspectionless.

+
+

RKT parity

+

If the “top face” of the LL is 180 degrees off from the rest of the puzzle, you have to use a special 4D algorithm to correct this. See RKT for algorithms.

+
+
+

RKT parity avoidance

+

You can avoid RKT parity by using 2-look PLL. When you get to PLL, correct any RKT debt you have. Then put the solved LL corner in the UIFR spot. Now you can do whichever A-perm you have (clockwise or anticlockwise). Finally, just solve the 3c with EPLL algorithms.

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Octachoroux

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+

Octachoroux

+
    +
  • Algorithms: 3D Roux + RKT Parity
    • +
  • Move count: way too many
    • +
+
+

This method is Rowan Fortier’s attempt to bring the Roux method to the 3x3x3x3.

+

Summary

+
    +
  1. 1st Block — Solve a 1x2x3x3 block using blockbuilding techniques.
    2. +
  2. 2nd Block — Solve a 1x2x3x3 block on the other side of the puzzle.
    3. +
  3. CMLC — Orient and permute the corners of the U cell.
    4. +
  4. L/R — Solve the Left and Right cells.
    5. +
  5. M slice - Permute the M slice.
    6. +
+

Steps

+

1st Block

+

Start with the “cross piece” of the 1st block. Then build four 2c3c and four 3c4c pairs around it, to complete the 1x2x3x3 block. No RKT is needed for this, as none of the other pieces have been solved yet.

+

2nd Block

+

Use the free M slice to help you build the pairs to make the 2nd 1x2x3x3 block on the opposite side of the puzzle from the 1st block.

+

CMLC

+

Use the same strategies from CFOP, but only for the 4C pieces. This step feels like solving the last cell of a 24, except you have to use 34 algorithms so that you don’t mess up other pieces.

+

L/R

+

Insert the UR and UL 2c pieces using setup moves and EPLL algorithms. Now set up the 3c pieces that need to go to L/R into the IDF spot with the L/R colour on the I cell and the U colour on the D cell. Then move the spot where that L/R edge needs to go above that edge and insert that piece using the RKT algorithm of M D2 M’ D2 (2RO’ IF’ RO2 IF 2RO IF’ RO2 IF). Repeat this for all of the edges (pretty tedious).

+

M slice

+

Now all that’s left to be solve is the M slice of the puzzle. Notice how the M slice is more like a 33 than the PLC of CFOP. The centers are already permuted too! However, this step is much more painful than normal RKT PLC due to the strange parities that can happen.

+
+

tricky situations

+
    +
  • A 3c piece can look “mirrored” in place.
    • +
  • A single 3c piece can be rotated wrong.
    • +
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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/methods/big-cube-reduction/index.html b/methods/big-cube-reduction/index.html new file mode 100644 index 00000000..c3b96c17 --- /dev/null +++ b/methods/big-cube-reduction/index.html @@ -0,0 +1,3933 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Reduction (Big Cube) - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Big Cube Reduction

+
+

44 with paired 1c centers

+

4×4×4×4 with centers paired

+
+

Just like 3D reduction of big cubes, higher dimensional reduction works by pairing up groups of pieces with more and more colors until the nd is reduced into an (n-1)d. Pairing up pieces is done in a very similar way to the 3D method; If you can solve 33, 43, and 34, then you can solve 44.

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Dimensional Reduction

+
+

Dimensional reduced 34

+

3×3×3×3 dimensionally reduced into 3×3×3

+
+

Dimensional reduction is a technique to reduce an nd puzzle into an nd-1 puzzle. It does this by pairing up n-piece long bars (e.g. to reduce 34 into 33, pair up 3-piece long bars).

+

To get to the dimensionally reduced state from a solved 34, do only turns that don’t change the colours of the I/O axis (moves that look like big 33 moves).

+
+Does RKT PLC count as Dimensional Reduction? +

RKT PLC does not count as dimensional reduction because certain states of the last cell are not reachable without incurring some form of RKT debt.

+
+

Below is Andrey’s log file of a 34 solve done with dimensional reduction. It’s essentially a CFOP solve, but with some advanced pseudo-slotting and commutators. The solve is similar to doing Pseudo F2L (where the pairs can be in any slot), and then doing OLC/PLC (except solving the PLC into the scrambled state of the rest of the puzzle) and then finishing by solving the reduced puzzle that remains.

+
+Andrey_34.log +
Andrey_34.log
MagicCube4D 3 0 315 {4,3,3} 3
+-0.18955569589822457 0.43991704061351233 -0.877805009971115 -2.7145128148370095E-16
+-2.839993659326744E-9 1.4847274022149884E-10 6.876845398745129E-10 -1.0
+-0.14794206390078768 -0.8966035341133775 -0.41739100175276544 8.357431997622695E-19
+0.970660488472316 -0.05074544307867573 -0.2350385417871584 -2.925836319069701E-9
+*
+104,1,1 12,-1,2 82,1,4 72,1,4 101,-1,1 58,1,4 169,1,4 136,1,2 174,-1,1 158,-1,2
+100,-1,2 44,-1,2 156,1,4 201,-1,1 141,-1,4 40,1,4 208,1,2 162,1,2 143,1,2 171,1,1
+208,-1,1 144,1,4 182,-1,1 56,-1,2 186,1,4 126,1,1 200,-1,2 120,-1,2 156,1,1 194,1,2
+166,-1,2 85,1,1 6,-1,1 143,-1,1 120,1,2 182,-1,2 147,1,4 203,1,1 71,-1,2 192,-1,2
+73,1,2 5,-1,2 159,-1,1 124,-1,2 5,1,2 184,1,4 122,1,1 213,-1,1 m| 102,-1,1
+24,1,1 49,-1,1 20,1,1 131,-1,1 24,1,1 186,1,1 157,1,1 22,1,1 49,-1,1 211,-1,4
+209,1,1 75,-1,1 197,1,1 75,1,1 212,1,4 131,-1,1 194,1,1 131,1,1 195,-1,4 102,-1,1
+211,1,1 102,1,1 214,-1,4 184,1,1 214,-1,1 184,-1,1 214,1,4 78,-1,1 210,1,1 78,1,1
+210,-1,4 209,-1,1 76,1,1 205,1,1 76,-1,1 212,-1,4 78,-1,1 199,1,1 78,1,1 191,-1,4
+202,1,1 52,-1,1 204,1,4 132,1,1 210,1,1 210,1,1 132,-1,1 15,1,1 128,1,1 74,-1,1
+106,-1,1 214,1,4 131,1,1 195,-1,1 131,-1,1 214,-1,4 214,-1,1 187,-1,1 196,-1,4 158,-1,1
+213,1,1 158,1,1 196,1,4 155,1,1 211,1,4 78,1,1 210,-1,1 78,-1,1 22,1,1 214,1,1
+77,1,1 212,1,1 77,-1,1 206,1,1 185,-1,1 212,-1,1 185,1,1 77,1,1 212,-1,1 77,-1,1
+190,1,1 6,-1,1 48,1,1 209,-1,1 48,-1,1 190,1,4 182,1,1 160,-1,1 52,-1,1 24,1,1
+157,1,1 213,1,4 157,-1,1 198,1,1 157,1,1 213,-1,4 157,-1,1 213,1,4 74,1,1 187,-1,1
+189,-1,4 105,1,1 211,1,4 105,-1,1 214,1,1 105,1,1 211,-1,4 105,-1,1 189,1,4 187,1,1
+105,1,1 211,1,4 105,-1,1 214,1,1 105,1,1 211,-1,4 105,-1,1 3,1,1 210,1,1 130,-1,1
+212,-1,4 130,1,1 199,1,1 130,-1,1 212,1,4 130,1,1 192,1,4 74,1,1 193,-1,4 198,1,1
+76,-1,1 210,-1,4 76,1,1 201,1,1 76,-1,1 210,1,4 76,1,1 193,1,4 101,1,1 187,-1,1
+187,-1,1 200,1,4 48,-1,1 209,-1,4 48,1,1 191,1,1 48,-1,1 209,1,4 48,1,1 200,1,4
+51,-1,1 199,1,4 51,1,1 213,-1,1 51,-1,1 199,1,4 51,1,1 182,-1,1 78,1,1 202,1,4
+78,-1,1 210,1,1 210,1,1 78,1,1 202,1,4 78,-1,1 128,1,2 160,1,1 128,1,2 160,-1,1
+182,-1,2 182,-1,2 132,-1,1 202,1,4 132,1,1 196,1,1 132,-1,1 202,1,4 132,1,1 211,1,1
+211,1,1 47,-1,1 201,1,1 101,1,1 74,1,1 201,1,1 128,-1,1 155,-1,1 200,1,1 47,1,1
+155,1,1 200,1,1 52,-1,1 202,1,1 74,-1,1 128,-1,1 202,1,1 182,-1,1 160,1,1 208,1,1
+182,-1,1 155,-1,1 208,1,1 155,1,2 206,1,1 182,1,1 101,1,1 206,1,1 47,1,1 74,-1,1
+207,1,1 182,1,1 128,1,1 207,1,1 187,-1,1 197,1,1 47,1,1 74,1,1 47,-1,1 74,-1,1
+47,1,1 74,1,1 197,1,1 133,-1,1 155,-1,1 155,-1,1 133,-1,1 52,1,1 133,1,1 52,-1,1
+205,1,1 74,-1,1 182,-1,1 74,1,1 182,1,1 74,-1,1 182,-1,1 205,1,1 133,-1,1 203,1,1
+155,-1,1 101,-1,1 155,1,1 101,1,1 155,-1,1 101,-1,1 203,1,1 52,-1,1 79,1,1 52,1,1
+52,1,1 208,1,1 182,-1,1 155,-1,1 182,1,1 155,1,1 182,-1,1 155,-1,1 208,1,1 155,-1,1
+74,-1,1 74,-1,1 182,1,1 74,-1,1 182,-1,1 101,1,1 182,1,1 101,-1,1 182,1,1 101,-1,1
+101,-1,1 74,1,1 74,1,1 47,-1,1 128,1,1 155,-1,1 155,-1,1 47,-1,1 47,-1,1 79,-1,1
+79,-1,1 47,-1,2 47,-1,2 106,1,1 74,-1,1 74,-1,1 47,1,1 133,1,1 133,1,1 79,1,1
+79,1,1 101,1,2 101,1,2 187,-1,1 47,1,1 47,1,1 101,1,1 101,1,1 47,1,1 47,1,1
+101,1,1 101,1,1 47,1,1 47,1,1 74,-1,1 74,-1,1 47,-1,2 74,1,1 74,1,1 47,1,2
+74,1,1 74,1,1 133,1,1 133,1,1.
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The move count is extremely low, which is pretty cool.

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Generalized CFOP

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This article is a generator for the steps of a dimension-generalized CFOP method. For CFOP on 34, see CFOP.

+

CFOP on: 35

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Cross

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F2L

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OLL

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PLL

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Methods

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This page lists solving methods by puzzle.

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It’s a good idea to learn Notation before trying to learn any of these methods

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+

Physical 2x2x2x2

+

While you can use virtual 24 methods on the physical 24, it isn’t recommended because you may have to use an excessive amount of gyros. These methods are specifically designed for the unique restrictive turning of the physical 24 in mind.

+ +

2x2x2x2

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3x3x3x3

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4x4x4x4+

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Last Layer Strategies

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This page is a work in progress.

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This page documents some common last layer strategies for hypercubes (variants of these techniques will work on other puzzles too).

+

OLL/PLL

+

First all of the pieces are oriented in increasing order by number of stickers.

+

To orient 2c pieces, use EOLL algorithms from 2-look OLL.

+

To orient the 3c pieces, use RKT to set up the E slice of the last cell into a configuration that looks like a possible OCLL case. Then use OCLL algorithms to solve that case.

+
+

Avoiding monotwist/flip

+

In higher dimensions, it’s possible to have a single 3c+ piece twisted in place. To avoid this, make sure that your last OCLL algorithm setup will solve all of the remaining pieces of that piece type. For example if you have 5 left to orient, don’t set up an alg that orients 4 because then you will be left over with 1. Instead, you can do an alg that orients 3, and then an alg that orients 2.

+
+

To orient the 4c pieces, use RKT to set up OCLL cases, but this time you have to perform the OCLL alg with RKT. Again, it’s possible to have just 1 4C left to orient, so be smart about your last setup and algorithm.

+

ّFor 4c pices in 2^4, because of a lack of a need to preserve 2 and 3cs, something that can be done is grouping 4 unoriented corners together in a 1x1x2x2 block; then all 4 of these pieces can be oriented with just a U OLL case (F R U R’ U’ F’) with big moves. Something similar can be done with the T and L OLL shapes as well. This way of orienting pieces is useful because not only are you solving more pieces at a time, but they don’t take many moves to do so compared to doing the algs in RKT. If after reaching the OLL step, there are 5 4cs that are misoriented, it may be preferable do something else instead (solve a group of 3, and then 2), as you would want to avoid a monoflip.

+

To orient 5c+ pieces, you have to use RKT to setup OCLL cases, and then do the algorithm using double/triple/etc RKT.

+

For PLL, first permute the 2c pieces using U-perms or other EPLL algorithms. Then use RKT to permute the last layer like a lower dimensional puzzle using RKT.

+

Direct LL

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In 4D+, you can use F2L isolations to help you orient and permute blocks of pieces at the same time.

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Partial Direct LL

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If you use direct LL techniques to just solve the F2L of the LL recursively until you’re left with a “square” of pieces that need orienting and permuting. This group of pieces is the same amount as in 3D, except they can be oriented wrong in many different ways. From here, you can use algorithms to orient them, and then just normal PLL algorithms to permute them.

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Notation

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Warning

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This page explains notation specifically for cell-turning virtual hypercube puzzles. Notation pages for other puzzles coming soon™️…

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Cells

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Yellow cell rotating

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Each side of an n4 hypercube is called a cell, and consists of a cubic grid of n3 stickers. Because each side is cubic, they can be rotated with cubic symmetry. A cube has 24 different possible orientations, meaning that each side of a 4D puzzle can be oriented in 24 different ways.

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There are 8 cells, six of them using the same letters as that in the 33: U (up), D (down), F (front), B (back), R (right), L (left). The one that you can see on the inside is called I (inside), and the one opposite of that (which is hidden) is called O (outside). This means we use the letters R L U D F B O I.

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Pieces

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Cell-turning hypercubes have 4 different types of pieces (with some slight variations for big hypercubes). These are the 1c (1-colored) centers, 2c ridges, 3c edges, and 4c corners. You can label each piece based on which cells it is on. The 3c piece that has stickers on the Inside, Right, and Up cells would be called the IUR piece.

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Twists

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Twists are notated based on which piece you click on to do the turn, with the first letter determining which sticker of that piece to click on. For example:

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  • RO' means to left click the R cell sticker of the RO piece.
    • +
  • IUR means to left click the I sticker of the IUR piece (edge twists are symmetrical, so it doesn’t matter if you left or right click).
    • +
  • FRUI means to right click the F sticker of the FRUI piece.
    • +
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For the slice layers, we use M E S P, where P is the slice that follows O (although with notation for n5 puzzles beginning to be fleshed out, P may end up being used by the new 5D sides of Posterior/Anterior). Wide moves are tricky, as we are now using the letter w for the 4th dimension axis. One way to get around this is to write the numbers you hold down on the keyboard (in MC4D/MPU etc) while clicking on a piece. On a 3-layered puzzle, holding down 2 does a slice move, and holding down 1 and 2 will do a wide move.

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  • {2}RO means to hold 2 while doing an RO move (essentially an M move).
    • +
  • {2-3}UO' means to hold 2 and 3 while doing a UO’ move.
    • +
  • {2-4}IF means to hold 2, 3, and 4
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Commutators

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Many algorithms are constructed using commutators and conjugates, so there is a compact notation for them using square brackets. See Commutators - Notation.

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Algorithms

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A shorter notation was developed to write specific algorithms, such as RKT cancels. All letters besides I and O correspond to their -O variants. R U R' U' would mean RO UO RO' UO'. For the wide O cell flips, they are notated with 3D rotations, such as {1-2}Oxz2. Using 3D rotations in this way is totally arbitrary, and its only purpose is for execution.

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Rotations

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3D notation for rotations doesn’t really generalize to higher dimensions. For example, we call an x rotation x because it rotates the puzzle “around the x-axis”. However, this is unhelpful because rotations don’t actually happen around an axis. Instead it’s better to think of rotations as happening within a 2D plane. When you’re doing an x, the whole puzzle is really being rotated within the zy plane. And writing rotations like this generalizes to higher dimensions, so that is what we use.

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First, we make a certain side from each axis be the “positive” side. There are standards for this in 3D (such as the right-handed rule). On the x-axis, going to the right is positive and going left is negative. This makes R the positive side from the x-axis. The same goes for the other axes: U, and F are the positive sides in 3D. In 4D we add two new sides, and have to decide which one is positive and which one is negative. Because of the projection, the side that we can’t see is closer to the 4D camera, making O the positive w-axis cell, and I the negative.

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To actually notate the rotations, write the letters of the 2 axes that form the plane that the puzzle is rotating in, in the order of which positive side on that axis goes to the positive side of the other axis. For example: rotating the positive y-axis side to the positive x-axis side (rotating U to R) would be written yx (and looks like z in 3D notation).

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  • yw: bring +y to +w (rotate U to O)
    • +
  • xz: bring +x to +z (rotate R to F (this is called y in 3D rotation notation))
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Note how we don’t have to use the ' prime symbol because you can just swap the letters (wy is the inverse of yw). You can also add a 2 to the end for double rotations, e.g. wx2, yz2 etc.

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This makes it a really nice system because it only ever uses 2 letters, no matter how many dimensions. You can also use it on n3 puzzles in order for more multi-dimensional consistency: xy means move x+ to y+ (R to U [z’ rotation]).

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Turn metrics

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There are many different ways to count the number of twists performed during a solve. All of these metrics (besides ETM) have their own Quarter Turn counterparts, where twists are broken up into 90° twists. For example, a 180° face twist takes two 90° twists, while a 120° edge twist takes three. Listed below are the metrics used by HSC. STM is used the most, but others are sometimes relevant.

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  • Consecutive twists of the same axis are combined, even with different layers.
    • +
  • Whole-puzzle rotations are not counted.
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  • Twists are counted as they are executed, including whole-puzzle rotations.
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  • Whole-puzzle rotations are not counted.
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  • Slice twists count as one move.
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  • Consecutive twists of the same axis and layers are combined.
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  • Whole-puzzle rotations are not counted.
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  • Noncontiguous slice twists are split into contiguous slice twists.
    • +
  • Consecutive twists of the same axis and layers are combined.
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  • Whole-puzzle rotations are not counted.
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  • Slice twists are split into contiguous outer-block twists.
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  • Consecutive twists of the same axis and layers are combined.
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+MC4DTM +

a.k.a. “whatever MC4D says”

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  • Twists are counted as they are executed.
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  • Whole-puzzle rotations counted only if they are executed as moves.
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  • Double rotations and certain reorientations are impossible to execute as one move.
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  • Any move that counts as 1 STM is possible with 1 or 2 MC4DTM moves.1
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    This is due to the 2-click theorem.^[citation needed]^ ↩

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Progression

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This is an outline of how to make your way through the world of hypercubing. Learning puzzles in another order could lead to confusion or an incomplete understanding in the long run.

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Hactar

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It’s very difficult to understand what’s really going on in a physical puzzle without first understanding virtual puzzles. We recommend starting with virtual 3^4 for the same reason that 3D cubers typically recommend starting with 3x3x3.

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Getting Started

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Completely new to twisty puzzles? Never solved a Rubik’s Cube? Here’s where to start:

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K-Card Game is a basic piece cycling puzzle using cards. Can you rearrange them into the correct order? Other things to pay attention to when playing:

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  • Cards in the left vs right hand
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  • What type of cycles of cards are possible
    • +
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Loopover is similar to sliding puzzles (such as the infamous 15-puzzle) but there’s no missing tile and the board loops over (hence the name). It might look overwhelming, but it’s actually recommended to start with a 5x5 size board. After that, try a 4x4 board and discover the differences between odd and even size boards.

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You only have 1 chance to try and solve a Rubik’s Cube by yourself with no tutorial. Definitely give that a shot first if you’re brave enough. Otherwise, look up some tutorials and try and get beginners 3D intuition manipulating the cube. Good luck!

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NxNxNxN’s

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First, make sure you know how to solve a 3x3x3 Rubik’s Cube. At the very minimum, knowing a beginner’s method is enough, but knowing more advanced methods, such as CFOP or Roux, or more advanced techniques, such as blockbuilding or intuitive F2L, could make getting into hypercubing easier.

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flowchart LR
+    A[3x3x3]
+    B[3x3x3x3]
+    C[2x2x2x2]
+    D[4x4x4x4]
+    E[5x5x5x5+]
+    F[Physical 2x2x2x2]
+    G[Virtual Physical 3x3x3x3]
+    A --> B
+    B --> C
+    B --> D
+    C --> F
+    D --> E
+    F --> G
+

Other 4D Puzzles

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There are many types of 4D puzzles, not just the hypercubes. These include hypercuboids, duoprisms, reqular 4D polytopes, and more. These puzzles require much more knowledge than just the 3x3x3. You’ll need to know how to solve 3D cuboids, and generally be able to figure out new puzzles using commutators.

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flowchart LR
+    Hypercuboids --> Duoprisms --> Polytopes
+
+    subgraph Polytopes
+    direction LR
+    Q[5-cell]
+    R[16-cell]
+    S[24-cell]
+    T[120-cell]
+    U[600-cell]
+    end
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+    subgraph Duoprisms
+    direction LR
+    Z["{5}x{4}"]
+    Y["{5}x{3}"]
+    X["{5}x{5}"]
+    end
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+    subgraph Hypercuboids
+    direction LR
+    D[1x3x3x3]
+    E[2x3x3x3]
+    F[2x2x2x3]
+    G[2x2x3x3]
+    end
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Non-Euclidean Puzzles

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While most puzzles in MagicTile aren’t 4D, they do share some similar concepts due to their strange geometries.

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flowchart LR
+    A[Torus Rubik]
+    B[Klein Bottle Rubik]
+    C[Hyperbolic Rubik]
+    E[Hemi-Megaminx]
+    D[Other]
+    A --> B
+    A --> C
+    B --> E --> D
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5D+ puzzles

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flowchart TD
+    subgraph 5D
+    direction LR
+    A["n⁴"]
+    B["3⁵"]
+    C["2⁵"]
+    D["4⁵"]
+    E[Other 4D]
+    F[5D simplex]
+    A --> B --> C --> D
+    E --> F
+    A --> E
+    end
+    subgraph 6D
+    direction LR
+    G["3⁶"]
+    H["Other n⁶"]
+    G --> H
+    end
+    subgraph 7D
+    direction LR
+    V["3⁷"]
+    Q["Other n⁷"]
+    V --> Q
+    end
+    5D --> 6D --> 7D
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2x2x2x2

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2×2×2×2 in Hyperspeedcube

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Shape: Tesseract

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Pieces: 16

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History

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The 2x2x2x2 (normally referred to as the 24) is a 4-dimensional twisty puzzle in the shape of a hypercube that is cut in half on each of the 4 axes.

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The first official 24 solve on the Hall of Fame was done by Jay Berkenbilt in 2004 using MC4D, although it was probably solved several times before that.

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Permutations

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The 16 4cs of the 2x2x2x2 can be placed in any even permutation. In order to identify positions that are a whole-puzzle rotation apart from each other, we fix one 4c, leaving 15 4cs that can be permuted. Each 4c can be placed in \(\frac{4!}{2}\) orientations, except the last, which can be placed in only \(4\) orientations. Thus, the number of permutations of the puzzle is +\(\frac{15!}{2} \cdot \left(\frac{4!}{2}\right)^{14} \cdot 4 \approx 3.36 \cdot 10^{27}.\)

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Physical version

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physical 2×2×2×2

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Since 2013, Melinda Green has been refining her physical 2x2x2x2. See her project home page for more details.

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See also

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3x3x3

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There’s not much we can say about the traditional Rubik’s cube that hasn’t already been said on Wikipedia or the Speedsolving.com Wiki.

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3x3x3 with 1D vision

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What if you were a 2D being trying to solve a 3D Rubik’s Cube? This is directly analagous to us 3D beings trying to solve a 4D cube, and MagicCube3D has a setting to view the 3x3x3 with 1D vision. The first known solver was Markk in December 2022.

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3x3x3x3

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3×3×3×3 in Magic Cube 4D

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Shape: Tesseract

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Pieces: 8 1c, 24 2c, 32 3c, 16 4c

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The 3x3x3x3 (normally referred to as the 34) is a 4-dimensional twisty puzzle in the shape of a hypercube that has 2 cuts along each of the 4 axes. It is a direct higher dimensional analogy of the 3x3x3 Rubik’s Cube.

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The 34 is recommended as one’s first introduction to solving higher dimensional puzzles.

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Pieces

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The 34 has 81 hypercubies, of which 72 are movable. It has 8 1c, 24 2c, 32 3c, and 16 4c pieces. The 3c pieces can be rotated in 6 different ways, and the 4c pieces in 12 different ways.

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Turning

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Each turn of the 34 is a rotation of one of its cubic cells (which can be oriented in any of 24 orientations of a cube). A normal move disturbs 8 4c, 12 3c, and 4 2c pieces.

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History

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The 34 has always been the main focus of the community. The first solve was done by Don Hatch in ~1988 using MC4D. After that, the Hall of Fame gradually began to increase in size. It eventually closed on December 7th, 2022 due to it reaching 500 solvers.

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Permutations

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The 16 4cs of the 3x3x3x3 can be placed in any even permutation. Each 4c can be placed in \(\frac{4!}{2}\) orientations, except the last, which can be placed in only \(4\) orientations, due to the existence of monoflip. The 32 3cs can be in any permutation, and can have \(3!\) orientations, except the last, which can only have \(3\). The 24 2cs can be in any permutation, and can have \(2!\) orientations, except the last, which is determined by the others. However, the permutation parity of the 2cs and 3cs are linked. Thus, the number of permutations of the puzzle is +\(\left[\frac{16!}{2} \cdot \left(\frac{4!}{2}\right)^{15} \cdot 4\right] \cdot \left[32! \cdot 3!^{31} \cdot 3\right] \cdot \left[\frac{24!}{2} \cdot 2!^{23}\right] \approx 1.76 \cdot 10^{120}.\)

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Speedsolving

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(See the leaderboards for the current records)

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There were a few hypercube speedsolving competitions during the early days of hypercubing, with the controversy at the time being if they were to allow macros or not. One of the first competitions happened in 2010, with the fastest time being 15:57 by Mateusz Burnicki (using prepared macros) +Some contestants recieved a t-shirt with an image of the 34 in MC4D on it.

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In mid 2017, The speedsolving scene was single handedly revived by Tetrian22, lowering his best from 37:07 to 10:11 in just under 10 months.

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Starting in November 2022, a 34 speedsolving frenzy began due to the popularity of the recently created Hyperspeedcube program, and its keyboard controls. The record bounced between Hactar and Grant as it smashed through all the barriers, finally lowering to sub-2 minutes in May 2023.

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3x3x3x3x3

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3×3×3×3×3 in Magic Cube 5D

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Shape: Penteract

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Pieces: 10 1c, 40 2c, 80 3c, 80 4c, 32 5c

+
+

The 3x3x3x3x3 or 35 is a 5-dimensional twisty puzzle in the shape of a penteract that has 2 cuts along each of the 5 axes. The 35 is recommended as one’s first introduction to solving 5D puzzles.

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Pieces

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The 35 has 243 hypercubies, of which 232 are movable. It has 10 1c, 40 2c, 80 3c, 80 4c, and 32 5c pieces. 1cs are called centers, 2cs ridges, 3cs peaks, 4cs edges, and 5cs corners.

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Turning

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Each turn of the 35 is a rotation of one of its hypercube cells. A hypercube can be reoriented in 192 different ways; a lot of ways just to turn a single side! However in most programs, only some of these moves are able to be done with 2 clicks.

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Programs

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There are 3 programs that currently have implementations of the 35: MPU, MC5D, and MC7D. When HSC2 releases, it will be the 35 program of choice with its amazing piece filters and keybind settings. Until then, the 3 programs all come with their different pros and cons.

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MPU

+
+

35 in MPU

+

3×3×3×3×3 in Magic Puzzle Ultimate

+
+

Magic Puzzle Ultimate objectively has the nicest view of the 35 out of all the programs. The projection is as symmetrical as possible, treating all dimensions equally. It lacks piece filters such that you can only toggle all pieces with a certain number of colours on or off, which makes finding pieces extremely time consuming. To turn the sides you have to click on 3 different 5c pieces.

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MC5D

+
+

35 in MC5D

+

3×3×3×3×3 in Magic Cube 5D

+
+

This program was designed specifically for 5D puzzles. The pros are that it has a feature to locate a certain piece, as well as an extensive macro system. However, the puzzle is projected in an interesting way that uses the 4D inside to outside projection twice. This makes many of the stickers overlap. Additionally it uses Euler angles for rotation, meaning that sometimes you get stuck at the poles when reorienting the puzzle with the mouse. There are some buttons in the bottom-right of the UI to rotate the puzzle to make up for this.

+

MC7D

+
+

35 in MC7D

+

3×3×3×3×3 in Magic Cube 7D

+
+

Magic Cube 7D was made by Andrey Astrelin, and has a different way of projecting higher dimensions. The dimensions higher than 4 are represented with stickers that are smaller than the others. Many of them are also duplicates of each other because they are attached to the other stickers of that piece. While not being the nicest to look at, it does have very useable piece filters and nice 2-click controls, making it a prime candidate to use for one’s first 35 solve.

+

Permutations

+

The 32 5cs of the 3x3x3x3x3 can be placed in any even permutation, and each 5c can be placed in \(\frac{5!}{2}\) orientations, including the last one. The 80 4cs can be in any even permutation, and can each have \(4!\) orientations, except the last which can only have \(\frac{4!}{2}\) orientations. The 80 3cs can be in any permutation, and can each have \(3!\) orientations, except the last which can only have \(\frac{3!}{2}\) orientations. Finally, the 40 2cs can be in any permutation with the same parity as that of the 3cs, and each one can have 2 orientations, except the last, whose orientation is determined by the rest of the 2cs.1 Thus, the number of permutations of the puzzle is +\(\left[\frac{32!}{2} \cdot \left(\frac{5!}{2}\right)^{32}\right] \cdot \left[\frac{80!}{2} \cdot \frac{4!^{80}}{2}\right] \cdot \left[80! \cdot \frac{3!^{80}}{2}\right] \cdot \left[\frac{40!}{2} \cdot 2^{39}\right] \approx 7.02 \cdot 10^{560}.\)

+

Speedsolving

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(See the leaderboards for the current records)

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A full solve of the 35 had not even been recorded until 2024-03-21 with Rowan Fortier’s first solve, uploading all 15 hours of footage to YouTube. Sadly this did not count as the first speedsolve because it was done over multiple days in different sessions. But a few days later, Luna managed to do it in one sitting, achieving a time of 3h 39m, automatically getting the first world record.

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    http://www.gravitation3d.com/magiccube5d/permutations.html ↩

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4D Skewb

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There are many puzzles that could be called a ‘4D skewb’. Each one has some characteristics similar to the skewb. By constructing the skewb in a certain way, and replacing 3D objects of the definition with similar 4D objects, you can construct a 4D puzzle that has some things in common with the skewb.

+

Cube-like

+

Vertex turning hypercube

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+

Vertex turning hypercube

+

Vertex turning hypercube

+
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The skewb is a vertex-turning cube with the cuts passing through the origin. If you replace the 3D cube in the definition with the 4D hypercube, you get the half-cut vertex-turning hypercube. This puzzle has 32 edge pieces, 64 X-centers, and 8 center pieces. Both the X-centers and the centers are 1c pieces.

+

In three dimensions, a plane perpendicular to the vertex axis passes through 1 corner, then 3, then 3, then 1 corner. This causes a plane passing through the origin to go between the two layers of 3 corners, which is why the skewb has corners. In four dimensions, a hyperplane perpendicular to the vertex axis passes through 1, 4, 6, 4, then 1 corner. Therefore, a hyperplane passing through the origin passes directly through 6 vertices, and there are no corner pieces on this 4D puzzle. Similarly, the pattern of cuts on one cell of the puzzle looks like a dino cube, not a skewb.

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Skewb lookalike

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+

Skewb lookalike

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Skewb lookalike

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You can place a cut to pass between the layers of 4 and 6 corners. If you do this, you get two cuts per axis and three layers per axis. This puzzle has 16 vertex pieces, 64 X-centers, and 24 ridge pieces. Like the skewb, the vertex pieces are separated into two orbits, but this puzzle has a middle layer on each axis. The cut pattern on the cells of this puzzle resembles a 3D skewb with additional trivial tips.

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By rectifying the hypercube, the trivial tip cuts on the cut patterns of the cells are removed, meaning the cells look like cuboctahedral skewbs.

+

Demi vertex turning hypercube

+
+

Demi vertex turning hypercube

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Demi vertex turning hypercube

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On the cube, the vertices can be separated into two tetrahedral subsets (the orbits of the skewb’s corners). Each cut corresponds to one corner in the tetrahedron. On the hypercube, a similar partitioning of the vertices results in two sets of 8, each at the vertices of the 16-cell, or dually, the facets of a differently-oriented hypercube. If you only cut along vertex axes corresponding to these cuts, you will get a puzzle with only 4 cuts, half as many as the vertex turning hypercube. This puzzle has 8 corners and 8 centers. Unlike on the skewb, only half of the vertices of the hypercube have a corner piece, and all the corner pieces are in one orbit.

+

Demi skewb lookalike

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Demi skewb lookalike

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Demi skewb lookalike

+
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If you only use the axes along half the vertices as above, but you use the cut depths of the skewb lookalike, it results in a puzzle that still has three layers per axis, but only half as many axes. The cut pattern on the cells of this puzzle also resembles a 3D skewb, but with only 4 trivial tips per cell. The puzzle has 8 corner pieces, 8 dual corner pieces of a different shape, 32 X-centers, and 24 ridge pieces.

+

As before, the hypercube can be rectified, which removes the trivial tips from the cut patterns of the cells.

+


+
+Extra turns +

In the two demi puzzles, the 8 vertices are arranged like the cells of a hypercube. This means they are actually shapemods of hypercubic puzzles. As such, they have an additional shapeshifting move in which you turn one layer by 90°. There is no analogous move on the skewb. In particular, the demi vertex turning hypercube is a shapemod of a 2x2x2x2, and the demi skewb lookalike is a shapemod of a 3x3x3x3, but with the 8 corner pieces that would be at the center of the cells missing.

+

Because the 16-cell axis system, or the vertex turning hypercube system, is composed of two disjoint hypercube axis systems, these puzzles also have additional shapeshifting moves in which you turn a layer 90°. Unlike those in the previous section, though, performing one of these turns would bandage the puzzle. If you unbandaged these puzzles, you would get additional cuts parallel to the cells, resulting in the axis system of a 24-cell.

+
+

Simplex-like

+

The skewb can be constructed in a different way: take four axes corresponding to a tetrahedron, and cut perpendicular to each axis through the origin. The tetrahedron is the 3D simplex, so if you replace it with the 5-cell, the 4D simplex, you get a new puzzle in 4D. This puzzle has several types of pieces, but since the shape of the puzzle does not yet exist, we can’t name them after their position. Thus, we will name the pieces by how many layers they are turned by. For the skewb, there are 4 pieces that are in one layer, 6 pieces that are in two layers, and 4 pieces that are in three layers. We will call these 1g, 2g, and 3g pieces. This 4D puzzle will have 5 1g pieces, 10 2g pieces, 10 3g pieces, and 5 4g pieces. The skewb has a symmetry (90° rotation of the puzzle) that swaps 1g and 3g pieces and sends 2g to 2g pieces, and like it, this 4D puzzle has a symmetry that swaps 1g and 4g pieces and swaps 2g and 3g pieces. Like the skewb, the cuts on this puzzle can each be offset in a consistent direction while preserving the functionality of the puzzle.

+

Constructing the skewb like this, we can describe how to construct its cubic shape. On each face of the cube, there is one 2g piece in its center. Thus, given our abstractly constructed skewb, we can construct a plane perpendicular to each 2g piece’s axis of symmetry. The shape bounded by all 6 of these planes is a cube.

+

2g carved simplex

+
+

2g carved simplex

+

2g carved simplex

+
+

Like in 3D, you can put a hyperplane perpendicular to the symmetry axis of each 2g piece. Since there are 10 2g pieces, this creates a polychoron with 10 triangular-bipyramidal cells called the joined 5-cell. On this puzzle, the 1g pieces are 4c at the tetrahedral vertices, the 2g pieces are 1c at the cell centers, the 3g pieces are 3c at the triangular edges, and the 4g pieces are 6c at the triangular-bipyramidal vertices. Unlike on the skewb, where the 1g and 3g pieces look alike, here, the 1g and 4g pieces are distinguishable by their number of colors, and so are the 2g and 3g pieces.

+

2g-3g carved simplex

+
+

2g-3g carved simplex

+

2g-3g carved simplex

+
+

You can also put hyperplanes perpendicular to the symmetry axis of both the 2g and 3g pieces. This gives the puzzle a 20-celled shape called the bijungato-10-cell. On this puzzle, the 1g and 4g pieces are both 4c at tetrahedral vertices, and the 2g and 3g pieces are both 1c at the cell centers. Unlike the previous shape but like the skewb, full symmetry of the puzzle is reflected in the shape.

+

Each cell of the puzzle either has two 1g and one 2g pieces, or two 4g and one 3g pieces. These two orbits of cells are only adjacent across the 2g-3g boundary. Because of this, this puzzle has multiple solved states, where each cell is a solid color, but the color schemes of the two orbits are rotated relative to each other.

+

2g-3g carved simplex, hemi colors

+

The puzzle can be recolored with 10 colors by giving opposite cells the same color. By asserting that the solved state has to have the two same-colored cells opposite to each other, it restricts the relative orientations of the two orbits to only one configuration, leading to there being only one solved state. Because of the geometry of the puzzle, a 1g-2g cell is opposite a 4g-3g cell, so the pieces on these cells are not swappable and coloring these two cells the same color leads to no ambiguity in the position.

+

2g-3g ridge carved simplex

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2g-3g ridge carved simplex

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2g-3g ridge carved simplex

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There are 30 square ridges between the 2g and 3g pieces. By constructing a shape bounded by the hyperplanes perpendicular to the symmetry axes of these ridges, you get a 30-cell shape with disphenoidal cells called the bi-10-cell. The 1g and 4g pieces are 12c at triakis-tetrahedral vertices, and the 2g and 3g pieces are 3c at triangular ridges. This shape of the puzzle also reflects the symmetry that swaps 1g and 4g and swaps 2g and 3g.

+

Prism

+
+

Prism

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Prism

+
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By starting with a skewb, you can extend it into the fourth dimension by taking the cartesian product of it with a line segment. This extends the cube to a cubic prism, and the skewb cut planes to hyperplanes all perpendicular to the original cube. This creates a skewb prism, which has exactly the same pieces as the skewb but with shapes that are the prisms of the originals. It allows all the moves of the skewb, and an additional set of moves that involve flipping the new dimension, which appear as reflection moves on the original skewb. By adding more cuts parallel to the original skewb, you can create multi-layer skewb prisms.

+

24-cell-like

+

24-cell skewb diamond lookalike

+
+

24-cell skewb diamond lookalike

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24-cell skewb diamond lookalike

+
+

The 24-cell’s cells are octahedra, which have the same symmetry as the skewb, and its dual the skewb diamond. Construct cutting planes parallel to the cells which pass through the centers of the adjacent cells. This creates a puzzle with 24 vertex pieces, 96 ridge pieces, and 144 X-centers. This puzzle is not half-cut, but each cell’s cut pattern looks like a skewb diamond with trivial tips. It is possible to apply RKT to a cell of this puzzle, in which case it acts like a skewb diamond.

+

By rectifying the 24-cell, the trivial tip cuts on the cut patterns of the cells are removed, meaning the cells look like cuboctahedral skewbs.

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24-cell skewb diamond lookalike, 16-cell cuts

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24-cell skewb diamond lookalike, 16-cell cuts

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24-cell skewb diamond lookalike, 16-cell cuts

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+

The 24-cell has a subset of cells that correspond to the cells of the 16-cell. If you only keep those cuts, you get a puzzle similar to the one above, but where 8 of the cells no longer appear to have trivial tips in their cut patterns. This puzzle has 16 large corner pieces, 8 small corner pieces, 96 ridge pieces, and 144 X-centers. Like before, RKT can be applied.

+


+
+

+

24-cell skewb diamond lookalike, 8-cell cuts

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24-cell skewb diamond lookalike, 8-cell cuts

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24-cell skewb diamond lookalike, 8-cell cuts

+
+

The 24-cell also has a subset of cells that correspond to the cells of the 8-cell, or hypercube. If you only keep those cuts, you get another puzzle similar to the one above, but where 16 of the cells do not appear to have trivial tips in their cut patterns, and 8 of the cells appear to have only the trivial tip cuts. This puzzle has 8 large corner pieces, 16 small corner pieces, 32 ridge pieces, and 8 center pieces. Unlike before, RKT cannot be applied to the skewb diamond cells. This puzzle is a shapemod of the 3x3x3x3 without corner pieces.

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Hemimegaminx

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Hemimegaminx

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Hemimegaminx in MagicTile

+

Shape: Hemi-dodecahedron

+

Pieces: 6 1c, 15 2c, 10 3c

+
+

The hemimegaminx is a twisty puzzle in the shape of a hemi-dodecahedron. In MagicTile, it inhabits the projective plane.

+

The puzzle can be constructed by identifying opposite faces on a megaminx. On this realization of the puzzle, opposite faces will turn in opposite directions. In order to be a true hemimegaminx, opposite faces should also be colored the same, so the puzzle will have 6 colors total.

+

Permutations

+

The 10 3cs of the hemimegaminx can be placed in any even permutation. Each 4c can be placed in \(3!\) orientations, except the last, which can be placed in only \(3\) orientations. The 15 2cs can be placed in any even permutation, and each one has \(2\) orientations, except the last, whose orientation is determined by the other pieces. Thus, the number of permutations of the puzzle is +\(\left[\frac{10!}{2} \cdot 3!^{9} \cdot 3\right] \cdot \left[\frac{15!}{2} \cdot 2^{14}\right] \approx 5.87 \cdot 10^{29}.\)

+

Physical version

+

Akkei's hemimegaminx

+

In 2019, after several prototypes, Oskar van Deventer built the first working hemimegaminx. In 2024, Akkei created another physical hemimegaminx and did the first known timed solve on it.

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Hypercuboids

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Introduction

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A hypercuboid is the multi-dimensional version of a cuboid.
+In a general context, we define a hypercuboid as an \(n\)-dimensional puzzle denoted by \(a_1 \times a_2 \times \dots \times a_n\).
+ For the sake of clarity and consistency, we will use \(a_1,a_2, \dots a_n\) as non-decreasing values.

+

Structure

+

A hypercuboid, as defined, is composed of \(2n\) cells, each of which is \((n-1)\)-dimensional.

+

Given \(k \geq 0\) and \(n \geq 1\), the elementary symmetric polynomial \(e_k(x_1, x_2, \dots, x_n) = \sum_{Y} \prod_{y\in Y} y\), where \(Y\) ranges over subsets of \(\{x_1, \dots, x_k\}\) where \(|Y| = k\). In other words, it is the sum of all terms, each of which are product of distinct \(x_i\) taken \(k\) at a time.

+
    +
  • For example: \(e_1(x_1,x_2,x_3,x_4,x_5) = x_1 + x_2 + x_3 + x_4 +x_5\), i.e. the sum of terms of 1 element, chosen in \(x_1, x_2, x_3, x_4, x_5\).
    • +
  • Another example: \(e_2(x_1,x_2,x_3,x_4) = x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4\), i.e., the sum of the products of all possible unordered pairs made with \(x_1, x_2, x_3, x_4\).
    • +
+

Note that \(e_k(x_1, \dots, x_n)\) has \(\binom{n}{k}\) terms. From this, we can also see that \(e_0(x_1, \dots, x_n)=1\).

+

Using the above notation, the \(a_1 \times a_2 \times \dots \times a_n\) hypercuboid, where \(a_i > 1\), has \(2^k\cdot e_{n-k}(a_1-2, \dots, a_n-2)\) pieces which are \(k\)-colored.

+

For example, consider the \(2 \times 3 \times 5 \times 7\) hypercuboid.

+
    +
  • +

    For 1-colored pieces we have:
    +\(2^1 \cdot e_3(0,1,3,5)=2^1\cdot (0\cdot1\cdot3 + 0\cdot 3 \cdot 5 + 1 \cdot 3 \cdot 5 + 0\cdot 1 \cdot5 )=\)
    +\(=2 \cdot (0+0+15+0)=30\) pieces.

    +
    • +
  • +

    For 2-colored pieces we have:
    +\(2^2 \cdot e_2(0,1,3,5)=2^2\cdot ( 0\cdot 1+ 0\cdot 3 +0 \cdot 5 + 1\cdot3 +1\cdot 5 + 3\cdot 5 )=\)
    +\(=4 \cdot (0+0+0+3+5+15)=92\) pieces.

    +
    • +
  • +

    For 3-colored pieces we have:
    +\(2^3 \cdot e_1(0,1,3,5)=2^3\cdot (0+1+3+5 )=\)
    +\(=8 \cdot 9=72\) pieces.

    +
    • +
  • +

    For 4-colored pieces we have:
    +\(2^4 \cdot e_0(0,1,3,5)=2^4\cdot 1= 16\) pieces.

    +
    • +
+

If \(a_1, \dots, a_m\) are all equal to \(1\) and \(a_{m+1} > 1\), the cuboid is a floppy cuboid. In this case, the number of \(k\)-colored pieces is equal to the number of \((k-2m)\)-colored pieces on the \(a_{m+1} \times \dots \times a_n\) hypercuboid, when \(k \geq 2m\). This includes the case where \(k = 2m\), in which case the floppy cuboid has pieces corresponding to the 0-colored pieces of the lower dimensional hypercuboid.

+

4D Hypercuboids

+

In 4 dimensions, a hypercuboid is denoted as \(a \times b \times c \times d\).
+\(a \times b \times c \times d\) is composed of 8 cells: 2 \((a \times b \times c)\)-cells, 2 \((a \times b \times d)\)-cells, 2 \((b \times c \times d)\)-cells and 2 \((a \times c \times d)\)-cells.
+In the following sections, we will denote some of these cells using the classic 3-dimensional puzzle names, in particular:

+
    +
  • “tower cell” will indicate a \(2 \times 2 \times 3\)-cell;
    • +
  • “domino cell” will indicate a \(2 \times 3 \times 3\)-cell;
    • +
  • ” \(n\) -cubic cell” will indicate a \(n \times n \times n\)-cell.
    • +
+

General solving strategies

+
    +
  • Hypercuboids in the form \(1 \times a \times b \times c\) can be solved by first orienting the \(a \times b \times c\)-cells, then solving the puzzle like a 3-dimensional \(a \times b \times c\).
    • +
  • Hypercuboids in the form \(2 \times a \times b \times c\) can be solved first by solving the \(a \times b \times c\)-cells and then solving the opposite, eventually adapting the solution for new possible cases.
    • +
  • If 2 dimensions have the same values, the puzzle can be seen as a duoprism.
    • +
  • If 3 dimensions have the same values, i.e. there is a couple of \(n\)-cubic cells, RKT can be used on these cells.
    • +
  • If 4 dimensions have the same values, we have a hypercube.
    • +
+

Some notable 4D hypercuboids

+

In some cases an idea of a possible solution method provided by Ema will be present but not spoiled.

+

1x3x3x3

+ + + + + + + + + + + + + + + + + + + +
Puzzle4c pieces3c pieces2c pieces1c pieces
1x1x3x31624122
+
+Solve idea (click to reveal) +
    +
  • Orient both cubic cells.
    • +
  • Solve 3^3 cube, paying attention to corner orientation.
    • +
+
+

2x2x2x3

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Puzzle4c pieces3c pieces2c pieces1c pieces
2x2x2x316800
+
+Solve idea (click to reveal) +
    +
  • Solve the middle 3-colored pieces of a tower cell (similar to solving a \(1 \times 2 \times 2 \times 2\) ).
    • +
  • Orient both \(2\)-cubic cells at the same time, slicing the solved part for exchanging pieces,being careful to use an even number of slice moves.
    • +
  • Use RKT to solve the cubic cells, using the same tower cells as R.
    • +
  • Fix tower cell middle layer.
    • +
+
+

2x2x3x3

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Puzzle4c pieces3c pieces2c pieces1c pieces
2x2x3x3161640
+
+Solve idea (click to reveal) +
    +
  • Solve a domino cell.
    • +
  • Orient the opposite domino cell, potentially re-solving the first cell.
    • +
  • Move pieces on the correct layers of the last cell.
    • +
  • Solve last domino cell using 3-dimensional cuboid algorithms an even number of times and conjugating between them.
    • +
+
+

2x3x3x3

+ + + + + + + + + + + + + + + + + + + +
Puzzle4c pieces3c pieces2c pieces1c pieces
2x3x3x31624122
+
+Solve idea (click to reveal) +
    +
  • Orient both 3-cubic cells at the same time.
    • +
  • Solve first cubic cell.
    • +
  • Solve the second cubic cell using RKT.
    • +
+
+

2x2x2x4

+ + + + + + + + + + + + + + + + + + + +
Puzzle4c pieces3c pieces2c pieces1c pieces
2x2x2x4161600
+

2x3x4x5

+ + + + + + + + + + + + + + + + + + + +
Puzzle4c pieces3c pieces2c pieces1c pieces
2x3x4x516484412
+

The smallest 4-dimensional “brick” hypercuboid.

+

4D hypercuboids in MPUlt

+

Here is a way to create your own 4D hypercuboid in MPUlt.
+The result would not be isometric, but still working.

+

Step 1: Recognize the form of your hypercuboid in one of the following

+
    +
  • \(a \times b \times c \times d\),
    • +
  • \(a \times a \times b \times c\),
    • +
  • \(a \times a \times b \times b\),
    • +
  • \(a \times b \times b \times b\),
    • +
  • \(a \times a \times a \times a\).
    • +
+

Step 2: Recognize the values of the letters, then substitute the letter with the corresponding string from the following table:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
ValueString
20.0
30.333 -0.333
40.5 0.0 -0.5
50.6 0.2 -0.2 -0.6
60.667 0.333 0.0 -0.333 -0.667
70.714 0.429 0.143 -0.143 -0.429 -0.714
80.75 0.5 0.25 0.0 -0.25 -0.5 -0.75
90.778 0.556 0.333 0.111 -0.111 -0.333 -0.556 -0.778
+

So if \(a=3\), you need to change “CUT-A” with “0.333 -0.333” in the general puzzle code, and so on.

+

Step 3: Insert the created code in “MPUlt_puzzles.txt” file, save and enjoy your puzzle.

+

Case axbxcxd

+

General code:
+

Puzzle NAME_AXBXCXD
+Dim 4
+NAxis 4
+Faces 1,0,0,0 0,1,0,0 0,0,1,0 0,0,0,1
+Group 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1
+Cuts CUT-A
+Axis 0,1,0,0
+Twists 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,0,1,0/0,0,0,1
+Cuts CUT-B
+Axis 0,0,1,0
+Twists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0
+Cuts CUT-C
+Axis 0,0,0,1
+Twists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0
+Cuts CUT-D
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Case axaxbxc

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General code: +

Puzzle NAME_AXAXBXC
+Dim 4
+NAxis 3
+Faces 1,0,0,0 0,0,1,0 0,0,0,1
+Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1
+Cuts CUT-C
+Axis 0,0,1,0
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0
+Cuts CUT-B
+Axis 0,0,0,1
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0
+Cuts CUT-A
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Case axaxbxb

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General code: +

Puzzle NAME_AXAXBXB
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0,0,1,0
+Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1
+Axis 1,0,0,0
+Twists 0,0,1,0/0,0,1,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0,1,1
+Cuts CUT-A
+Axis 0,0,1,0
+Twists 1,0,0,0/1,1,0,0 0,0,0,1/1,0,0,0 0,0,0,1/1,1,0,0
+Cuts CUT-B
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Case axbxbxb

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General code: +

Puzzle NAME_AXBXBXB
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0,0,0,1
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1
+Cuts CUT-B
+Axis 0,0,0,1
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0
+Cuts CUT-A
+

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Case axaxaxa

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General code: +

Puzzle NAME_AXAXAXA
+Dim 4
+NAxis 1
+Faces 1,0,0,0
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,-1,0/0,0,0,1 0,2,-1,-1/0,1,1,-2
+Cuts CUT-A
+

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5D+ Hypercuboids

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These hypercuboids haven’t been studied yet, except for some “simpler” versions with lots of \(1\)’s.

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Puzzles

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In hypercubing, we generalize the notion of a twisty puzzle beyond 3D space. The best introduction to hypercubing is the standard 3x3x3x3 4D Rubik’s cube.

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See the navigation sidebar for a list of puzzles that have wiki pages.

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NxNxNxN

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4x4x4x4

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3×3×3×3 in Hyperspeedcube

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Shape: Tesseract

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NxNxNxN, or N4 is a generic term for a 4-dimensional twisty puzzle in the shape of a hypercube with N layers per axis. It is a direct higher dimensional analogy of the NxNxN Rubik’s Cube.

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This page is concerned with the case where N is greater than 3. The 2x2x2x2 and 3x3x3x3 have their own pages.

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Pieces

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The N4 has \(N^4 - (N-2)^4\) hypercubies. If N is even, all hypercubies are movable, and if N is odd, all but 8 are movable. It has \(8(N-2)^3\) 1c, \(24(N-2)^2\) 2c, \(32(N-2)\) 3c, and \(16\) 4c pieces. These pieces come in many subtypes.

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  • 1c
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    • Centers: These pieces are at the centers of the facets. When \(N \geq 3\) is odd, 8 of these pieces exist and they are immovable. They are not present when \(N\) is even.
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    • T-centers: These pieces exist in orbits of 48 between the facet centers and the ridge centers. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. They are not present when \(N\) is even.
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    • Y-centers: These pieces exist in orbits of 96 between the facet centers and the edge centers. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. They are not present when \(N\) is even.
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    • X-centers: These pieces exist in orbits of 64 between the facet centers and the corners. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. When \(N \geq 4\) is even, there are \(\frac{N-2}{2}\) orbits.
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    • Semi-oblique centers: These pieces exist in orbits of 192. There are several subtypes, each of which have \(\frac{(N-3)(N-5)}{4}\) orbits when \(N \geq 7\) is odd, and \(\frac{(N-2)(N-4)}{4}\) when \(N \geq 6\) is even.
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      • TY-centers: These pieces are between the facet centers, ridge centers, and edge centers.
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      • TX-centers: These pieces are between the facet centers, ridge centers, and corners.
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      • YX-centers: These pieces are between the facet centers, edge centers, and corners.
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    • Oblique centers: These pieces exist in orbits of 192 off all hyperplanes of symmetry. They come in two chiralities. When \(N \geq 9\) is odd, there are \(\frac{(N-3)(N-5)(N-7)}{8}\) orbits of each chirality. When \(N \geq 8\) is even, there are \(\frac{(N-2)(N-4)(N-6)}{8}\) orbits of each chirality.
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  • 2c
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    • Middle ridges: These pieces are at the centers of the ridges. When \(N \geq 3\) is odd, they come in one orbit of 24. They are not present when \(N\) is even.
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    • T-ridges: These pieces exist in orbits of 96 between the ridge centers and the edge centers. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. They are not present when \(N\) is even.
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    • X-ridges: These pieces exist in orbits of 96 between the ridge centers and the corners. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. When \(N \geq 4\) is even, there are \(\frac{N-2}{2}\) orbits.
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    • Oblique ridges: These pieces exist in orbits of 192, but they are not chiral. When \(N \geq 7\) is odd, there are \(\frac{(N-3)(N-5)}{4}\) orbits of each chirality. When \(N \geq 6\) is even, there are \(\frac{(N-2)(N-4)}{4}\) orbits of each chirality.
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  • 3c
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    • Middle edges: These pieces are at the centers of the edges. When \(N \geq 3\) is odd, they come in one orbit of 32. They are not present when \(N\) is even.
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    • Wings: These pieces exist in orbits of 64 between the edge centers and the corners. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. When \(N \geq 4\) is even, there are \(\frac{N-3}{2}\) orbits of each chirality.
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    • Corners: These pieces are at the corners. When \(N \geq 2\), they come in one orbit of 16.
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Turning

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Each turn of the N4 is a rotation of one of its cubic cells, or a slice layer between two parallel cuts. Each layer turned can be oriented in any of 24 orientations of a cube.

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Tymon’s 1x2x2x2

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Tymon’s 1x2x2x2

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Tymon's 1×2×2×2

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4D Shape: Hypercuboid

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Physical Shape: Cube

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Pieces: 8 5c

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Magnets: 384

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Completed: 2024 Jan 11

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History

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Tymon F began some initial tests for this puzzle in October 2023. The first 2 pieces were fully assembled and magnetized by December 2023. Finally (conincidentally exactly 1 year after he discovered hypercubing), Tymon completed the 1x2x2x2 hypercuboid on 2024-01-11. After that, he started work on creating the cuboids up to 1x3x3x3.

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Moves

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Legal moves of a layer include basic 2x2x2 90° twists, 180° twists in any plane which also rotate pieces in 4d axis, or a combination of both.

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Tymon's 1×2×2×2 with one half exposed

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Solving

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Beside basic stickers (white, yellow, orange, red, green, blue) every piece on the puzzle has pink and purple stickers which indicate in which 4d orientation the piece is. In the solved state every piece must have the same 4d axis orientation. +The 1x2x2x2 is not much more difficult than the 2x2x2. It can be solved by: orienting 5c pieces in 4d axis, except “OLL parity” (because of the extra axis, a single 5c can be missoriented) the rest of the puzzle can be solved like a 3d 2x2x2. Tymon made a YouTube video showing an example solve of the puzzle.

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Tymon’s 1x2x2x3

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Tymon’s 1x2x2x3

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Tymon's 1×2×2×3

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4D Shape: Hypercuboid

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Physical Shape: Cuboid

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Pieces: 8 5c, 4 4c

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Magnets: 512

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Completed: 2024 Jan 15

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History

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A few days after completing the physical 1x2x2x2 Tymon F finished assembling four new 4c pieces that combined with the previous 5c pieces to create a physical analog of the 1x2x2x3 puzzle. This one was another important milestone on the way to create a physical 1x3x3x3 puzzle.

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Moves

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Legal moves of a layer include basic 2x2x2 90° and 180° twists that rotate pieces without changing their 4d orientation, 4d 180° twists that besides moving pieces in 3d, rotate them in 4d axis, or a combination of both.

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Tymon's 1×2×2×3 with one half exposed

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Solving

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Besides basic stickers (white, yellow, orange, red, green, blue) every piece on the puzzle has pink and purple stickers which indicate in which 4d orientation the piece is. In the solved state every piece must have the same 4d axis orientation. The 1x2x2x3 is a bit more complicated than 2x2x3. It can be solved by: orienting 4c piece in 4d axis, orienting 5c pieces in 4d axis, solving the rest of the puzzle like a 23. Tymon made a YouTube video showing an example solve of the puzzle.

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Tymon’s 1x2x3x3

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Tymon’s 1x2x3x3

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Tymon's 1×2×3×3

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4D Shape: Hypercuboid

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Physical Shape: Cuboid

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Pieces: 8 5c, 8 4c, 2 3c

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Magnets: 688

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Completed: 2024 Jan 24

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History

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Over a week after completing the physical 1x2x3x3, Tymon F finished assembling another four 4c pieces and two new 3c pieces that combined with the 1x2x3x3 to create the physical 1x2x3x3 puzzle. This was the last puzzle made with the goal of creating a physical 1x3x3x3.

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Moves

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Legal moves of a layer include normal 2x3x3 90° and 180° twists that rotate pieces without changing their 4d orientation, 4d 180° twists that change the pink/purple stickers, or a combination of both.

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Solving

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Besides basic stickers (white, yellow, orange, red, green, blue) every piece on the puzzle has pink and purple stickers which indicate in which 4d orientation the piece is. In the solved state every piece must have the same 4d axis orientation. +The 1x2x3x3 puzzle is much more complicated than smaller hypercuboids, but it can be solved by: orienting 3c pieces in 4d axis, orienting 4c piece in 4d axis, orienting 5c pieces in 4d axis, solving the rest of the puzzle like 3d 2x3x3. Tymon made a YouTube video showing an example solve of the puzzle.

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Tymon’s 1x3x3x3

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Tymon’s 1x3x3x3

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Tymon's 1×3×3×3

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4D Shape: Hypercuboid

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Physical Shape: Cube

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Pieces: 8 5c, 12 4c, 6 3c, 1 2c (2 1c)

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Magnets: 936

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Completed: 2024 Jan 26

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History

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Two days after finishing the 1x2x3x3 Tymon F completed his final goal, creating a physical analog of the 1x3x3x3 puzzle. To do this he had to assemble another four 4c pieces, four 3c pieces and one 2c piece that represents two 1c pieces on the virtual puzzle. This ended Tymon’s journey of creating all physical puzzles from the AxBxCx1 family.

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Moves

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Legal moves of a layer include 90° twists, 180° twists in any plane, or a combination of both.

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Tymon's 1×3×3×3 with one half exposed

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Solving

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Every piece on the puzzle has pink and purple stickers which indicate which 4d orientation the piece is. In the solved state every piece must have the same 4d axis orientation. +1x3x3x3 is harder than the 3d Rubik’s cube, but it can be solved by: orienting 3c pieces in 4d axis, orienting 4c pieces in 4d axis, orienting 5c pieces in 4d axis, except “OLL parity” (because of the extra axis, a single 5c can be missoriented), then solving the rest of the puzzle like a 3d Rubik’s cube. Tymon made a YouTube video explaining the puzzle as well as showing an example solve.

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Gyro

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This puzzle has two 1c pieces that are physicaly represented by one 2c piece (the core). Technically the puzzle is solved only when all of the other pieces have the pink sticker on the outside cell (because pink is outside on the core piece). To be able to solve the puzzle with purple color facing outwards, Tymon created a gyro algorithm that flips all pieces inside out and puts the core outside of the puzzle. This represents that pink/purple axis is flipped and now the puzzle is only solved when all of the pieces are oriented with purple on the ouside. The gyro can be reversed to get back to default projection.

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1×3×3×3 gyro gif

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OLL Parity

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Becuase of the extra freedom of the 4th dimension a monoflip (single missoriented corner) can occur. It can be solved by doing one of the 4d moves, to set up a valid OLL case or just rotate corners and undoing 4d move. A single corner can also be rotated with a simple commutator.

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Alternative Solved state

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The original solved state well represents which sticker on a piece is on the inside and which one is on the outside, but some people prefer a solved state with edges rotated 4 dimensionaly. This is because then the overall look of the puzzle is less chaotic and more friendly to non hypercubers.

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Tymon's 1×3×3×3 alternative solve state

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Physical 2x2x2x2 Canonical Moves & Notation

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Info

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Watch Melinda Green’s video for a quick overview of the canonical moves

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The Canonical Moveset is a set of moves agreed upon by the community for solving the physical 24. To get accepted into the official Hall of Fame, you must follow the canonical moves exactly. Make sure that you have already solved a virtual 24, and are familiar with how the pieces of the physical 24 correspond. Also make sure that you know your 3D rotations (x y z) very well.

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Canonical Moves

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Simple Rotations

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Simple Rotations

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Simple puzzle rotations on Melinda's 2×2×2×2

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A lot of the whole puzzle reorientations can be reached without the need for the gyro algorithm. These are called simple rotations, and consist of rotaing the L and R cells together in opposing directions (as to not change the state of the puzzle, only its orientation).

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zy yz yw wy zw wz

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Cell Twists

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The left and right cells of the puzzle can be twisted into any reorientation of a cube. In other words, there are 23 different twists of the L/R cells:

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Left cell twists: Ly Ly' Ly2 Lx2 Lz2 Lx2,y Lx2,y' Lx Lx,y Lx,y' Lx,y2 Lx' Lx',y Lx',y' Lx',y2 Lz Lz,y Lz,y' Lz,y2 Lz' Lz',y Lz',y' Lz',y2

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Right cell twists: Ry Ry' Ry2 Rx2 Rz2 Rx2,y Rx2,y' Rx Rx,y Rx,y' Rx,y2 Rx' Rx',y Rx',y' Rx',y2 Rz Rz,y Rz,y' Rz,y2 Rz' Rz',y Rz',y' Rz',y2

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Each move here either starts with an L or an R, followed by the rotations (separated by commas).

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Inside/Outside Twists

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Ix2 twist

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Ix2 move on physical 2×2×2×2

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The I and O cells are the sides with the next most turning freedom after L and R. This was referred to as an “axial twist” in Melinda’s video. The canonical moves for the I/IO cells are Ix Ix' Ix2 Ox Ox' Ox2

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These twists can be difficult to perform for speedsolving, so most people tend to split it up by temporarily doing 2 illegal 90 degree twists that yield the same result. Be careful though, as this can lead to an illegal state if you accidentally screw up while doing it this way.

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Slab Twists

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U2 twist

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U2 move on physical 2×2×2×2

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The remaining canonical twists involve picking up a 2x2x4 “slab” off the puzzle, and rotating it 180 degrees in the same plane. These slabs you can pick up are the U, F, D, and B cells. Because the slabs can only be turned 180 degrees those moves will just be referred to as U2 F2 D2 B2

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Gyro

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If you try to scramble the puzzle with all the moves above, you will quickly notice that the 2 colours on the x-axis aren’t mixing with the other colours. This is because the simple rotations and slab twists are restricting what we can do because of the symmetry of the physical puzzle. To fix this, we need a series of illegal moves that rotate the puzzle 4-dimensionally in a way that changes the x-axis. This is called the Gyro, and will allow us to access all the rest of the puzzle rotations (xz zx yx xy xw wx)

+

Rowan performing the Gyro algorithm

+

There are several different algorithms for this, some of which gyro different axes. Melinda has several videos about different gyro algorithms, the shortest known one being 6 snaps. Below is a common gyro algorithm that several people in the community use.

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    +
  • Take the left endcap off and put it on the right so it becomes the right endcap (this brings the puzzle into the inverted state)
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  • Ly Ry'
    • +
  • Take the right endcap off and put it on the left so it becomes the left endcap (this brings the puzzle back into the normal state)
    • +
  • Rx2 B2 D2 Lx2
    • +
+

Expanded Canonical Moveset

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The expanded canonical moveset is what is accepted for the Hypercubing.xyz leaderboards. These moves all correspond perfectly with twists on the virtual puzzle, but weren’t allowed in the canonical moveset because people agreed not to for some reason.

+

Extra slab twists

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Several new slab twists are added. For the U cell, this means going from having just Uy2 to now having Uy2 Ux,y2 Ux',y2 Ux Ux' Ux2 Uz2

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Extra I/O cell twists

+

Iz2 Iy2 Oz2 Oy2 are added (although they are very hard to fingertrick and perform quickly).

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Scramble notation

+

Scrambles use comma-separated cell twists and the special move #. # represents a Uy2 move followed by a “left-to-right restack,” which consists of taking the left endcap off and putting it on the right side of the puzzle. This is not a legal move during solves, but does result in a legal puzzle state so it is used during scrambles.

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Physical 2x2x2x2 Image Generator

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Generate an image of a physical 2x2x2x2 by inputting moves using canonical moves notation, or by specifying the color per sticker. You can also do this directly from the URL by adding ?&moves= or ?&stickers= at the end, and using - to separate (and H instead of # for gyro).

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+Generate by move input +

Moves that the generator will accept: +zy yz xz zx yx xy yw wy xw wx zw wz Ly Ly' Ly2 Lx2 Lz2 Lx2,y Lx2,y' Lx Lx,y Lx,y' Lx,y2 Lx' Lx',y Lx',y' Lx',y2 Lz Lz,y Lz,y' Lz,y2 Lz' Lz',y Lz',y' Lz',y2 Ry Ry' Ry2 Rx2 Rz2 Rx2,y Rx2,y' Rx Rx,y Rx,y' Rx,y2 Rx' Rx',y Rx',y' Rx',y2 Rz Rz,y Rz,y' Rz,y2 Rz' Rz',y Rz',y' Rz',y2 Ix Ix' Ix2 Ox Ox' Ox2 U2 F2 B2 D2 # + +

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+Generate by sticker input +

Colors: +

W = white
+Y = yellow
+R = red
+O = orange
+G = green
+B = blue
+M = magenta (purple)
+P = pink
+S = silver (gray)
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+Piece/Sticker order: +LUBO LUBI LUFI LUFO LDBO LDBI LDFI LDFO RUBI RUBO RUFO RUFI RDBI RDBO RDFO RDFI

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Melinda’s 2x2x2x2

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Melinda’s 2x2x2x2

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Melinda's 2×2×2×2

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4D Shape: Tesseract

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Physical Shape: Rectangular prism

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Pieces: 16 4c

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Magnets: 384

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Permutations: \(3.3*10^{27}\)

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+

See Melinda’s 2x2x2x2 website for her project overview.

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History

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Invention

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Many hypercubers, including Melinda Green (one of the developers of MC4D) wanted to create a physical 34, but sadly it just seemed too complicated. Eventually, she decided to focus on the 24 due to its simplicity of only having 16 4c pieces.

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Throughout 2013 and 2014, Melinda was in contact with Oskar van Deventer, and together they tried to think of any possible mechanism for a physical 24. Originally they were determined to find a good mechanism that didn’t have to rely on magnets, but after several failed attempts a magnetic mechanism was the only solution. In 2017, Melinda built the first prototype and shared an unlisted YouTube video showing it off to the mailing list members.

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A few months later, Melinda built a 2nd prototype using 3D printing from a company called Shapeways (which just so happened to offer 8 colors, 6 of them being standard Rubik’s Cube colors, plus pink and purple (this is where the standard 4D colour scheme comes from)).

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Popularization

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Professor Erno Rubik inspecting a physical 24 at G4G 2018

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Erno Rubik inspecting Melinda's 2×2×2×2

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At the Gathering For Gardner conference of 2018, Roice Nelson got to show his physical 24 to Erno Rubik. It was reported that his only comment was something along the lines of “none of the derivative puzzles matter and that only his original invention is important”.

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Also in 2018, the YouCuber CanChrisSolve? made a series of videos where he solved the physical 24, which have now amassed over 100,000 views combined! A month later, a Spanish YouCuber called TheMaoiSha made a video explaining the physical 24 which now sits at over 800,000 views, exposing the world of hypercubing to more and more people.

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Because of all this recognition, many new people bought the puzzle, but it was still pretty expensive because of the costs of 3d printing and the labor of manual assembly. Eventually Melinda moved away from 3D printing in favour of injection molding, which greatly reduced the price while increasing the quality dramatically.

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+

Where can I buy one?

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Read this section of Melinda’s website.

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+

How Does it Work?

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+

Animations showing how the virtual and physical 24 transform into each other

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+Physical to virtual 2×2×2×2 transformation 1 +Physical to virtual 2×2×2×2 transformation 2 +

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+

Each twist and rotation on the virtual 24 is possible to do on the physical puzzle, albeit with certain moves requiring some extra setup. See the canonical moves page for a more detailed explanation.

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Also see Melinda’s video on the official canonical moves.

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In December of 2017, Melinda made a video fully explaining the puzzle, including its canonical moves, scrambling, and more.

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Physical 2x2x2x2 Scramble Generator

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See Scramble notation. In particular, # represents a Uy2 followed by a right-to-left restack.

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The starting orientation is:

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  • R = red
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  • U = white
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  • F = green
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Scrambles: 5

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Scrambles

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2x2x2x2x2

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2x2x2x2x2

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Grant's new 2x2x2x2x2

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5D Shape: Hypercube

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Physical Shape: “Rectangular prism”

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Pieces: 32 5c

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Designed: 2023 Feb 13

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The Physical 2x2x2x2x2 (25) is a valid design that represents its virtual counterpart, designed by Akkei, Grant, and Andreas. Unlike the restricted 2x2x2x2x2, this puzzle would have the full freedom and moves of the virtual puzzle.

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History

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In the Hypercubers Discord, Akkei and Andreas were discussing how to represent the symmetries of a 5D 5c piece in 3D. They came upon a certain colouring of the rhombic triacontahedron that worked.

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Akkei's 2×2×2×2×2

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Grant

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I see

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HHHHHHHHHMMMMMMMMM

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I seeeee

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Akkei quickly finished rendering the full puzzle, with all 32 pieces: +Akkei's 2×2×2×2×2

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A few hours later, Grant figured out a simpler way to arrange the stickers that still worked: +Grant's 2×2×2×2×2

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And 2 days later, after an extensive vc discussion, he rendered this different version where the pieces are oriented in a more readable way: +Grant's new 2×2×2×2×2

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This works because the rhombic triacontahedron can be viewed as 5 cubes circumscribed within it.

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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/puzzles/physical/2x2x2x3/index.html b/puzzles/physical/2x2x2x3/index.html new file mode 100644 index 00000000..e735cdc6 --- /dev/null +++ b/puzzles/physical/2x2x2x3/index.html @@ -0,0 +1,4021 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Grant’s 2×2×2×3 - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Grant’s 2x2x2x3

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Grant’s 2x2x2x3

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Grant's 2×2×2×3

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4D Shape: Hypercuboid

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Physical Shape: Rectangular prism

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Pieces: 16 4c, 8 3c

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Magnets: 1152

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Completed: 2022 Feb 3

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Grant’s 2x2x2x3 is a physical 4D twisty puzzle that was designed and 3D printed by Grant S. It works almost the same as its virtual counterpart, with the restrictions of only certain moves being canonical, plus the extra gyro algorithms needed to rotate the puzzle 4 dimensionally.

+

History

+

In December of 2021, Melinda green posted this image to the Hypercubers Discord server, jokingly calling it a 2x2x2x3:

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Melinda's 2×2×2×3

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After Melinda posted that, Luna started sketching out some ideas of possible layouts for an actual 2x2x2x3.

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Physical 3C piece rendering

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She invented the design for the 3c pieces by dividing the edges of a cube into 12 sections like this, grouping 4 sets of 3 together symmetrically.

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This is when Grant joined the server, and shared his 3D printed 2x2x2x2. Rowan then challenged him to build Luna’s design for the 2x2x2x3. After a lot of time tweaking the designs, printing them, magnetizing them, and assembling them, the 2x2x2x3 was completed on February 3rd 2022.

+

Grant's physical 2×2×2×3 shortly after finishing assembly

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How Does it Work?

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It follows the same basic principles as Melinda’s 2x2x2x2, except gyros get more complicated. Grant made a YouTube video showing off the legal twists, and how it matched the virtual puzzle in MPU exactly.

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Grant’s 2x2x3x3

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Grant’s 2x2x3x3

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Grant's 2×2×3×3

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4D Shape: Hypercuboid

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Physical Shape: Rectangular prism

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Pieces: 16 4c, 16 3c, 4 2c

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Magnets: 1728

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Completed: 2022 May 14

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+

History

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After the invention of grant’s 2x2x2x3, hypercubers were already thinking about what was next. The 2x2x3x3 has an extra middle layer that the 2x2x2x3 doesn’t have, which introduces the 2c piece to the design. Hactar designed the new 2c piece, and then Grant made the first rendering of what a physical 2x2x3x3 would look like.

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Grant's 2×2×3×3

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It took a while to get the designs just right, but after printing and assembling 4 of the new 2c pieces and 8 more 3c pieces, the puzzle was completed on May 14th, 2022.

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Functionality

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In order to prove that it was fully functional, it had to be shown that moves matched the virtual puzzle, and that it was possible to gyro the puzzle into different orientations.

+

The gyro starts by removing the middle layer, and then gyroing just like the 2x2x2x3, which involves separating that middle layer and then gyroing like a 2x2x2x2. Next, the middle layer has some 4 dimensional black magic done to it, and then finally the puzzle is reassembled into the gyroed state. After doing this, Grant realized that some of the 2x2x3 cell moves aren’t accessible, meaning that we actually need a 2nd type of gyro! This secondary gyro involves centering a 2x2x3 cell, and then making the middle layer stick out. Now the puzzle was fully functional.

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Grant’s 2x3x3x3

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Grant’s 2x3x3x3

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Grant's 2×3×3×3

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4D Shape: Hypercuboid

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Physical Shape: Rectangular prism

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Pieces: 16 4c, 24 3c, 12 2c, 2 1c

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Magnets: 2496

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Completed: 2022 Jul 6

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The next logical step to take after Grant built his 2x2x3x3 was the 2x3x3x3, which once again requires a new piece type: the 1c. But this was no big deal because a 1-colored piece only has 1 color.

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The production went pretty smoothly, and after printing 8 more 3c and 2c pieces, and 2 of the new 1c pieces, the puzzle was completed on July 6th, 2022.

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Now given what we’ve seen before with the previous gyro algorithms, you’d think that the 2x3x3x3 needs an extremely long complicated recursive gyro, and although the design of the puzzle has at least 15 different possible types of gyros, it turns out that it actually doesn’t need one at all! All the moves are accessible from this state, which makes for a really nice solving experience.

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The 4th ever physical 4D puzzle was constructed, bringing us closer and closer to our ultimate goal: the physical 3x3x3x3.

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Grant’s 3x3x3x3

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Physical 3x3x3x3

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physical 3×3×3×3

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4D Shape: Tesseract

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Physical Shape: Rectangular prism with 8 pieces sticking out

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Pieces: 8 1c, 24 2c, 32 3c, 16 4c

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Magnets: 3,840

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Completed: 2022 July 22

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Permutations: \(1.7\times 10^{120}\)

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History

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With the expansion of Melinda’s 2x2x2x2 through the hypercuboid series, only 1 puzzle remained unmade: the physical 3x3x3x3. A true 4-dimensional analog of the classic 3x3x3 Rubik’s Cube.

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Melinda Green

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I can’t believe progress is being made towards the holy grail. The cost and effort doesn’t matter since one will have to be made if it’s at all possible. (The main reason to be sure it’s right) It also doesn’t matter how clumsy it is to operate, though of course to get it into multiple people’s hands, you’ll want to explore every possible way to improve it.

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At first, it was not clear how to expand the physical 2x3x3x3 into the 3x3x3x3, as it would need extra pieces that would no longer make it a nice cuboid shape. This is because we have one more slice layer than the number of dimensions we’re trying to simulate. For example a 3x3x3 has 3 slice layers, so to represent it in 2D space, some of the pieces have to stick out like this:

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2D exploded view of the 3×3×3

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After months and months of effort, Grant completed assembling the puzzle on July 22nd, 2022. To this day, Grant has not scrambled or solved the physical 3x3x3x3 due to it being extremely big, heavy, and awkward to use. The most that was done on it was a checkerboard algorithm, but he did make a video showing off all of the hypercuboids and their legal twists.

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Grant holding the physical 3×3×3×3

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In April 2023, Akkei made a new program that simulated the physical 34, and then Hyperespy became the first person to ever fully solve the physical 34. The actual gyros are pretty painful to do, but luckily the program does them for you :)

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In just under 6 months, the community went from having 1 physical 4d puzzle, to having 5!

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On 2024-07-19, The Cube Dude became the first person ever to solve the physical 34 in real life.

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Dominik’s 4D Pyraminx

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Dominik’s 4D Pyraminx

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Dominik's 4D Pyraminx

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4D Shape: 5-cell

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Physical Shape: Tetrahedron + small Octahedron

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Pieces: 5 4c, 10 3c

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Magnets: 180 +Completed: 2023 Dec 28

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Introduction

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History

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Based on some ideas for the 4D pyraminx, Dominik first created a paper model in mid-December of 2023. After some suggestions from Melinda Green he created the prototype with magnets which was the first functioning version of this design. In April 2024 he finished an improved 3d printed version of the puzzle.

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How does it work?

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The puzzle can be moved like a pyraminx but with some additional legal moves. The edges that are in the position of the trivial tips can’t legally be twisted.

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There are a few ways to project a 5-cell into 3d space. In a vertex-first projection the “hidden” cell is on the “outside” of the others while in a cell-first projection it is on the “inside”. My design for the 4D pyraminx represents the latter way. In a solved state we have 4 colors on the outside and one on the inside of the puzzle.

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Projections

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2D visualizations of 3D perspective projections of the first iteration of a pentatope-based fractal: a) vertex-first, b) cell-first, c) face-first, d) edge-first.

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VertexFirst CellFirst

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(pictures: projections Source, vertex first pyraminx, cell first pyraminx Source)

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The Pieces

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There are ten tetrahedron-shaped 3-colored edge pieces where the fourth face is split into three colors.

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EdgePiece

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Then there are five octahedron-shaped 4-colored center pieces where four of the faces are split into three colors of the adjacent faces.

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CenterPiece

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The five 4-colored trivial tips can also be represented by tetrahedrons but with the current design they would need to be a separate arrangement. So this puzzle without the trivial tips represents just the truncated pyramnix.

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TrivialTips

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Chirality

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For the pieces to fit together in the correct color arrangement they need to have the same chirality. There is a left-handed and a right-handed version of the pieces and the centers and trivial tips need to have the same order of colors. Since the edges are allowed to be in more states the chirality doesn’t matter as much but if the chiralities don’t line up one color will be “favored” in a certain orientation.

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Moves

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There are some legal moves that don’t represent a movement in the 4D puzzle.

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Edge Migration

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You can move edges to a corresponding position on the “free center” and back.

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EdgeMigration EdgeMigrationBack

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Reorient Edges

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Edges can be in two states in relation to a center. They can align with the split face towards the center which will be call a “correct” state or the split face can be away from the center which will be called an “incorrect” state. In the latter case the edge can legally be oriented in three different ways.

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IncorrectEdge CorrectEdge

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OrientEdge

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Next we have moves which do represent a movement in the 4D puzzle. Unlike the normal 3D pyraminx the tips can’t legally be twisted. Other than that the four different moves of turning two layers of the pyraminx is legal. In the 4D version of the puzzle we have 20 different of such slice moves. Here are the four axes of rotations that are allowed for each 2-layered pyramid.

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Move1 +Move2 +Move3 +Move4

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If you perform such a move on each of the four sides of the tetrahedron you get a total of 16 moves. For the missing four moves we can “equip” the free octahedron with the edges by swapping them onto it and perform the rest of the rotations that way.

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Equip

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Move5 +Move6 +Move7 +Move8

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After the rotation you have can move the edges back.

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Unequip +…

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This is not necessary for a solve, though, since you have access to all the edge pieces through the 16 other moves.

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To keep the puzzle in a legal state sometimes after certain rotations we need to make edge reorientations. When performing a non-standard slice move an edge that isn’t in a correct state regarding the center that has rotated can after a turn show a split face.

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UnorientedEdge

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If it isn’t in a tip position it needs to be reoriented for otherwise the puzzle can get into an illegal or impossible state and also the color alignment becomes unknown for practical purposes when the puzzle is scrambled.

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How does this work? Before the rotation the colors A and B are on the outside.

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EdgeCorrect

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After a non-standard rotation either A or B will be on the inside and then there are two possible cases. Firstly, if the edge is “correct” then the split face will remain touching the rotated center and the full face with color C will show on the outside.

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EdgeCorrectAfterRotation

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The other case is when the edge is “incorrect”. Here the split face will be on the outside and the full color C will be touching the center.

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EdgeIncorrect +EdgeIncorrectAfterRotation

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As a rule you can now rotate the edge in the “down” direction in regards to the the rotation (where the center would be in the “up” direction) as shown here.

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EdgeRorientation +UpDown

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This way we can simulate a 3-cycle of the faces A, B and C with the 4-sided shape of the tetrahedron.

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Gyro

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While it isn’t necessary for a solve it is possible to reorient the whole puzzle by what’s usually called a gyro. There might be better ways to do it but here is one way that can be done in four steps.

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Step 1

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First you have to align the free octahedron with one of the “outer” edges. Regarding the colors it’s best to look at the colors of the adjacent octahedron. In this step you have two cases. Either the edge is already correct the you don’t have to do anything or the edge is incorrect then you have to reorient it towards the free octahedron.

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Align +OrientTipEdge

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Step 2

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Now you have to separate one 2-layered pyramid next to the aligned free octahedron. Here you have to watch out for the three edges facing you. In case they are “incorrect” you have to reorient them towards you. Then you can move it together towards the free octahedron.

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Separate +Orient1 +MoveTogether

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Step 3

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Here you have to look at what’s left of the “upper pyramid”. The topmost piece and the “inner” piece have to be reoriented in case they are incorrect. This time the direction you have to reorient them towards is “down”, which means it’s just like you would do in a normal slice move. One handy thing is that for the “inner” piece you simply have to look on the inside by only taking these three pieces off and if you see a whole face you can just leave the piece there. If the face is 3 colored you can just take it and rotate it together with the other pieces and it will be correct.

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HighlightedPieces +Orient2 +Flip1 +Flip2

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Step 4

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The last step only involves the last edge. Again, if it’s incorrect you first have to reorient it, this time towards the “down” direction. And then you take the piece and move it “over the Pyraminx”, like this, and simply put it in the right spot.

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Orient3 +MoveOver

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That’s all there is to the gyro and here we have all in one movement.

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Gyro

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Comparison of Moves with MC4D

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Dominik’s PyraminxMagicCube 4D
Move1MC4DMove1
Move2MC4DMove2
Move3MC4DMove3
Move4MC4DMove4
Move5MC4DMove5
Move6MC4DMove6
Move7MC4DMove7
Move8MC4DMove8
GyroMC4DGyro
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Example solve

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Here is a link to a video of an example scramble and solve of the puzzle:

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Also you can watch a tutorial of how to solve it here:

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Notes for a solve

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The 4D pyraminx can end up with a single edge flipped incorrectly unlike the 3D pyraminx. This state can be solved with an algorithm that flips two edges, the right slice move and an additional two edges flip. But if a piece was moved in a wrong way accidentally it can also end up in a state that’s not solvable like this. The reason for this is that there are three states that flipping an edge moves between: (a) solved state, (b) one edge correct and the other two flipped and (c) all edges incorrect.

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Going from (c) to (a) requires two flips so the method above can flip edge A from (b) to (a) or (c) but then edge B will go to state (b). In the following tree diagram it can be seen how all 2 flip moves only lead to looping between unsolved states:

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Tree

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Physical Puzzles

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Warning

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It is highly recommended to learn how virtual puzzles work first before trying to understand the physical puzzles. See Why not start with the physical 2x2x2x2?.

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Physical 3x3x3x3

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Grant's physical 3×3×3×3

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Physical puzzles refer to higher dimensional puzzles that are physically built in real life (using only 3 dimensions). This involves clever designs, strange symmetry-abusing tricks, and lots and lots of magnets. Often, these puzzles are too impractical to use, defeating their sole purpose. For an explanation of how these puzzles work, see the theory page.

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For a documentary about the history of physical puzzles as of December 2022, see Rowan’s video.

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Physical puzzles that have been built

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Designs that haven’t been built yet

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  • AxBxCxD (any hypercube or hypercuboid with layers bigger than 1)
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  • 1x1xAxB series
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  • Markk’s duoprisms
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  • {5}x{4} duoprism
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  • Akkei’s 2x2x2x2x2
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Restricted 2x2x2x2x2

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Restricted 2x2x2x2x2

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Grant holding his restricted 2×2×2×2×2

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5D Shape: Penteract

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Physical Shape: Rectangular prism

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Pieces: 32 5c

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Magnets: 768 (without buffer)

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The restricted 2x2x2x2x2 or 25 (also called the physical domino reduced 2x2x2x2x2) is a physical puzzle that represents the state space of a virtual 25 puzzle, with one of the axes only being able to do 180 degree turns. This puzzle is like the 5D equivalent of scrambling a 23 with only <U,D,R2,F2,L2,B2>, which makes the solve way easier than the non-restricted version.

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The first known restricted 25 was made by Grant Staten which he showed off in a YouTube video. Hyperespy also made a version using foldable paper cubes to cover the pieces.

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Hyperespy's Restricted 2×2×2×2×2

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Pieces and Twists

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There are 32 5c pieces, but each piece can only be oriented in 12 ways instead of 60 because of the axis restriction. And just like Melinda’s physical 24, this puzzle has certain legal moves that a solver must follow. Each twist of this puzzle is like rotating a tesseract. You can take off half of it and do any reorientation like you would on the physical 24. This includes simple rotations and gyros. The only thing you can’t do is a 90 degree twist of the square layers (this results in a legal puzzle state, but is not analogous to any move on the virtual 25). See Hyperespy’s video for a full overview of the canonical moves and a way to scramble the puzzle quickly.

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How to make one

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Buffer pieces magnet layout

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A drawing of the magnet layout for the buffer pieces of a restricted 2×2×2×2×2

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Order a 24 DIY kit from Melinda Green. Then assemble it with a mirrored colour scheme from your current 24, but with the same magnet layout. For the 5D stickers 192 are needed in total, with half (96) of them being black and the other half grey. This means that each normal physical 24 piece gets 6 stickers on it. A good option is to order a couple of custom sticker sets online such as these.

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Grant used buffer pieces when making his version in order to make it obvious to avoid doing illegal twists. If you want to add buffer pieces, you need 8 monocolored hypercubies with a custom magnet layout. Although there’s only a few different illegal twists that are even possible so it’s not too tricky to avoid altogether.

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+ + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/search/search_index.json b/search/search_index.json new file mode 100644 index 00000000..ceba5821 --- /dev/null +++ b/search/search_index.json @@ -0,0 +1 @@ +{"config":{"lang":["en"],"separator":"[\\s\\-]+","pipeline":["stopWordFilter"]},"docs":[{"location":"","title":"Welcome","text":""},{"location":"#hypercubing","title":"Hypercubing","text":"

  • Introduction to 4D

    Learn how the fourth dimension works

  • Frequently Asked Questions

    Get answers to common questions

  • Introduction to Hypercubing

    Jump into hands-on hypercubing

  • Software

    Download hypercubing programs

  • Discord server

    Chat with other hypercubers

  • Progression

    Solve puzzles to build specific skills

Hypercubing is the hobby of solving twisty puzzles (such as the Rubik\u2019s cube) in higher dimensions. The ways that twisty puzzles move are mathematically well defined, and can be generalized to higher spatial dimensions. These puzzles can then be visualized and simulated using computer software.

The most well known 4D shape is the hypercube (also called the tesseract, 8-cell, octachoron, or 4-cube). It has 8 cubic sides that are called cells. Turning any of the cells involves rotating it like a cube to any of 24 orientations.

The short article Abstracting Rubik\u2019s Cube introduces a number of the hypercubing puzzles.

"},{"location":"contributing/","title":"Contributing","text":"

Welcome! If you\u2019d like to contribute, join the #wiki channel on the Discord server and ask what you can do to help or let us know what changes you want to make.

If you aren\u2019t familiar with creating pull requests on GitHub, see this quick video tutorial.

"},{"location":"contributing/#setup","title":"Setup","text":"
  1. Install Python 3.9 or later
  2. Install Git
  3. Clone the repository and install requirements:
git clone https://github.com/Hypercubers/hypercubing.xyz\ncd hypercubing.xyz\npython3 -m pip install -r requirements.txt --user\n

Run python3 -m mkdocs serve and go to http://127.0.0.1:8000/ in your web browser. As you edit files, the page will automatically refresh to show your changes.

"},{"location":"contributing/#file-layout","title":"File layout","text":"

Markdown files should have lowercase names, with hyphens to separate words. Choose shorter names when possible.

"},{"location":"contributing/#pages","title":".pages","text":"

Some folders have a .pages file in them, which changes the order of pages in the navigation sidebar on the left. This uses the mkdocs-awesome-pages-plugin. Most of the time, you shouldn\u2019t have to worry about this.

"},{"location":"contributing/#syntax","title":"Syntax","text":"

This site uses Markdown.

For linking between files within the site, use an absolute link like this with no trailing .md:

[Melinda's physical 2^4^](/puzzles/physical/2x2x2x2/index.md) was the first 4D puzzle to have a 3D physical design.\n

This site uses the Material for MkDocs theme; in particular, this gives us admonitions, tables, footnotes, and subscripts & superscripts. We also have MathJax, which uses $ symbols:

The $n$th Fibonacci number is given by $F_n = F_{n-1} + F_{n-2}$.\n

The \\(n\\)th Fibonacci number is given by \\(F_n = F_{n-1} + F_{n-2}\\).

"},{"location":"contributing/#adding-images","title":"Adding images","text":"

We are hosting images using a private Nextcloud instance hosted on DigitalOcean. Contact HactarCE on Discord to get access to add images to be hosted there. Once you have an image somewhere in the assets folder, you can embed it in a page. When including any image, be sure to include a brief text description of the image for screen readers. For example:

![Erno Rubik inspecting Melinda's 2x2x2x2](https://assets.hypercubing.xyz/img/phys/melinda_2x2x2x2_erno_inspects.jpg)\n
"},{"location":"contributing/#naming-images","title":"Naming images","text":"
  • Organize images into folders when possible. If you\u2019re adding many related images, consider making a new folder for them.
  • Use underscores when naming.
  • Use only lowercase, except when the filename includes twist notation.
  • Try to stay consistent with existing filenames.
"},{"location":"contributing/#abbreviations","title":"Abbreviations","text":"

Abbreviations are listed in includes/abbreviations.md. Be careful adding new ones, lest you create another Grant Standingslice incident.

"},{"location":"contributing/#style","title":"Style","text":"

When naming a puzzle with a product in its name, prefer the cross symbol x over x (e.g. 3x3x3x3 vs. 3x3x3x3). By using the unicode x in the source file, the site will automatically make it searchable using x. In an environment where you cannot use unicode x, use the HTML escape &times;.

"},{"location":"discord/","title":"Discord","text":"

How did you get here?

This page is supposed to redirect to the Discord server.

"},{"location":"faq/","title":"FAQ","text":""},{"location":"faq/#other-faqs","title":"Other FAQs","text":"
  • MC4D FAQ on superliminal.com
  • quickfur\u2019s 4D FAQ on qfbox.info
"},{"location":"faq/#table-of-contents","title":"Table of contents","text":"
  • What is hypercubing?
    • Isn\u2019t the 4th dimension time?
    • How can we visualize 4D?
    • Where can I interact with other hypercubers?
  • Virtual puzzles
    • What program should I download?
    • How do I learn to solve 3x3x3x3?
    • What methods exist for the 3x3x3x3?
    • Why not start with the physical 2x2x2x2?
    • Why not start with the virtual 2x2x2x2?
    • What is God\u2019s number for [puzzle]?
  • Physical puzzles
    • What is a physical 4D puzzle?
    • How can I buy a physical 2x2x2x2?
    • How can I buy other physical puzzles?
    • Can I download 3D files for the physical 2x2x2x2?
    • What physical 4D puzzles have been built?
    • What physical 4D puzzles are possible?
  • Hyperspeedcube
    • Does Hyperspeedcube run on my OS?
    • I get an error when I try to run Hyperspeedcube
    • How do I use keybinds in Hyperspeedcube?
    • How do I use piece filters in Hyperspeedcube?
  • Hyperspeedcube 2
    • When will Hyperspeedcube 2 be ready?
    • What features are planned for Hyperspeedcube 2?
    • Does Hyperspeedcube 2 have [feature] yet?
    • Can I download the latest development build?
    • I\u2019m having trouble with Hyperspeedcube 2?
    • Where can I follow the latest development updates?
  • Speedsolving
    • What are the speedsolving records for 4D puzzles?
    • Why not use speedrun.com?
    • I don\u2019t know full OLL/PLL/ZBLL/etc. Can I still get fast at 4D?
    • What 4D algorithms are there?
    • How can I get faster at solving 3x3x3x3?
    • How could the 3x3x3x3 record be improved?
  • Does this puzzle exist?
    • 2D Rubik\u2019s Cube
    • 4D Square-1
    • 4D Skewb
    • 8-dimensional and higher
    • 3D Rubik\u2019s Clock
    • How do I make a 4D [thing]?
"},{"location":"faq/#what-is-hypercubing","title":"What is hypercubing?","text":"

Hypercubing is the hobby of solving twisty puzzles (like the Rubik\u2019s cube) in higher dimensions and exotic geometries.

"},{"location":"faq/#isnt-the-4th-dimension-time","title":"Isn\u2019t the 4th dimension time?","text":"

While time is one dimension in 4D spacetime, it behaves completely differently from the three spatial dimensions. Hypercubing deals with higher dimensions of space, where all dimensions are interchangeable.

"},{"location":"faq/#how-can-we-visualize-4d","title":"How can we visualize 4D?","text":"

Our eyes see only a 2D projection of the 3D world, and we\u2019re able to reconstruct the 3D world around us with only minimal difficulty. Using mathematics, we can project a 4D shape onto 3D, and then project that 3D shape onto 2D to be displayed on a computer screen. With enough practice, it\u2019s possible to reason intuitively about higher-dimensional space using these visualizations.

"},{"location":"faq/#where-can-i-interact-with-other-hypercubers","title":"Where can I interact with other hypercubers?","text":"
  • The Hypercubers Discord Server has the most active community of hypercubers and contains the latest updates on developing projects and speedsolving strategies.
  • The Hypercubing Google Group is a good option for those who prefer mailing lists or less frequent updates.
  • The r/Hypercubers subreddit is mostly inactive.

Before the Google Group or Discord server, there was a Yahoo Groups mailing list. Those messages are archived here.

"},{"location":"faq/#virtual-puzzles","title":"Virtual puzzles","text":""},{"location":"faq/#what-program-should-i-download","title":"What program should I download?","text":"

For getting started, we recommend Hyperspeedcube. See the software page for a list of hypercubing software and feature comparisons.

"},{"location":"faq/#how-do-i-learn-to-solve-the-3333","title":"How do I learn to solve the 3x3x3x3?","text":"

We highly recommend that you figure out how to solve the 34 on your own. It\u2019s a fantastic puzzle and you\u2019ll learn a lot!

  1. Learn how to solve a 3x3x3, ideally intuitively. There are plenty of tutorials online, although you should try it on your own first. It\u2019s a good challenge!
  2. Download Hyperspeedcube or use the web version.
  3. Follow this video to get aquainted with the program:

Once you feel comfortable solving 33 from a full scramble and 34 from a 1-move scramble, you know everything you need to do a full solve. If you get stuck, you can ask for help on the Hypercubers Discord server.

"},{"location":"faq/#what-methods-exist-for-the-3333","title":"What methods exist for the 3x3x3x3?","text":"

Many 3D methods can just be scaled up and used on the 4D cube. Some notable methods are:

  • Layer-by-layer
  • By piece type
  • CFOP
  • 3-Block (4D FreeFOP)
  • Octachoroux (4D Roux)

As of 2024, 3-Block is the most popular speedsolving method and what\u2019s used in the world record, but CFOP is also competitive.

"},{"location":"faq/#how-do-i-start-learning-to-solve-4d-puzzles","title":"How do I start learning to solve 4D puzzles?","text":"

First, download Hyperspeedcube or MC4D and start experimenting with the 34! Try to solve one-move scrambles and keep practicing that until you\u2019re comfortable. Once you can solve one-move scrambles with ease, pick a method to learn.

"},{"location":"faq/#why-not-start-with-the-physical-2222","title":"Why not start with the physical 2x2x2x2?","text":"

You will never understand physical puzzles before understanding virtual puzzles. Computer simulations are the most direct way to experience 4D puzzles, and it\u2019s what all physical puzzles are emulating. It\u2019s effectively impossible to develop new physical puzzles without first understanding the equivalent virtual puzzles.

Grant Staten

I think that approaching hypercubing by starting with only physical puzzles does not lead to an understanding of how the puzzles actually work. Even if you can solve them.

Before I touched a virtual puzzle, I had:

  • almost gotten sub-2 on a physical 24 I\u2019d built
  • started building physical 2x2x2x3
  • rendered physical 2x2x2x3, 2x2x3x3, 2x3x3x3 (piggybacking off of Luna\u2019s work and just following patterns in place)

However, looking back, I 100% honestly did not yet understand how any of those puzzles actually worked at the time.

I have absolutely nothing against someone only focusing on physical puzzles. But in my experience you don\u2019t gain an actual understanding of how the puzzles work through solely using the physical puzzles.

"},{"location":"faq/#why-not-start-with-the-virtual-2222","title":"Why not start with the virtual 2x2x2x2?","text":"

The 24 is particularly disorienting for beginners because half of the puzzle turns at once. As a result, while the 24 strategy is technically simpler, it\u2019s actually more challenging to wrap your head around, especially when you\u2019re new to 4D puzzles. Just like how the 33 is a better starting puzzle in 3D, you can learn lots of important concepts from the 34 that will help you with other 4D puzzles.

Sergej Volkov

I regret not starting with a 34. I first solved virtual 24 using Rowan\u2019s physical method and it was extremely painful and did not really help to develop any 4D intuition. I ended up just drawing the physical representation of the puzzle on a piece of paper.

"},{"location":"faq/#what-is-gods-number-for-puzzle","title":"What is God\u2019s number for [puzzle]?","text":"

See God\u2019s Number.

"},{"location":"faq/#physical-puzzles","title":"Physical puzzles","text":""},{"location":"faq/#what-is-a-physical-4d-puzzle","title":"What is a physical 4D puzzle?","text":"

The physical 4D puzzles are puzzles that are perfectly analogous to the virtual 4D puzzles, but implemented in the physical world. See these links:

  • Physical Puzzles on this site
  • Physical Puzzle on the Superliminal Wiki
  • Rowan Fortier\u2019s video about the history of physical hypercubes
"},{"location":"faq/#how-can-i-buy-a-physical-2222","title":"How can I buy a physical 2x2x2x2?","text":"

See the Ordering Melinda\u2019s 2x2x2x2 on the Superliminal website. Also see Melinda\u2019s 2x2x2x2 on the Superliminal site, which includes the history, statistics, and Hall of Fame.

"},{"location":"faq/#how-can-i-buy-other-physical-puzzles","title":"How can I buy other physical puzzles?","text":"

Melinda\u2019s 24 is the only physical puzzle for sale. The physical 34 and hypercuboids are currently one-of-a-kind. If you want one, you need to design and 3D print it yourself.

"},{"location":"faq/#can-i-download-3d-files-for-the-physical-2222","title":"Can I download 3D files for the physical 2x2x2x2?","text":"

No. Melinda has put a lot of work into her physical 24 designs, and invested quite a bit of her own capital into selling prototypes at a loss and getting them mass-produced, so we respect her wishes to not make those files public. If you want to create your own files modeling Melinda\u2019s physical 24 and 3D print them, that is fine, but we ask that you do not make the files avaliable for download unless Melinda is OK with it.

"},{"location":"faq/#what-physical-4d-puzzles-have-been-built","title":"What physical 4D puzzles have been built?","text":"

See Physical Puzzles for a comprehensive list.

"},{"location":"faq/#what-physical-4d-puzzles-are-possible","title":"What physical 4D puzzles are possible?","text":"

While it\u2019s always possible to just arrange the stickers on a table, the real challenge is in finding a design that is piece-based instead of sticker-based and fits in a compact shape that isn\u2019t too horrendous to turn. This requires some out-of-the-box thinking and, in extreme cases, application of group theory. We currently have several renderings for physical puzzles that haven\u2019t been built in real life yet; see the Physical Puzzles page for an incomplete list.

"},{"location":"faq/#hyperspeedcube","title":"Hyperspeedcube","text":""},{"location":"faq/#does-hyperspeedcube-run-on-my-os","title":"Does Hyperspeedcube run on my OS?","text":"

Hyperspeedcube runs natively on Windows, macOS, and Linux. There is also a web version, which runs on Chromebooks or other devices where the downloaded version does not work. For mobile phones, it is possible to run Hyperspeedcube in a browser but not recommended. See Software for alternatives.

"},{"location":"faq/#i-get-an-error-when-i-try-to-run-hyperspeedcube","title":"I get an error when I try to run Hyperspeedcube","text":"

See Hyperspeedcube - Troubleshooting.

"},{"location":"faq/#how-do-i-use-keybinds-in-hyperspeedcube","title":"How do I use keybinds in Hyperspeedcube?","text":""},{"location":"faq/#how-do-i-use-piece-filters-in-hyperspeedcube","title":"How do I use piece filters in Hyperspeedcube?","text":""},{"location":"faq/#hyperspeedcube-2","title":"Hyperspeedcube 2","text":""},{"location":"faq/#when-will-hyperspeedcube-2-be-ready","title":"When will Hyperspeedcube 2 be ready?","text":"

When it\u2019s done. Hopefully before summer 2025.

"},{"location":"faq/#what-features-are-planned-for-hyperspeedcube-2","title":"What features are planned for Hyperspeedcube 2?","text":"

See Hactar\u2019s website.

"},{"location":"faq/#does-hyperspeedcube-2-have-feature-yet","title":"Does Hyperspeedcube 2 have [feature] yet?","text":"

See Hactar\u2019s website.

"},{"location":"faq/#can-i-download-the-latest-development-build","title":"Can I download the latest development build?","text":"

You can access the latest development builds either by donating to Hactar on Ko-fi (any amount one time should work) or building it yourself from the source code, which takes some time to set up but generally isn\u2019t too hard.

"},{"location":"faq/#im-having-trouble-with-hyperspeedcube-2","title":"I\u2019m having trouble with Hyperspeedcube 2?","text":"

Please do not ask Hactar for help building or using development builds of Hyperspeedcube 2 unless the build on Ko-fi is broken. Everything in the program is subject to change, and any time spent helping an individual user is time that could instead be spent writing documentation.

"},{"location":"faq/#where-can-i-follow-the-latest-development-updates","title":"Where can I follow the latest development updates?","text":"

See the Hyperspeedcube 2.0 Development Updates thread in #hyper-forum on the Hypercubers Discord Server. Once Hyperspeedcube 2.0 is ready for general use, there will be an announcement that pings the @Hyperspeedcube Update role (and possibly @everyone).

"},{"location":"faq/#speedsolving","title":"Speedsolving","text":""},{"location":"faq/#what-are-the-speedsolving-records-for-4d-puzzles","title":"What are the speedsolving records for 4D puzzles?","text":"

See the leaderboards. To get on the leaderboard, read the submission guidelines and submit a video of your solve to this form.

"},{"location":"faq/#why-not-use-speedruncom","title":"Why not use speedrun.com?","text":"

Speedrun.com does not allow \u201cgeneric Rubik\u2019s Cube simulators.\u201d We applied and were rejected.

"},{"location":"faq/#i-dont-know-full-ollpllzblletc-can-i-still-get-fast-at-4d","title":"I don\u2019t know full OLL/PLL/ZBLL/etc. Can I still get fast at 4D?","text":"

Absolutely! Most 4D speed methods are highly intuitive, and world-record times often use just 2-look OLL and PLL. Executing algorithms is a very negligible part of the solve compared to the massive amounts of pair or block building.

Hactar (mid-2024)

Ok I actually ran the numbers:

  • My 2:05.30 WR solve had full-step 2-look OLL + 2-look PLL and took 8 seconds
  • My 1:56.42 WR solve had an easy full OLL (OCLL skip) + U perm (CPLL skip) and took 4 seconds

Consider that the second solve is basically the best case for a 2-look last layer (the goal of full OLL+PLL), using algorithms that have really nice RKT cancels and are easy to execute with my keybinds and I\u2019ve practiced them a ton, but it was still only 4 seconds faster, which is ~2% of the total solve. Additionally, my experience is that it takes much more practice to execute 4D algorithms at max speed with a keyboard compared to executing 3D algorithms at max speed on a 3^3.

Based on all that, I can confidently say I don\u2019t think full OLL+PLL will ever be meaningfully better than RKT-canceled 2-look OLL + 2-look PLL, with one exception: There\u2019s a handful of full OLL algorithms (namely the ones composed of fruruf and sune) that I think are worth using if you know them from 3D, but aren\u2019t at all crucial. I do use these during 4D solves when I recognize them, and with some effort we may be able to find a few more cases with easy cancels, but I don\u2019t think it\u2019s worth learning these just for 4D solves.

"},{"location":"faq/#what-4d-algorithms-are-there","title":"What 4D algorithms are there?","text":"

There\u2019s so many cases for each step of the solve that creating a complete algorithm set is basically impossible, and there\u2019s so many options for moves that algorithm explorers are infeasible. Almost every algorithm we have is based on an algorithm from 3D, and the only search program we have is a sort of optimizer for one very specific kind of algorithm derived from 3D.

"},{"location":"faq/#how-can-i-get-faster-at-solving-3x3x3x3","title":"How can I get faster at solving 3x3x3x3?","text":"

First, learn 3-Block! Pairs are more popular since the recognition is easier, but solutions using triplets tend to be shorter; as far as we know, both are viable. If you\u2019re getting times in the 8-15 minute range, do slowsolves where you focus on efficient solutions to F2L pairs/triplets.

Hactar

From most to least significant:

  • 1-key-per-move keybinds that prioritize R and I cells. This is more than 2x faster than default keybinds, because you do not need to time the release of keys.
  • 1-key-per-move RKTbinds with RKT cancels. These two optimizations complement each other so well, because it spreads out the work among all your fingers.
  • Single keys for some 180-degree turns, because repeated keys are slow! By adding keys for x2 and y2, you can reorient a cell into any orientation with just two keypresses.
  • RKT-canceled triggers during F2L-3. I leave debt on U and have muscle memory for the common RKT-canceled trigger R (flip) U' R' (flip) (where (flip) = {1-2}Ozx2), its inverse (flip) R U (flip) R'*, and their back and left-handed forms. This + U moves accounts for basically everything you do in F2L other than R U2 R', which you can usually avoid or in worst case just (flip) R (flip) U2 (flip) R' (flip).
"},{"location":"faq/#how-could-the-3x3x3x3-record-be-improved","title":"How could the 3x3x3x3 record be improved?","text":"

Hactar (mid-2024)

Here\u2019s what I think would make a big difference, from most to least significant:

  • Lookahead during F2L-4. This is essentially unexplored.
  • Optimal solutions for F2L-b cases. My gut says this could save an average of ~3 moves per pair over my solutions during left block, and ~5 moves per pair during right block, which adds up to ~30 STM saved. I think there\u2019d be some value in a comprehensive study of 3-block F2L cases and what techniques are effective in solving them, and research into the psychology of how to recognize them. For some examples of what I mean: I remember reading somewhere that on 3^3 it\u2019s better to find an F2L corner and then search for the matching edge, rather than the other way around. I also know that once you find the edge, you can recognize its orientation to know whether you can solve the pair using just or whether you need less-ergonomic moves. Figuring out and documenting these sort of tricks for 4D would help.

    I think both of those improvements could save 15-30 seconds, bringing it from ~1:15 to ~0:45. I also feel like general lookahead/efficiency improvements in PLC might save 10-15 seconds but I can\u2019t put my finger on exactly what they would be. I know my OLC feels very variable during solves, but in practice I can brute-force pretty much anything into a valid final case by spending a second or two flipping one edge/corner.

    "},{"location":"faq/#does-this-puzzle-exist","title":"Does this puzzle exist?","text":""},{"location":"faq/#2d-rubiks-cube","title":"2D Rubik\u2019s Cube","text":"

    Depending on how you define \u201c2D Rubik\u2019s cube,\u201d it might or might not exist.

    A 3D Rubik\u2019s cube has 6 square faces and each face twists within its 2D plane (with 4 possible rotations). A 4D Rubik\u2019s cube has 8 cubic cells and each cell twists within its 3D plane (with 24 possible rotations). By analogy, a 2D Rubik\u2019s cube has 4 edges and each edge can be rotated within its 1D line \u2026 but there\u2019s no way to do a rotation in 1D. So if twists have to be rotations, then a 2D Rubik\u2019s cube doesn\u2019t have any turns and so isn\u2019t a puzzle (or is a trivial one).

    Using different definitions, we can create a few different puzzles that could reasonably be called a 2D Rubik\u2019s cube:

    • If we allow reflections instead of rotations, we get the Reflesquare, which generalizes to the Reflecube.
    • If we allow translations instead of rotations, we get Loopover.
    • If we allow circular cuts instead of flat cuts, we get various MagicTile puzzles, including some that are actually equivalent to an ordinary 3D Rubik\u2019s cube!
    "},{"location":"faq/#4d-square-1","title":"4D Square-1","text":"

    Square-1 is fundamentally a bandaged dodecagonal prism. There are so many ways to extend that into 4D that there isn\u2019t really a canonical \u201c4D square-1\u201d

    "},{"location":"faq/#4d-skewb","title":"4D Skewb","text":"

    There\u2019s so many ways to generalize a skewb to 4D that we have a whole page full of them!

    "},{"location":"faq/#8-dimensional-and-higher","title":"8-dimensional and higher","text":"

    Above 5 dimensions, cube puzzles aren\u2019t more difficult or interesting, just more tedious and computationally expensive. But there might be some wild hyperpuzzles yet to be discovered up there, say one based on the very special geometry of the E8 Lattice!

    "},{"location":"faq/#3d-rubiks-clock","title":"3D Rubik\u2019s Clock","text":"

    Instead of rotating circles in 2D, you can rotate spheres in 3D. This is a more interesting puzzle than the traditional Rubik\u2019s Clock because moves don\u2019t commute. No one\u2019s written a program yet to simulate it but you totally could!

    "},{"location":"faq/#how-do-i-make-a-4d-thing","title":"How do I make a 4D [thing]?","text":"

    Generalising Things to 4D: A Handy Guide

    1. Understand and define the thing you\u2019re generalising
    2. Find where your definitions reference or assume something dimension-specific
    3. Rewrite your definitions to avoid dimension-specific references or assumptions
    4. Find what 4D object fits your new definitions (there may be one, several, or none)
    "},{"location":"glossary/","title":"Glossary","text":"

    This is a glossary of terms used in the hypercubing community. We take a mostly prescriptivist1 approach: terminology has a great effect on how we think about puzzles, and we try to be mindful when naming concepts and inventing notation to ensure that they encourage better understanding of puzzles and are useful in as many contexts as possible.

    Before you propose new terminology \u2026

    We\u2019ve wrestled in the past with poor terminology that actively hurt understanding. First, gain hands-on experience and intuition for the thing you want to describe, and then see what terms are actually needed. There\u2019s no value in making up words for the pieces on a 7-dimensional puzzle, for example, if there\u2019s no need to communicate about them.

    "},{"location":"glossary/#puzzle-elements","title":"Puzzle elements","text":"

    A 1-dimensional turning axis is not always well-defined for higher-dimensional puzzles, because rotations generally happen in a plane, not around an axis.

    "},{"location":"glossary/#polytope-elements","title":"Polytope elements","text":"

    For an \\(N\\)-dimensional polytope: (Some of these terms are from Polytope - Wikipedia)

    • vertex = rank 0, single point
    • edge = rank 1, line connecting two vertices
    • face = rank 2, polygon constructed from edges
    • cell = rank 3, polyhedron constructed from faces
    • \u2026
    • \\(N\\)-face = rank \\(N\\), polytope constructed from rank \\(N-1\\) elements
    • peak = \\(N-3\\) face
    • ridge = \\(N-2\\) face
    • facet = rank \\(N-1\\), polytope constructed from rank \\(N-2\\) elements

    In 4D, we prefer facet rather than cell. In simple terms: on most puzzles, a facet is the thing with a single color.

    "},{"location":"glossary/#pieces","title":"Pieces","text":"

    Basic definitions for an \\(N\\)-dimensional hypercubic puzzle:

    • corner = piece with \\(N\\) colors (4 colors in 4D)
    • edge = piece with \\(N-1\\) colors (3 colors in 4D)
    • peak or 3c = piece with 3 colors (5D+)
    • ridge or 2c = piece with 2 colors (4D+)
    • center or 1c = piece with 1 color

    We prefer words rather than 1c, 2c, etc. because the words generalize better to other, non-facet-turning puzzles and describe how a piece behaves rather than how it looks.

    "},{"location":"glossary/#moves","title":"Moves","text":"
    • axis or turning axis = ray start from the center of the puzzle, around which puzzle elements rotate during twists
    • twist or move or turn = movement of pieces that changes the puzzle state
    • rotation or full-puzzle rotation = rotation of the whole puzzle that does not change its state
    "},{"location":"glossary/#solving","title":"Solving","text":""},{"location":"glossary/#actions","title":"Actions","text":"

    An action is sequence of moves that preserves invariants of the stage. Usually, an action keeps certain pieces solved. For example, when a 4n has been reduced to a 3n using big cube reduction, outer layer moves are the only actions. In this case, the actions are reduced moves. Another common set of actions is RKT.

    "},{"location":"glossary/#parity","title":"Parity","text":"

    There is no community consensus on the definitions of parity. Below are some proposed definitions:

    • group theory parity = a case where the puzzle is in an unexpected coset of a subgroup of index 2
      • It is often more broady applied to a case where the puzzle is in an unexpected coset of a subgroup of any index.
    • cuber parity = a case that is difficult to solve that the solver didn\u2019t expect
      • Melinda\u2019s definition: a local maximum, where the puzzle is largely solved but requires many moves to fix
      • Hactar\u2019s definition: a case which cannot be solved using the actions expected at this stage in the solve

    None of these definitions are satisfactory. According to most of these definitions, RKT parity is not parity at all, but is more accurately called \u201cRKT error.\u201d According to Melinda\u2019s definition, most PLL cases are parity. The first definition given for \u201ccuber parity\u201d is highly subjective, but is the only one that captures its current use.

    Open questions

    • Is there a definition for \u201ccuber parity\u201d that captures the way it\u2019s naturally used?
    • Is there a catchy term we can use instead of \u201cRKT parity\u201d? Melinda proposes \u201cRKT error.\u201d
    "},{"location":"glossary/#f2l","title":"F2L","text":"

    F2L is a very general solving strategy that works by building a small block of pieces and then inserting the block into its solved position. F2L stands for \u201cfirst two layers\u201d because it was originally developed to solve the first two layers of 33, but in hypercubing we use it for many other puzzles.

    "},{"location":"glossary/#f2l-axes","title":"F2L axes","text":"
    • free axes = axes which affect only unsolved pieces; can be turned freely
    • side axes = axes which affect some unsolved pieces and some solved pieces; can be turned, but must be turned back to restore solved pieces
    • base axes = non-free and non-side axis that is not completely solved; usually mostly solved, rarely turned during F2L
    • top axis = the free axis currently being worked on

    Examples

    • In F2L on a 33, D is the only base axis, U is the top axis (the only free axis), and R, L, F, & B are all side axes.
    • When beginning F2L on a megaminx there are, 6 free axes, 5 side axes, and 1 base axis.
    • Near the end of F2L on a megaminx there are, 1 free axis, 5 side axes, and 5 base axes.

    We use the letter T to represent the top axis, R & F to represent intersecting side axes, and R & L to represent non-intersecting side axes.

    "},{"location":"glossary/#f2l-blocks","title":"F2L blocks","text":"

    An F2L block or pair is a group of pieces that is paired and solved as one unit. There\u2019s usually a head and body, where the head intersects with more twisting axes than the body.

    The base sticker of a head is the sticker which will be facing the base axis when it is solved. The facing direction of the head of a block is whatever direction its base sticker is facing. The facing direction of the body of a block is the same as the head, when they are paired. This notion of which direction a head or body faces gives a way to describe edge orientation before the pieces have been paired, which is helpful especially in 4D+ where edge orientation is otherwise difficult to define.

    Examples

    • On the 33 an F2L pair consists of a corner (the head) and an edge (the body).
    • On the 34, an F2L-a pair consists of an edge (the head) and a ridge (the body).
    • paired = fully assembled
    • split pair = one move away from paired, or can be paired as part of inserting the block
    "},{"location":"glossary/#f2l-action-terminology","title":"F2L action terminology","text":"
    • breaking the base = unsolving some pieces that were solved
    • restoring the base = re-solving some pieces
    • push = a twist of a side axis that breaks the base and puts new pieces on top
    • pull = a twist of a side axis that restores the base and puts new pieces on top
    • overpush = push again after pushing (e.g., R U R U R2\u2019)
    • overpull = push as a continuation of a pull (e.g., R U R2\u2019 U\u2019 R)
    • push pair = formation of a pair via a push
    • pull pair = formation of a pair via a pull
    • hide = to remove a piece from the top (using a push or pull)
    • reveal = to bring a piece to the top (using a push or pull)
    • rebase or reorient = to reorient a piece to face a different direction (i.e., change where its base sticker is facing)
    • cap = to twist T to form a pair (where the head is on top and the body is not on the top)
    • uncap = to separate the head and body of a pair by twisting T

    Open question

    What should we call a move like RT on 34, which doesn\u2019t change the set of pieces on T and might or might not unsolve some pieces?

    "},{"location":"glossary/#methods-in-higher-dimensions","title":"Methods in higher dimensions","text":"

    In higher and higher dimensions, it gets annoying to have to say stuff like \u201cpermuting the last cell of the last cell of the\u2026\u201d etc. To avoid this, we simply add a hyphen and the rank of the thing you\u2019re solving at the end. Examples:

    • PLL-4 is the PLL step on a rank-4 object, which permutes a rank-3 object. With CFOP on 34, it consists of permuting the 2c\u2019s, then permuting the rest like a 33.
    • For F2L, you put the number before the letter at the end e.g. F2L-5a, F2L-6d, etc.
    • If you were solving a 36 with pure CFOP and you were solving the F2L of the final cube with triple RKT, that would be F2L-3 of PLL-4 of PLL-5 of PLL-6.
    "},{"location":"glossary/#puzzle-descriptors","title":"Puzzle descriptors","text":"

    This section is a work-in-progress.

    • Solid vs. tiling vs. soup
    • Doctrinaire
    • Reduced
    • Bandaged
    • Unbandaged
    • Shapeshifting
    • Sliding vs. twisting
    • Circle
    • Super
    • Real
    • Complex
    • Stickermod
    • Shapemod
    • Cuboid terms (tower, brick, floppy, domino, pancake)
    • Other common puzzle families: weirdling, bubbloid, rotate-gap, sliding-gap (15-puzzle), loopover
    "},{"location":"glossary/#cut-depth","title":"Cut depth","text":"

    This section is a work-in-progress.

    • Shallow cut
    • Half cut
    • Deep cut
      • Deeper than adjacent
      • Deeper than origin
    • Semideep cut?

    1. Go ahead, run us over with the descriptivist bus.\u00a0\u21a9

    "},{"location":"history/","title":"History of Hypercubing","text":""},{"location":"history/#1988-2002-the-beginning-of-hypercubing","title":"1988-2002: The Beginning of Hypercubing","text":"

    MagicCube4D in 2001

    Shortly after the Rubik\u2019s Cube was invented in 1974, people began thinking about different versions of the puzzle. The idea of higher dimensional analogues was intriguing to many people. This article from 1982 is one of the earliest known documents theorizing properties of the 4D Rubik\u2019s Cube. In 1988 Melinda Green and Don Hatch wrote MagicCube4D, and the first solutions to the 34 puzzle came shortly thereafter. The Superliminal website was created some time around 2001, and Charles Doan has an excellent video about an old version of MC4D. Communication was very difficult because of how extremely niche hypercubing was.

    "},{"location":"history/#2002-present-mailing-list-years","title":"2002-Present: Mailing List Years","text":"

    In August 2003, the Hypercubing Yahoo Groups mailing list was created. Anyone could subscribe to the list and join the discussion via email. This greatly improved the speed of sharing knowledge. Many new people joined and discussed methods, puzzles, and even had some speedsolving competitions! It was moderately active, up until Yahoo groups began removing past content in 2019. Click here to view an archive of all the past messages. After that, the Hypercubing Google Groups mailing list was created, with pretty much the same members and type of discussions as before. It wasn\u2019t until 2021 that a Discord server was created to speed up communication even more.

    "},{"location":"history/#2010-2017-andreys-influence","title":"2010-2017: Andrey\u2019s Influence","text":"

    Andrey Astrelin was a Russian programmer that became known to the hypercubing community when he claimed to have programmed and solved a 34 before MC4D was created in 1988. He had no proof for this, but was able to program a new simulator in only 129 lines of code, which certified his legitimacy. Andrey\u2019s main legacy came in the form of creating many programs for the community, including Magic Puzzle Ultimate, Magic Cube 7D, Magic Simplex 5D, and Magic Hyperbolic Tile. These programs unlocked a massive amount of higher dimensional puzzles that were probably never seen or solved before, enriching the hypercubing community. Unfortunately Andrey passed away in 20171, but his contributions to the hypercubing community will never be forgotten.

    "},{"location":"history/#2017-present-making-physical-4d-puzzles","title":"2017-Present: Making Physical 4D Puzzles","text":"

    Throughout 2013 and 2014, Melinda Green got in contact with Oskar van Deventer to try and think of any possible mechanism for a physical 24. After failing to find a reasonable physical mechanism, they ended up going with only using magnets in order to stick the hypercubies together. In 2017 the first design was finalized, and Melinda began selling the puzzles by 3D printing the parts from Shapeways. The quality of the puzzle improved in 2022 when Melinda secured a deal with a company to mass produce the pieces via injection moulding.

    In late 2021, some speculation began as to if you could expand the physical 24 into a physical 2x2x2x3 hypercuboid. After Rowan learned that Grant had 3D printed his own physical 24, he challenged him to build a physical 2x2x2x3. Grant completed the challenge, and began designing and printing many more physical puzzles, all the way up to the physical 3x3x3x3. Many physical puzzle designs have still yet to be built\u2026

    Physical puzzles page

    • 2013-2015: Brainstorming ideas to make the 2x2x2x2 in 3D space
    • 2017 Feb 08: First puzzle made with magnetic dice mechanism
    • 2017 Apr 26: 2nd 24 made using 3D printing from Shapeways
    • 2022 Feb 03: Grant finishes the 2x2x2x3
    • 2023 May: First brainstorming of floppyhypercubes
    • 2022 May 14: Grant finishes the 2x2x3x3
    • 2022 Jul 06: Grant finishes the 2x3x3x3
    • 2022 Jul 22: Grant finishes the 3x3x3x3
    • 2022 Aug: Markceluna designs a physical simplex
    • 2024 Jan: Tymon finishes making 1x2x2x2 through 1x3x3x3 hypercuboids
    • 2024 Jan: Dietl builds a physical simplex based on a completely different design than Markceluna\u2019s
    "},{"location":"history/#2022-present-the-hyperspeedsolving-revolution","title":"2022-Present: The Hyperspeedsolving Revolution","text":"

    On 2021 Jun 10, Rowan Fortier created the Hypercubers Discord server, and in late 2021 Hactar began working on the Hyperspeedcube program, which was a massive upgrade over older programs like MC4D. Features like piece filters and customizable keybinds were instrumental to lowering solving times. November 2022 saw the beginning of the race for the 34 world record. The record was traded almost exclusively by Grant and Hactar, smashing all the barriers from sub-10 minutes all the way down to sub-2! There were also a few smaller record races for bigger layered n4s.

    In early 2024, the first speedsolves of the 35 were done. A lot of hypercubers were waiting for HSC 2 to come out so they could use keybinds and better piece filters to solve in 5D, but using MC7D or MPU isn\u2019t that bad at all.

    • 2022 Jan: HSC v0.1 includes 33 and 34, filtering by piece type (not color), and customizable keybinds.
    • 2022 Aug: HSC v0.8 adds modern piece filters. HSC v0.9 adds multiple keybind sets.
    • 2022 Nov 06: First 34 sub-10:00 by Hactar
    • 2022 Nov 18: First 34 sub-9:00 by Rowan
    • 2022 Nov 19: First 34 sub-8:00 by Hactar
    • 2022 Nov 22: First 34 sub-7:00 by Grant
    • 2022 Nov 23: First 34 sub-6:00 by Hactar
    • 2022 Nov 26: First 34 sub-5:00 by Hactar
    • 2022 Dec 07: First 34 sub-4:00 by Grant
    • 2022 Dec 22: First 34 sub-3:00 by Hactar
    • 2023 May 13: First 34 sub-2:00 by Hactar

    1. Read Andrey\u2019s biography here \u21a9

    "},{"location":"intro-4d/","title":"Introduction to 4D","text":"

    This page aims to teach you the fundamentals of understanding higher dimensional space, some of the shapes that exist in them, and how we can turn them into twisty puzzles to play with.

    "},{"location":"intro-4d/#what-is-the-4th-dimension","title":"What is the 4th dimension?","text":"

    The dimensionality of a space/object is the number of coordinates needed to describe its position. First, think of a 1-dimensional number line. To describe a position on the number line, all you need is one set of numbers for the coordinate. In 2-dimensional space like a graph, you need 2 sets of numbers to represent a point, such as \\((3,4)\\). This continues on to higher dimensions, where you need more and more numbers in order to locate a point.

    The 4th dimension is a space where you need 4 sets of numbers to describe the coordinates of something. As 3D beings in a 3D universe, we can\u2019t interact with or see the 4th dimension. But computers can easily simulate higher dimensions, and project it down to 2D so we can see it on a computer screen.

    Dimension Levels (by NerdBoy1392 - Own work, CC BY-SA 3.0, link)

    "},{"location":"intro-4d/#understanding-4d-shapes","title":"Understanding 4D shapes","text":"

    Hypercube rotating

    Puzzles are (usually) based on physical geometric shapes, so you have to understand what 4D shapes there are. 4D shapes have sides, or cells, that take the form of 3D shapes, and enclose some amount of \u201chypervolume\u201d. As the most popular 4D puzzles are the hypercubes, so it\u2019s important to at least understand its geometry.

    A hypercube (also known as a tesseract, 8-cell, octachoron, or 4-cube) is a shape with 8 cubic sides. 3 sides meet at each edge, giving it the Schl\u00e4fli symbol {\\(4,3,3\\)}. Watch the animation below, as the tesseract unfolds itself into its 3-dimensional net.

    But of course, there are many other shapes besides the tesseract. Check out these resources for a deeper understanding:

    • Perfect Shapes in Higher Dimensions - Numberphile
    • Regular polytope
    • Regular 4-polytope
    "},{"location":"intro-4d/#making-puzzles-out-of-shapes","title":"Making puzzles out of shapes","text":"

    To make twisty puzzles out of higher dimensional shapes, you have to slice it up! Adding 2 evenly spaced planar cuts along each dimension, we get the 3x3x3x3, or 34.

    Congratulations, you have completed the tutorial!

    "},{"location":"jokes/","title":"Hypercubing community in-jokes","text":"

    Like any niche internet community, we\u2019ve accumulated some in-jokes over the years. In an effort to be welcoming to newcomers, here\u2019s a list of them!

    This page is specifically for in-jokes that have been used by multiple people over a relatively long time, and wouldn\u2019t be obvious to newcomers.

    "},{"location":"jokes/#my-carrerr","title":"My carrerr","text":"

    On 2022-10-31, Grant did a one-handed solve of the physical 24 on a whim and immediately set a world record of 3:55.62, beating Rowan\u2019s prior record by two and a half minutes and breaking three minute barriers. In response:

    Rowan: I worked so hard for my sub-7:00 Rowan: wth is this Grant: first try too Grant: lol Grant: it\u2019s uploading rn Rowan: wow Grant: first sub 6, sub 5, and sub 4 Luna: rowan in shambles Luna: what will he do Rowan: my carrerr Rowan: it\u2019s over

    "},{"location":"jokes/#sticker-soup","title":"Sticker soup","text":"

    Hactar: You can always just make a soup of flat stickers with rules about how to move them around. It just becomes less and less usable. Physical 24 is special because the rules are fairly straightforward.

    In the wake of the Rowan\u2019s video The Quest to Build a 4D Rubik\u2019s Cube, many people joined the server proposing \u201c3D physical\u201d puzzles, which are flat 2D constructions of 3D puzzles. Generally, hypercubers are only interested in these as a way to explain how 4D physical puzzles work, since \u201c3D physical\u201d puzzles aren\u2019t mathematically interesting in their own right and they\u2019re more difficult to use than real 3D puzzles. They tend to become a \u201csoup\u201d of stickers that move according to arbitrary rules, a generalization of twisty puzzles that throws away all geometric relations.

    Luna: how many times do I have to say this Luna: dimensional \ud83d\udc4f analogies \ud83d\udc4f don\u2019t \ud83d\udc4f work \ud83d\udc4f for \ud83d\udc4f physical \ud83d\udc4f puzzle \ud83d\udc4f design \ud83d\udc4f

    Designing physical versions of 4D+ puzzles is generally very difficult and is usually mathematically impossible to do well. In the end, most end up becoming sticker soup.

    "},{"location":"jokes/#dunning-kruger-graph","title":"Dunning-Kruger graph","text":"

    From Dunning-Kruger effect - Wikipedia

    The Dunning\u2013Kruger effect is a cognitive bias whereby people with low ability, expertise, or experience regarding a type of task or area of knowledge tend to overestimate their ability or knowledge.

    The Dunning-Kruger effect is especially common in hypercubing, which lies at the intersection of puzzle theory and higher-dimensional geometry, two niche mathematical fields. Of course, you don\u2019t need to be an expert in math to solve 4D puzzles, but if you have no 4D mathematical foundations then your understanding will probably be wrong at first.

    The Dunning-Kruger effect is sometimes described using this graph, where \u201ccompetence\u201d increases to the right and \u201cconfidence\u201d increases upward:

    Of course, in reality the effect is not quite so extreme, and varies from person to person.

    "},{"location":"jokes/#jambling","title":"Jambling","text":"

    On 2023-01-03, in Hypercubers voice chat, a very sleepy Hactar mispronounced the word \u201cjumbling\u201d as \u201cjambling.\u201d Since then, some hypercubers have been using the phrases \u201cjambling\u201d or \u201cjamble\u201d instead of \u201cjumbling\u201d.

    "},{"location":"jokes/#fake-anderson","title":"Fake Anderson","text":"

    On 2023-01-12, in Hypercubers voice chat, Hactar and Luna were conversing with Anderson Taurence. During that time, a user named Jedi Cabanna Boy (Andreas Thompson) joined and continued the conversation, and Anderson left. But, their voices were similar enough that Hactar, who was distracted speedsolving 54, didn\u2019t notice the switch, and repeatedly addressed Jedi as Anderson. Luna noticed and did not correct him.

    Once Hactar finally realized what was happening, Jedi changed their nickname on the server to Anderson (Beta), and later \ud83c\udd71a n d e r s o n.

    "},{"location":"jokes/#cactus","title":"Cactus","text":"

    On 2023-02-19, several members were brainstorming possible shapes for physical 5D puzzle pieces and named one of them \u201cCactus.\u201d

    Banderson: then yeah I think (compound of four cubes)-generated tetragonal icositetrahedron is good; C4CTi or something Banderson: C4QTi ? troller: \ud83c\udf35 Banderson: oh wait no C4Ct.I Banderson: or better yet I\u2019ll let akkei make a shorthand/pick a better name to describe it lol troller: Cacti Rowan: squeez cheezahedron Grant: Your shape, your name akkei Akkei: i still like the Akkei: we Do Not Have A Scheme for the cactus Grant: Cactus LOL \u2026 Grant: I don\u2019t care what actual name we end up giving it, I\u2019m calling it cactus in this channel

    "},{"location":"jokes/#silence-derivative","title":"Silence, derivative","text":"

    In early 2018, at the 13th Gathering for Gardner, Melinda\u2019s physical 2x2x2x2 was shown to Ern\u0151 Rubik himself. It was reported that his only comment was that none of the derivative puzzles matter and that only his original invention is important.

    On the night of 2023-06-25, Kitcen lay awake, unable to sleep, their mind plagued with the thought of Ern\u0151 Rubik with lasers coming out of his eyes to blow up a physical 24 while denouncing it and calling it \u201cderivative.\u201d

    30 minutes later, Akkei responded wordlessly with this image:

    "},{"location":"jokes/#grant-standingslice","title":"Grant Standingslice","text":"

    In earlier versions of hypercubing.xyz, individual letters such as R, F, etc. showed tooltips indicating their use in common notation. On 2023-01-06, Hactar posted a screenshot of the leaderboards page in which the \u201cS\u201d in \u201cGrant S.\u201d had a tooltip that said \u201cStanding slice (parallel to F).\u201d

    Hactar: Fun fact: the \u201cS\u201d in Grant\u2019s name stands for \u201cStanding slice\u201d

    "},{"location":"jokes/#mimcaf-and-celse-pign","title":"Mimcaf and Celse pign","text":"

    Many hypercubers play Minecraft and Celeste, and have adopted common misspellings of those games. Akkei was the first to use \u201cmimcaf\u201d on 2022-12-03, and Rowan was the first to use the phrase \u201ccelse pign\u201d on 2023-09-30 as a misspelling of \u201cCeleste ping.\u201d

    "},{"location":"jokes/#that-may-i-can","title":"That may I can","text":"

    On 2024-11-09, OmnipotentEntity sent the following message in #offtopic-or-memes:

    I don\u2019t often remember my dreams. Last night I dreamt that xkcd started including puns of the phrase \u201cthat may I can.\u201d And it was supposed to be a reference to something. And everyone but me got it, and no one would explain it to me.

    Several members then responded with various images punning on the phrase.

    "},{"location":"notation/","title":"Notation","text":"

    Warning

    This page explains notation specifically for cell-turning virtual hypercube puzzles. Notation pages for other puzzles coming soon\u2122\ufe0f\u2026

    "},{"location":"notation/#cells","title":"Cells","text":"

    Yellow cell rotating

    Each side of an n4 hypercube is called a cell, and consists of a cubic grid of n3 stickers. Because each side is cubic, they can be rotated with cubic symmetry. A cube has 24 different possible orientations, meaning that each side of a 4D puzzle can be oriented in 24 different ways.

    There are 8 cells, six of them using the same letters as that in the 33: U (up), D (down), F (front), B (back), R (right), L (left). The one that you can see on the inside is called I (inside), and the one opposite of that (which is hidden) is called O (outside). This means we use the letters R L U D F B O I.

    "},{"location":"notation/#pieces","title":"Pieces","text":"

    Cell-turning hypercubes have 4 different types of pieces (with some slight variations for big hypercubes). These are the 1c (1-colored) centers, 2c ridges, 3c edges, and 4c corners. You can label each piece based on which cells it is on. The 3c piece that has stickers on the Inside, Right, and Up cells would be called the IUR piece.

    "},{"location":"notation/#twists","title":"Twists","text":"

    Twists are notated based on which piece you click on to do the turn, with the first letter determining which sticker of that piece to click on. For example:

    • RO' means to left click the R cell sticker of the RO piece.
    • IUR means to left click the I sticker of the IUR piece (edge twists are symmetrical, so it doesn\u2019t matter if you left or right click).
    • FRUI means to right click the F sticker of the FRUI piece.

    For the slice layers, we use M E S P, where P is the slice that follows O (although with notation for n5 puzzles beginning to be fleshed out, P may end up being used by the new 5D sides of Posterior/Anterior). Wide moves are tricky, as we are now using the letter w for the 4th dimension axis. One way to get around this is to write the numbers you hold down on the keyboard (in MC4D/MPU etc) while clicking on a piece. On a 3-layered puzzle, holding down 2 does a slice move, and holding down 1 and 2 will do a wide move.

    • {2}RO means to hold 2 while doing an RO move (essentially an M move).
    • {2-3}UO' means to hold 2 and 3 while doing a UO\u2019 move.
    • {2-4}IF means to hold 2, 3, and 4
    "},{"location":"notation/#commutators","title":"Commutators","text":"

    Many algorithms are constructed using commutators and conjugates, so there is a compact notation for them using square brackets. See Commutators - Notation.

    "},{"location":"notation/#algorithms","title":"Algorithms","text":"

    A shorter notation was developed to write specific algorithms, such as RKT cancels. All letters besides I and O correspond to their -O variants. R U R' U' would mean RO UO RO' UO'. For the wide O cell flips, they are notated with 3D rotations, such as {1-2}Oxz2. Using 3D rotations in this way is totally arbitrary, and its only purpose is for execution.

    "},{"location":"notation/#rotations","title":"Rotations","text":"

    3D notation for rotations doesn\u2019t really generalize to higher dimensions. For example, we call an x rotation x because it rotates the puzzle \u201caround the x-axis\u201d. However, this is unhelpful because rotations don\u2019t actually happen around an axis. Instead it\u2019s better to think of rotations as happening within a 2D plane. When you\u2019re doing an x, the whole puzzle is really being rotated within the zy plane. And writing rotations like this generalizes to higher dimensions, so that is what we use.

    First, we make a certain side from each axis be the \u201cpositive\u201d side. There are standards for this in 3D (such as the right-handed rule). On the x-axis, going to the right is positive and going left is negative. This makes R the positive side from the x-axis. The same goes for the other axes: U, and F are the positive sides in 3D. In 4D we add two new sides, and have to decide which one is positive and which one is negative. Because of the projection, the side that we can\u2019t see is closer to the 4D camera, making O the positive w-axis cell, and I the negative.

    To actually notate the rotations, write the letters of the 2 axes that form the plane that the puzzle is rotating in, in the order of which positive side on that axis goes to the positive side of the other axis. For example: rotating the positive y-axis side to the positive x-axis side (rotating U to R) would be written yx (and looks like z in 3D notation).

    • yw: bring +y to +w (rotate U to O)
    • xz: bring +x to +z (rotate R to F (this is called y in 3D rotation notation))

    Note how we don\u2019t have to use the ' prime symbol because you can just swap the letters (wy is the inverse of yw). You can also add a 2 to the end for double rotations, e.g. wx2, yz2 etc.

    This makes it a really nice system because it only ever uses 2 letters, no matter how many dimensions. You can also use it on n3 puzzles in order for more multi-dimensional consistency: xy means move x+ to y+ (R to U [z\u2019 rotation]).

    "},{"location":"notation/#turn-metrics","title":"Turn metrics","text":"

    There are many different ways to count the number of twists performed during a solve. All of these metrics (besides ETM) have their own Quarter Turn counterparts, where twists are broken up into 90\u00b0 twists. For example, a 180\u00b0 face twist takes two 90\u00b0 twists, while a 120\u00b0 edge twist takes three. Listed below are the metrics used by HSC. STM is used the most, but others are sometimes relevant.

    ATM
    • Consecutive twists of the same axis are combined, even with different layers.
    • Whole-puzzle rotations are not counted.
    ETM
    • Twists are counted as they are executed, including whole-puzzle rotations.
    STM
    • Whole-puzzle rotations are not counted.
    • Slice twists count as one move.
    • Consecutive twists of the same axis and layers are combined.
    BTM
    • Whole-puzzle rotations are not counted.
    • Noncontiguous slice twists are split into contiguous slice twists.
    • Consecutive twists of the same axis and layers are combined.
    OBTM
    • Whole-puzzle rotations are not counted.
    • Slice twists are split into contiguous outer-block twists.
    • Consecutive twists of the same axis and layers are combined.
    MC4DTM

    a.k.a. \u201cwhatever MC4D says\u201d

    • Twists are counted as they are executed.
    • Whole-puzzle rotations counted only if they are executed as moves.
    • Double rotations and certain reorientations are impossible to execute as one move.
    • Any move that counts as 1 STM is possible with 1 or 2 MC4DTM moves.1

    1. This is due to the 2-click theorem.^[citation needed]^\u00a0\u21a9

    "},{"location":"progression/","title":"Progression","text":"

    This is an outline of how to make your way through the world of hypercubing. Learning puzzles in another order could lead to confusion or an incomplete understanding in the long run.

    Hactar

    It\u2019s very difficult to understand what\u2019s really going on in a physical puzzle without first understanding virtual puzzles. We recommend starting with virtual 3^4 for the same reason that 3D cubers typically recommend starting with 3x3x3.

    "},{"location":"progression/#getting-started","title":"Getting Started","text":"

    Completely new to twisty puzzles? Never solved a Rubik\u2019s Cube? Here\u2019s where to start:

    K-Card Game is a basic piece cycling puzzle using cards. Can you rearrange them into the correct order? Other things to pay attention to when playing:

    • Cards in the left vs right hand
    • What type of cycles of cards are possible

    Loopover is similar to sliding puzzles (such as the infamous 15-puzzle) but there\u2019s no missing tile and the board loops over (hence the name). It might look overwhelming, but it\u2019s actually recommended to start with a 5x5 size board. After that, try a 4x4 board and discover the differences between odd and even size boards.

    You only have 1 chance to try and solve a Rubik\u2019s Cube by yourself with no tutorial. Definitely give that a shot first if you\u2019re brave enough. Otherwise, look up some tutorials and try and get beginners 3D intuition manipulating the cube. Good luck!

    "},{"location":"progression/#nnnns","title":"NxNxNxN\u2019s","text":"

    First, make sure you know how to solve a 3x3x3 Rubik\u2019s Cube. At the very minimum, knowing a beginner\u2019s method is enough, but knowing more advanced methods, such as CFOP or Roux, or more advanced techniques, such as blockbuilding or intuitive F2L, could make getting into hypercubing easier.

    flowchart LR\n    A[3x3x3]\n    B[3x3x3x3]\n    C[2x2x2x2]\n    D[4x4x4x4]\n    E[5x5x5x5+]\n    F[Physical 2x2x2x2]\n    G[Virtual Physical 3x3x3x3]\n    A --> B\n    B --> C\n    B --> D\n    C --> F\n    D --> E\n    F --> G
    "},{"location":"progression/#other-4d-puzzles","title":"Other 4D Puzzles","text":"

    There are many types of 4D puzzles, not just the hypercubes. These include hypercuboids, duoprisms, reqular 4D polytopes, and more. These puzzles require much more knowledge than just the 3x3x3. You\u2019ll need to know how to solve 3D cuboids, and generally be able to figure out new puzzles using commutators.

    flowchart LR\n    Hypercuboids --> Duoprisms --> Polytopes\n\n    subgraph Polytopes\n    direction LR\n    Q[5-cell]\n    R[16-cell]\n    S[24-cell]\n    T[120-cell]\n    U[600-cell]\n    end\n\n    subgraph Duoprisms\n    direction LR\n    Z[\"{5}x{4}\"]\n    Y[\"{5}x{3}\"]\n    X[\"{5}x{5}\"]\n    end\n\n    subgraph Hypercuboids\n    direction LR\n    D[1x3x3x3]\n    E[2x3x3x3]\n    F[2x2x2x3]\n    G[2x2x3x3]\n    end
    "},{"location":"progression/#non-euclidean-puzzles","title":"Non-Euclidean Puzzles","text":"

    While most puzzles in MagicTile aren\u2019t 4D, they do share some similar concepts due to their strange geometries.

    flowchart LR\n    A[Torus Rubik]\n    B[Klein Bottle Rubik]\n    C[Hyperbolic Rubik]\n    E[Hemi-Megaminx]\n    D[Other]\n    A --> B\n    A --> C\n    B --> E --> D
    "},{"location":"progression/#5d-puzzles","title":"5D+ puzzles","text":"
    flowchart TD\n    subgraph 5D\n    direction LR\n    A[\"n\u2074\"]\n    B[\"3\u2075\"]\n    C[\"2\u2075\"]\n    D[\"4\u2075\"]\n    E[Other 4D]\n    F[5D simplex]\n    A --> B --> C --> D\n    E --> F\n    A --> E\n    end\n    subgraph 6D\n    direction LR\n    G[\"3\u2076\"]\n    H[\"Other n\u2076\"]\n    G --> H\n    end\n    subgraph 7D\n    direction LR\n    V[\"3\u2077\"]\n    Q[\"Other n\u2077\"]\n    V --> Q\n    end\n    5D --> 6D --> 7D
    "},{"location":"videos/","title":"YouTube Videos & Playlists","text":"

    List of popular videos and playlists related to hypercubing.

    "},{"location":"videos/#youtube-videos","title":"YouTube Videos","text":"
    • Demystifying the 4D Rubik\u2019s Cube
    • Cracking the 4D Rubik\u2019s Cube with simple 3D tricks
    • How to Solve a 4D Rubik\u2019s Cube | Beginner\u2019s Method Tutorial
    • The Quest to Build a 4D Rubik\u2019s Cube
    "},{"location":"videos/#youtube-playlists","title":"YouTube Playlists","text":"
    • Introduction to Hypercubing
    • Hypercubing Crash Course
    • Melinda\u2019s 2x2x2x2 UWRs in order
    • 3x3x3x3 UWRs
    • 4D Twisty Puzzles
    • Melinda\u2019s 2x2x2x2
    • Grant\u2019s hypercubing videos
    "},{"location":"drafts/","title":"Drafts","text":"

    Put drafts for pages in this directory

    "},{"location":"drafts/grip_theory/","title":"Grip theory","text":""},{"location":"drafts/grip_theory/#rkt-rank","title":"RKT rank","text":"

    As above, the existence of RKT was caused by the existence of a certain stabilizer of two grips. However, not all such stabilizers will enable RKT. On the 3x3x3 grip system, there is a stabilizer of R and L that can only rotate the anti-middle layer (the intersection of the R and L layers) and cannot cause the R layer to emulate another puzzle. In order for the emulation to work, there must be a twist whose axis is R that does not fix L. In general, RKT is possible on a puzzle when there are two grips \\(a\\) and \\(b\\) where there is a non-identity element that fixes \\(a\\) and \\(b\\), and a different element that fixes \\(a\\) but moves \\(b\\).

    This can be generalized even further. We can define an RKT rank-0 puzzle to be one with no twists, i.e. one where the stabilizer of any grip is the whole group. Then, we proceed recursively: an RKT rank-\\((n+1)\\) puzzle is one where there is some grip \\(a\\) such that the \\(a\\)-stabilizer of the grip group is an RKT rank-\\(n\\) puzzle, after removing grips that are stabilized by every element of the stabilizer. This definition correlates with the geometric rank of a puzzle; the RKT rank of a 3n puzzle is \\(n-2\\) for \\(n\\geq 2\\).

    TODO THIS NEEDS DEVELOPMENT

    "},{"location":"koans/","title":"Index","text":"

    Welcome, neophyte!

    Here you will find a selection of short stories of other Learners such as yourself, who have become enlightened on the path of Hypercubing. To start, you might want to read about Neophyte Cherto\u2019s Application of Algorithms.

    This is a work of fiction. Resemblence to any persons, real or fictitious, is entirely coincidental. Resemblence to any puzzles, real or complex, is probably intentional.

    "},{"location":"koans/adaptability/","title":"Adaptability","text":"

    Hactar & Luna, 2024-04-15

    Neophyte Grigor had been learning under Master Yuwi for several months, strengthening his skills with the 3x3x3. Over time, he grew frustrated; he had come to the Temple of Intuition to learn to solve more exciting puzzles.

    \u201cMaster Yuwi, thank you for your mentorship. You have taught me many new methods of solving a 3x3x3, but are these useful? I had already learned the method CFOP from the Temple of the Swift Solve, and could solve the cube in just 20 seconds. These new methods are slower and more confusing; what is their use?\u201d

    Master Yuwi nodded. \u201cThe techniques I have taught you are slower. Certainly your Method of the Swift Solve is superior.\u201d She then handed him an Astroid Attack. \u201cI trust you can solve this using CFOP.\u201d

    The neophyte stared at the puzzle for a moment. It was like nothing he had ever seen before. Certainly the Orientation and Permutation algorithms of CFOP would be of no use here. Nonetheless, when he applied the slower methods of Commutators and Blockbuilding, he was able to completely solve the puzzle piece by piece. By the end, he had understood exactly the purpose of each and every move in his solution.

    At that moment, Grigor was enlightened.

    "},{"location":"koans/algorithms/","title":"Applications of Algorithms","text":"

    Hactar, 2024-04-15

    Neophyte Cherto toiled into the late hours of the night, eager to complete their latest assignment: producing a written guide to their own solution to Rubik\u2019s Puzzle of Four Dimensions. They had solved the puzzle once before, but Master Nazir had suggested that they solve it again before beginning their writing, for there was a case they had not yet encountered.

    As Cherto approached the end of the Orientation of the Last Cell of their 3x3x3x3, they found that only a single piece was twisted in place. Cherto, being wise in the ways of Isolation and the construction of Commutators, was easily able to solve this case. They carefully noted the sequence of moves they had performed, added it to their guide, and then continued on.

    Some days later, another disciple came upon Cherto and addressed them. \u201cI am Neophyte Nima. I have nearly solved my puzzle, save for a single 3-color piece twisted in place. Its stickers are permuted in a cycle, and I cannot seem to rotate it without rotating another piece. Master Nazir said that you had recently solved this case and suggested that I learn from the solution from you.\u201d

    Cherto excitedly withdrew the latest draft of their solution guide and began copying their algorithm onto a slip of parchment for Nima to read. They began: U R2 ...

    Later that day, Master Nazir found Nima still struggling on the same monotwist case, and inquired whether she had found help from Cherto. She replied: \u201cCherto was kind enough to provide their method, but I believe they copied it wrong, for my puzzle becomes scrambled when I apply their algorithm!\u201d

    \u201cCherto has given you an algorithm?\u201d Master Nazir peered over her shoulder. \u201cI see you struggle with the application of Cherto\u2019s algorithm. Did they explain the derivation of this algorithm to you?\u201d Nima responded that they had not. Master Nazir left the room, leaving Nima to her struggles.

    Shortly after, Master Nazir returned with Neophyte Cherto. Cherto watched Nima diligently applying their algorithm to the hemimegaminx in front of her, unaware of its origins.

    At that moment, Cherto was enlightened.

    "},{"location":"koans/behold/","title":"Behold!","text":"

    Hactar, 2024-04-15

    Neophyte Horatio had been studying hyperbolic tilings under Master Nikolai for nearly a month, and was learning well the ways of the petal, the fourfold corner, and orientability. He woke up to find an envelope outside his door containing a message from his mentor, Master Nikolai:

    Neophyte Horatio, I have fallen ill and cannot attend your instruction today. Enclosed is a hyperbolic tiling which I am sure you have solved before. I expect you will have it solved by tomorrow. - Master Nikolai

    The neophyte unfolded the puzzle and gasped in horror. The polygons of this tiling were eight-sided, and six met at each vertex! He had learned by now that each additional polygon at a vertex immensely increased the complexity of the cut interactions, so surely this puzzle would require some extensive sequences to isolate pieces. But he had no room! The puzzle had only six colors, and often the same face met multiple times at one vertex! Nonetheless, he began building an edge, a petal, another edge, another petal, \u2026 but as he inserted the second petal, the first petal popped out. Try as he might, he could not remove the second petal without replacing the first as well.

    Horatio sat and pondered. He had heard fragments from Neophyte Atenea, who had been studying The Theory of Grips, and had an idea. He counted the colors, the cuts, the interactions between pieces, \u2026

    At that moment, Horatio was enlightened.

    He solved the rest of the puzzle trivially, using not a single hyperbolic technique.

    "},{"location":"koans/projection/","title":"Projection","text":"

    Rowan, 2024-06-12

    Neophyte Reyhan had recently seen Neophyte Horatio ascend the dimensional barrier by conquering the legendary 3x3x3x3x3 puzzle, and wanted to join him in this accomplishment. But Reyhan was confused by the spell Horatio used for the construction of his puzzle. It seemed to show stickers and pieces in strange locations that she couldn\u2019t make sense of.

    Later that night, Reyhan went to the altar and crafted her dimensional spell. Finally after several days of solving, she had solved the puzzle! Reyhan excitedly rushed to find Horatio, to tell him about her tremendous achievement, but Horatio was also confused about Reyhan\u2019s puzzle.

    \u201cWhy do you have so many duplicate stickers?\u201d he asked. \u201cFurthermore, even with those duplicate stickers, you still cannot see every piece at the same time.\u201d

    That\u2019s when Master Nilu strolled past, casually solving a 3x3x3x3x3 (formed using yet another different spell). Both Reyhan and Horatio ran up to Nilu and started arguing about whose puzzle was correct.

    \u201cMy neoyphytes\u201d, Nilu sighed as she guided them towards the Temple of the Sacred Hyperstone.

    \u201cReyhan, you will stand to the left of the Hyperstone. Horatio, stand on the right. I will stand in front.\u201d After a brief pause to allow a moment of reflection, Master Nilu inquired \u201cnow, what are we looking at?\u201d

    Reyhan and Horatio each described how the sacred hyperstone looked from their viewpoints\u2026

    \u201cBut it is the same stone!\u201d Nilu exclaimed.

    At that moment, Reyhan and Horatio were enlightened.

    "},{"location":"koans/soup/","title":"Soup","text":"

    Hactar, Luna, Banderson, et al., 2024-04-15

    One autumn day, Neophyte Cherto approached their mentor, Master Nazir, with a new design for a puzzle. Cherto had heard of the Shrine of Manifestation, where the Priests of High Symmetry imbue 4-dimensional or even 5-dimensional symmetry unto ordinary 3-dimensional shapes, and decided to try their hand at designing such a spell. They had counted the stickers and the pieces of a 4D puzzle, and had constructed a 3D shape with the same stickers.

    Master Nazir, having been presented with the neophyte\u2019s design, sighed deeply and gently shook his head as they walked through the garden together.

    \u201cYoung neophyte, how will the pieces twist? Have you accounted for the orientations of each piece? Do they change according to the moves?\u201d

    Neophyte Cherto scrambled to fix the issues. \u201cWell we can adjust the orientation of the pieces after each twist!\u201d

    \u201cAnd how, young neophyte, is that any better a puzzle than the soup we eat for lunch?\u201d

    Cherto\u2019s face fell. They knew that when incantations failed at the Shrine of Manifestation, the polytopes on the altar melted into a formless blob that the elders called \u201csoup.\u201d

    \u201cBut surely there must be a way to make this work! I have even constructed an analogous 2D representation of a similar 3D puzzle\u2013\u201c

    Master Nazir hushed them and gestured toward a nearby tree. \u201cSurely there must be a way for you to climb this tree! I have even routed a path through a similar tree.\u201d

    \u201cBut that tree is dead.\u201d Cherto paused, then continued more confidently. \u201cThe branches may be weak; it cannot be compared to another tree.\u201d

    At that moment, Cherto was enlightened.

    "},{"location":"koans/square-prism/","title":"Square Prism","text":"

    Hactar & Luna, 2024-04-15

    Master Nazir was working on a new puzzle, a novel cut of a square-pentagon duoprism, when he noticed Neophyte Sama standing awkwardly, as though she were waiting for something.

    \u201cDo you have a question for me?\u201d Master Nazir asked, not bothering to look up from his dihedral angle calculations.

    \u201cY- yes. Master, what is this shape? I did not know a 4-dimensional shape could be constructed only from cubes and pentagonal prisms.\u201d

    \u201cYou recognize cubes in this shape?\u201d

    \u201cAre these cells not cubes? They appear to be cubes.\u201d

    Master Nazir stood up from his work and led the neophyte around the room. The puzzle workshop extended quite far in all directions, and several monks from other temples were working on other designs throughout.

    He asked, \u201cIs this room a cube?\u201d

    \u201cIt certainly appears to be, if the ceiling is as tall as the length of the wall.\u201d

    \u201cAre you able to measure it?\u201d

    \u201cNo.\u201d

    \u201cWhy not?\u201d

    \u201cI cannot walk on the wall.\u201d

    \u201cAnd yet you walk on the floor.\u201d

    At that moment, Sama was enlightened.

    "},{"location":"koans/twice-spun/","title":"Twice Spun","text":"

    Hactar, 2024-04-15

    One day Acolyte Quenta found herself wandering the gardens, as she often did, and upon crossing a small bridge she came upon a curious puzzle. It appeared in the form of an ordinary 3x3x3, but with an odd coating on the stickers that presented a gradient over each piece that tilted strangely as she observed it from different angles. She took the cube and scrambled it, intending to return it once she had solved it.

    As she twisted the puzzle, she became sure that it was playing tricks on her. Even whole-puzzle rotations seemed to behave oddly. She had placed a blue piece next to an orange piece, but upon spinning the puzzle in her hands, the blue piece became orange and the orange piece had become blue! After struggling for a half-hour, managing only to pair a handful of pieces, she resolved to find Master Eiji, who was an expert in variations of the 3x3x3.

    Upon showing the mysterious puzzle to Master Eiji, he led her inside to a bare room with a column in the center. \u201cWalk around the column and return to me. Then you shall learn how to solve this puzzle,\u201d he instructed. Quenta gave him a quizzical look, but then walked around the column\u2013 and found that Master Eiji had disappeared! Furthermore, the room was slightly different; she could not see the entrance they had come in from, and instead there sat a bench against the wall. She walked another full circle around the column, and then found Master Eiji.

    \u201cMaster Eiji!\u201d she exclaimed. \u201cWhat strange geometry is this room constructed from? It is neither Spherical nor Hyperbolic, as the light does not seem to bend. But it cannot be Euclidean, because I circled the room twice before returning to you\u2013\u201d she stopped.

    At that moment, Quenta was enlightened.

    "},{"location":"leaderboards/history/","title":"World Record History","text":"View Options Time Format 4D5DMagicTileMisc HypercubesHypercuboidsPhysicalOther 3x3x3x32x2x2x24x4x4x45x5x5x56x6x6x67x7x7x7 SingleAo5BlindfoldedNo Filters Name Time Date Program Andrew Farkas (Hactar) 1m 56s 420ms1:56.42 2023-05-13 HSC Grant Staten 2m 03s 830ms2:03.83 2023-04-21 HSC Andrew Farkas (Hactar) 2m 07s 300ms2:07.30 2023-04-08 HSC Grant Staten 2m 11s 330ms2:11.33 2023-03-16 HSC Andrew Farkas (Hactar) 2m 17s 970ms2:17.97 2023-03-11 HSC Grant Staten 2m 21s 010ms2:21.01 2023-03-08 HSC Andrew Farkas (Hactar) 2m 26s 460ms2:26.46 2023-02-18 HSC Andrew Farkas (Hactar) 2m 29s 730ms2:29.73 2023-01-11 HSC Andrew Farkas (Hactar) 2m 39s 470ms2:39.47 2023-01-08 HSC Andrew Farkas (Hactar) 2m 41s 220ms2:41.22 2023-01-03 HSC Andrew Farkas (Hactar) 2m 42s 810ms2:42.81 2022-12-30 HSC Grant Staten 2m 59s 340ms2:59.34 2022-12-23 HSC Andrew Farkas (Hactar) 2m 59s 970ms2:59.97 2022-12-22 HSC Grant Staten 3m 06s 060ms3:06.06 2022-12-21 HSC Grant Staten 3m 13s 970ms3:13.97 2022-12-21 HSC Grant Staten 3m 16s 440ms3:16.44 2022-12-20 HSC Grant Staten 3m 21s 520ms3:21.52 2022-12-15 HSC Grant Staten 3m 23s 260ms3:23.26 2022-12-14 HSC Grant Staten 3m 34s 430ms3:34.43 2022-12-14 HSC Grant Staten 3m 45s 690ms3:45.69 2022-12-13 HSC Grant Staten 3m 55s 320ms3:55.32 2022-12-10 HSC Grant Staten 3m 58s 290ms3:58.29 2022-12-07 HSC Grant Staten 4m 02s 600ms4:02.60 2022-12-03 HSC Grant Staten 4m 09s 190ms4:09.19 2022-12-03 HSC Grant Staten 4m 10s 840ms4:10.84 2022-12-02 HSC Andrew Farkas (Hactar) 4m 30s 880ms4:30.88 2022-12-01 HSC Grant Staten 4m 43s 860ms4:43.86 2022-11-30 HSC Andrew Farkas (Hactar) 4m 50s 730ms4:50.73 2022-11-29 HSC Grant Staten 4m 52s 940ms4:52.94 2022-11-27 HSC Andrew Farkas (Hactar) 4m 58s 140ms4:58.14 2022-11-26 HSC Grant Staten 5m 01s 920ms5:01.92 2022-11-26 HSC Grant Staten 5m 23s 820ms5:23.82 2022-11-25 HSC Grant Staten 5m 25s 140ms5:25.14 2022-11-25 HSC Andrew Farkas (Hactar) 5m 32s 980ms5:32.98 2022-11-23 HSC Grant Staten 6m 36s 380ms6:36.38 2022-11-23 HSC Andrew Farkas (Hactar) 6m 38s 920ms6:38.92 2022-11-22 HSC Grant Staten 6m 46s 210ms6:46.21 2022-11-22 HSC Andrew Farkas (Hactar) 7m 02s 170ms7:02.17 2022-11-21 HSC Grant Staten 7m 36s 320ms7:36.32 2022-11-21 HSC Andrew Farkas (Hactar) 7m 43s 330ms7:43.33 2022-11-19 HSC Andrew Farkas (Hactar) 8m 02s 530ms8:02.53 2022-11-19 HSC Andrew Farkas (Hactar) 8m 15s 350ms8:15.35 2022-11-19 HSC Andrew Farkas (Hactar) 8m 23s 590ms8:23.59 2022-11-18 HSC Rowan Fortier 8m 58s 820ms8:58.82 2022-11-18 HSC Andrew Farkas (Hactar) 9m 05s 820ms9:05.82 2022-11-06 HSC Tetrian 10m 11s 870ms10:11.87 2019-05-04 MC4D Name Time Date Program Andrew Farkas (Hactar) 2m 03s 180ms2:03.18 2024-05-23 HSC Andrew Farkas (Hactar) 2m 18s 180ms2:18.18 2023-05-12 HSC Andrew Farkas (Hactar) 2m 20s 190ms2:20.19 2023-04-22 HSC Andrew Farkas (Hactar) 2m 26s 160ms2:26.16 2023-04-22 HSC Grant Staten 2m 28s 050ms2:28.05 2023-04-10 HSC Andrew Farkas (Hactar) 2m 33s 100ms2:33.10 2023-04-08 HSC Andrew Farkas (Hactar) 2m 35s 290ms2:35.29 2023-04-08 HSC Grant Staten 2m 35s 940ms2:35.94 2023-03-10 HSC Andrew Farkas (Hactar) 2m 39s 640ms2:39.64 2023-02-18 HSC Andrew Farkas (Hactar) 3m 02s 780ms3:02.78 2022-12-31 HSC Grant Staten 3m 52s 000ms3:52.00 2022-12-16 HSC Name Time Date Program Sebastian Yuste 1h 21m 17s 680ms1:21:17.68 2022-03-17 MC4D Name Time Date Program PKEarth 4m 48s 970ms4:48.97 2024-08-10 HSC Grant Staten 5m 16s 010ms5:16.01 2023-07-03 HSC Grant Staten 5m 33s 390ms5:33.39 2023-07-03 HSC Tetrian 10m 11s 870ms10:11.87 2019-05-04 MC4D SingleAo5Blindfolded Name Time Date Program Bilal Mourad 16s 310ms16.31 2024-12-13 HSC Bilal Mourad 18s 210ms18.21 2024-12-13 HSC Bilal Mourad 18s 790ms18.79 2024-12-03 HSC PKEarth 19s 630ms19.63 2024-09-08 HSC PKEarth 22s 240ms22.24 2024-09-01 HSC PKEarth 22s 600ms22.60 2024-08-31 HSC Bilal Mourad 23s 600ms23.60 2024-08-28 HSC Adam Marcellus Kelly 23s 680ms23.68 2024-08-07 HSC Adam Marcellus Kelly 25s 790ms25.79 2024-06-29 HSC Adam Marcellus Kelly 26s 370ms26.37 2024-06-29 HSC Adam Marcellus Kelly 26s 490ms26.49 2023-09-23 HSC Grant Staten 28s 510ms28.51 2023-05-24 HSC Grant Staten 31s 640ms31.64 2023-05-23 HSC Grant Staten 33s 340ms33.34 2023-05-20 HSC Grant Staten 34s 170ms34.17 2023-05-20 HSC Grant Staten 34s 590ms34.59 2023-05-16 HSC Grant Staten 34s 950ms34.95 2023-05-11 HSC Grant Staten 36s 230ms36.23 2023-03-08 HSC Adam Marcellus Kelly 39s 400ms39.40 2023-01-16 HSC Adam Marcellus Kelly 43s 540ms43.54 2023-01-16 HSC Grant Staten 44s 750ms44.75 2023-01-02 HSC Grant Staten 48s 520ms48.52 2023-01-01 HSC Grant Staten 49s 980ms49.98 2023-01-01 HSC Grant Staten 50s 950ms50.95 2022-12-21 HSC Tetrian 1m 00s 800ms1:00.80 2021-07-24 MC4D Name Time Date Program Bilal Mourad 22s 390ms22.39 2024-12-26 HSC Bilal Mourad 22s 730ms22.73 2024-12-21 HSC Bilal Mourad 23s 410ms23.41 2024-12-21 HSC Bilal Mourad 23s 450ms23.45 2024-12-17 HSC Bilal Mourad 23s 530ms23.53 2024-12-16 HSC Bilal Mourad 23s 710ms23.71 2024-12-11 HSC Bilal Mourad 24s 430ms24.43 2024-12-09 HSC PKEarth 24s 650ms24.65 2024-10-01 HSC PKEarth 25s 380ms25.38 2024-09-14 HSC PKEarth 26s 880ms26.88 2024-09-08 HSC PKEarth 27s 160ms27.16 2024-09-07 HSC PKEarth 28s 030ms28.03 2024-09-01 HSC Adam Marcellus Kelly 28s 080ms28.08 2024-08-28 HSC Bilal Mourad 29s 670ms29.67 2024-08-03 HSC Adam Marcellus Kelly 31s 310ms31.31 2024-07-16 HSC Adam Marcellus Kelly 33s 040ms33.04 2024-07-01 HSC Andrew Farkas (Hactar) 33s 940ms33.94 2024-06-27 HSC Bilal Mourad 39s 820ms39.82 2024-06-25 HSC Andrew Farkas (Hactar) 40s 170ms40.17 2024-06-25 HSC Bilal Mourad 41s 890ms41.89 2024-06-22 HSC Andrew Farkas (Hactar) 43s 440ms43.44 2024-06-14 HSC Bilal Mourad 48s 120ms48.12 2024-06-14 HSC Bilal Mourad 48s 510ms48.51 2024-06-14 HSC Bilal Mourad 48s 920ms48.92 2024-06-11 HSC Bilal Mourad 50s 720ms50.72 2024-06-08 HSC Bilal Mourad 55s 590ms55.59 2024-06-07 HSC Bilal Mourad 59s 290ms59.29 2024-06-05 HSC Bilal Mourad 1m 01s 930ms1:01.93 2024-06-05 HSC Bilal Mourad 1m 05s 370ms1:05.37 2024-06-04 HSC Bilal Mourad 1m 09s 430ms1:09.43 2024-06-04 HSC Bilal Mourad 1m 10s 990ms1:10.99 2024-06-03 HSC Bilal Mourad 1m 13s 310ms1:13.31 2024-06-03 HSC Bilal Mourad 1m 14s 340ms1:14.34 2024-06-03 HSC Bilal Mourad 1m 15s 690ms1:15.69 2024-06-02 HSC Bilal Mourad 1m 19s 540ms1:19.54 2024-06-02 HSC Bilal Mourad 1m 26s 900ms1:26.90 2024-05-31 HSC Name Time Date Program Kevin Liu 10m 12s 770ms10:12.77 2024-10-12 MC4D TheGrayCuber 14m 41s 550ms14:41.55 2019-10-12 MC4D Single Name Time Date Program Andrew Farkas (Hactar) 7m 52s 610ms7:52.61 2024-04-24 HSC Andrew Farkas (Hactar) 7m 55s 180ms7:55.18 2024-04-22 HSC Grant Staten 7m 55s 770ms7:55.77 2023-08-01 HSC Grant Staten 8m 30s 660ms8:30.66 2023-07-31 HSC Grant Staten 8m 32s 560ms8:32.56 2023-07-27 HSC Grant Staten 8m 52s 970ms8:52.97 2023-04-26 HSC Grant Staten 8m 55s 680ms8:55.68 2023-04-25 HSC Grant Staten 9m 47s 420ms9:47.42 2023-04-25 HSC Grant Staten 9m 50s 700ms9:50.70 2023-02-23 HSC Grant Staten 10m 03s 550ms10:03.55 2023-02-18 HSC Grant Staten 10m 33s 210ms10:33.21 2023-02-16 HSC Grant Staten 10m 44s 230ms10:44.23 2023-01-30 HSC Grant Staten 11m 13s 680ms11:13.68 2023-01-29 HSC Grant Staten 11m 16s 710ms11:16.71 2023-01-17 HSC Andrew Farkas (Hactar) 11m 22s 350ms11:22.35 2023-01-13 HSC Andrew Farkas (Hactar) 11m 39s 600ms11:39.60 2023-01-13 HSC Andrew Farkas (Hactar) 11m 42s 850ms11:42.85 2023-01-13 HSC Andrew Farkas (Hactar) 12m 25s 220ms12:25.22 2023-01-13 HSC Andrew Farkas (Hactar) 12m 28s 140ms12:28.14 2023-01-13 HSC Grant Staten 12m 33s 750ms12:33.75 2023-01-12 HSC Andrew Farkas (Hactar) 13m 13s 650ms13:13.65 2023-01-12 HSC Grant Staten 13m 17s 780ms13:17.78 2023-01-08 HSC Grant Staten 13m 21s 360ms13:21.36 2023-01-06 HSC Grant Staten 14m 00s 760ms14:00.76 2023-01-05 HSC Grant Staten 14m 10s 440ms14:10.44 2023-01-05 HSC Andrew Farkas (Hactar) 14m 38s 910ms14:38.91 2023-01-05 HSC Grant Staten 14m 39s 130ms14:39.13 2023-01-04 HSC Grant Staten 15m 18s 840ms15:18.84 2023-01-04 HSC Andrew Farkas (Hactar) 15m 28s 210ms15:28.21 2023-01-04 HSC Grant Staten 15m 44s 920ms15:44.92 2023-01-03 HSC Grant Staten 15m 59s 750ms15:59.75 2023-01-03 HSC Andrew Farkas (Hactar) 17m 14s 760ms17:14.76 2023-01-03 HSC Grant Staten 17m 21s 340ms17:21.34 2022-12-31 HSC Grant Staten 18m 53s 840ms18:53.84 2022-12-29 HSC Grant Staten 19m 17s 700ms19:17.70 2022-12-28 HSC Grant Staten 19m 57s 080ms19:57.08 2022-12-24 HSC Grant Staten 20m 18s 480ms20:18.48 2022-12-23 HSC Grant Staten 21m 37s 070ms21:37.07 2022-12-20 HSC Grant Staten 23m 50s 750ms23:50.75 2022-12-17 HSC Luna Harran 29m 54s 820ms29:54.82 2022-12-04 HSC Grant Staten 32m 46s 550ms32:46.55 2022-12-04 HSC Luna Harran 44m 40s 420ms44:40.42 2022-11-23 HSC Tetrian 1h 12m 07s 650ms1:12:07.65 2019-05-19 MC4D Single Name Time Date Program Andrew Farkas (Hactar) 19m 35s 750ms19:35.75 2024-05-16 HSC Grant Staten 19m 57s 610ms19:57.61 2023-07-27 HSC Grant Staten 20m 17s 280ms20:17.28 2023-07-26 HSC Grant Staten 20m 48s 930ms20:48.93 2023-07-22 HSC Grant Staten 21m 40s 870ms21:40.87 2023-07-20 HSC Grant Staten 21m 44s 120ms21:44.12 2023-06-02 HSC Grant Staten 22m 11s 780ms22:11.78 2023-05-29 HSC Grant Staten 23m 19s 100ms23:19.10 2023-05-27 HSC Grant Staten 23m 42s 580ms23:42.58 2023-05-13 HSC Grant Staten 25m 11s 080ms25:11.08 2023-05-06 HSC Grant Staten 27m 08s 970ms27:08.97 2023-04-27 HSC Grant Staten 28m 14s 310ms28:14.31 2023-03-17 HSC Grant Staten 28m 58s 950ms28:58.95 2023-02-21 HSC Grant Staten 29m 58s 510ms29:58.51 2023-02-01 HSC Grant Staten 31m 06s 980ms31:06.98 2023-01-23 HSC Grant Staten 33m 47s 210ms33:47.21 2023-01-20 HSC Grant Staten 34m 33s 540ms34:33.54 2023-01-17 HSC Grant Staten 39m 03s 510ms39:03.51 2023-01-12 HSC Grant Staten 41m 59s 320ms41:59.32 2023-01-05 HSC Grant Staten 49m 52s 500ms49:52.50 2023-01-04 HSC Andrew Farkas (Hactar) 55m 13s 040ms55:13.04 2023-01-04 HSC Grant Staten 57m 43s 650ms57:43.65 2022-12-30 HSC Luna Harran 1h 31m 38s 500ms1:31:38.50 2022-12-07 HSC Single Name Time Date Program Grant Staten 56m 03s 940ms56:03.94 2023-06-02 HSC Grant Staten 1h 00m 31s 150ms1:00:31.15 2023-05-29 HSC Grant Staten 1h 03m 23s 500ms1:03:23.50 2023-05-23 HSC Grant Staten 1h 04m 57s 880ms1:04:57.88 2023-05-15 HSC Grant Staten 1h 12m 22s 580ms1:12:22.58 2023-05-10 HSC Grant Staten 1h 19m 30s 890ms1:19:30.89 2023-05-09 HSC Grant Staten 1h 23m 25s 520ms1:23:25.52 2023-03-03 HSC Grant Staten 1h 29m 29s 900ms1:29:29.90 2023-02-02 HSC Grant Staten 1h 38m 45s 280ms1:38:45.28 2023-01-18 HSC Grant Staten 2h 16m 26s 540ms2:16:26.54 2023-01-12 HSC Luna Harran 3h 15m 46s 110ms3:15:46.11 2022-12-31 HSC Single Name Time Date Program Grant Staten 1h 58m 21s 230ms1:58:21.23 2023-07-18 HSC Grant Staten 2h 11m 41s 970ms2:11:41.97 2023-05-31 HSC Grant Staten 3h 27m 17s 510ms3:27:17.51 2023-01-22 HSC Andrew Farkas (Hactar) 4h 39m 59s 940ms4:39:59.94 2023-01-21 HSC 1x3x3x32x2x2x32x2x3x3 Single Name Time Date Program Tymon Fro 57s 460ms57.46 2023-11-11 MPU Andrew Farkas (Hactar) 1m 00s 930ms1:00.93 2023-05-31 MPU Luna Harran 1m 02s 910ms1:02.91 2023-05-31 MPU Luna Harran 1m 24s 340ms1:24.34 2023-05-31 MPU Grant Staten 1m 27s 390ms1:27.39 2023-03-20 MPU Single Name Time Date Program Rowan Fortier 2m 58s 900ms2:58.90 2024-06-20 MPU Rowan Fortier 3m 19s 700ms3:19.70 2024-06-20 MPU Tymon Fro 4m 22s 580ms4:22.58 2024-06-11 MPU Emanuele Battistin 5m 52s 720ms5:52.72 2024-05-26 MPU Single Name Time Date Program Emanuele Battistin 22m 12s 690ms22:12.69 2024-06-12 MPU Physical 2x2x2x2Physical 3x3x3x3Virtual Physical 3x3x3x3 SingleAo5One-HandedBlindfolded Name Time Date The Cube Dude 31s 580ms31.58 2024-12-24 The Cube Dude 49s 570ms49.57 2024-12-22 The Cube Dude 53s 030ms53.03 2024-09-22 Grant Staten 54s 840ms54.84 2022-11-08 Grant Staten 56s 650ms56.65 2022-10-03 Grant Staten 1m 06s 040ms1:06.04 2022-09-18 Grant Staten 1m 07s 570ms1:07.57 2022-08-12 Grant Staten 1m 23s 280ms1:23.28 2022-08-07 Rowan Fortier 1m 27s 170ms1:27.17 2022-08-02 Rowan Fortier 1m 28s 140ms1:28.14 2022-06-20 Rowan Fortier 1m 46s 240ms1:46.24 2022-05-07 Rowan Fortier 1m 56s 750ms1:56.75 2021-12-11 Rowan Fortier 2m 00s 660ms2:00.66 2021-12-11 Rowan Fortier 2m 03s 580ms2:03.58 2021-12-09 Rowan Fortier 2m 05s 270ms2:05.27 2021-12-06 Rowan Fortier 2m 16s 540ms2:16.54 2021-11-15 Connor Lindsay 2m 26s 000ms2:26.00 2019-08-11 Name Time Date The Cube Dude 59s 890ms59.89 2024-12-27 The Cube Dude 1m 06s 250ms1:06.25 2024-09-08 Name Time Date Tymon Fro 2m 56s 980ms2:56.98 2023-10-08 Grant Staten 3m 55s 620ms3:55.62 2022-10-01 Rowan Fortier 6m 25s 120ms6:25.12 2022-07-01 Name Time Date Asa Kaplan 47m 14s 000ms47:14.00 2022-08-08 Single Name Time Date The Cube Dude 56m 22s 110ms56:22.11 2024-07-28 Single Name Time Date Program Grant Staten 9m 58s 950ms9:58.95 2023-05-03 AKKEI-SIM Grant Staten 10m 30s 070ms10:30.07 2023-04-20 AKKEI-SIM Grant Staten 10m 54s 210ms10:54.21 2023-04-20 AKKEI-SIM Grant Staten 10m 58s 780ms10:58.78 2023-04-18 AKKEI-SIM Grant Staten 11m 30s 660ms11:30.66 2023-04-12 AKKEI-SIM Grant Staten 12m 05s 250ms12:05.25 2023-04-09 AKKEI-SIM Grant Staten 14m 33s 060ms14:33.06 2023-04-04 AKKEI-SIM Grant Staten 17m 11s 750ms17:11.75 2023-03-29 AKKEI-SIM 3-Layer Simplex Single Name Time Date Program Grant Staten 1m 17s 700ms1:17.70 2023-03-15 MC4D Rowan Fortier 1m 30s 460ms1:30.46 2023-01-26 MC4D Tetrian 1m 43s 440ms1:43.44 2021-07-24 MC4D 3x3x3x3x32x2x2x2x24x4x4x4x4 Single Name Time Date Program Rowan Fortier 1h 52m 29s 730ms1:52:29.73 2024-07-01 MC7D Rowan Fortier 2h 14m 16s 580ms2:14:16.58 2024-06-29 MC7D Rowan Fortier 2h 30m 30s 480ms2:30:30.48 2024-06-26 MC7D Luna Harran 2h 43m 59s 430ms2:43:59.43 2024-06-26 MC7D Rowan Fortier 3h 01m 48s 630ms3:01:48.63 2024-06-25 MC7D Luna Harran 3h 17m 59s 660ms3:17:59.66 2024-04-07 MC7D Luna Harran 3h 39m 01s 240ms3:39:01.24 2024-03-22 MC7D Single Name Time Date Program Tymon Fro 48m 22s 310ms48:22.31 2024-07-02 MPU Tymon Fro 58m 41s 410ms58:41.41 2024-07-02 MPU Tymon Fro 1h 13m 07s 260ms1:13:07.26 2024-04-01 MPU Luna Harran 1h 48m 43s 500ms1:48:43.50 2024-03-23 MPU Single Name Time Date Program Luna Harran 15h 20m 24s 490ms15:20:24.49 2024-04-26 MC7D HemimegaminxCanon-Cut Klein Quartic Single Name Time Date Program Milo Jacquet 1m 31s 670ms1:31.67 2024-03-31 MT Milo Jacquet 1m 47s 690ms1:47.69 2024-03-06 MT Grant Staten 1m 51s 940ms1:51.94 2023-03-19 MT Luna Harran 2m 06s 520ms2:06.52 2023-03-18 MT Luna Harran 2m 11s 570ms2:11.57 2023-03-18 MT Grant Staten 2m 19s 990ms2:19.99 2023-03-18 MT Luna Harran 2m 54s 050ms2:54.05 2023-03-18 MT Luna Harran 4m 30s 570ms4:30.57 2023-03-18 MT Single Name Time Date Program Ethan Davis 22m 32s 810ms22:32.81 2024-09-23 MT Milo Jacquet 24m 14s 410ms24:14.41 2024-04-04 MT 3x3x3 in 2D projection with 1D Vision Single Name Time Date Program Grant Staten 3m 22s 200ms3:22.20 2023-07-05 MC3D Grant Staten 4m 20s 210ms4:20.21 2023-07-04 MC3D Grant Staten 5m 10s 710ms5:10.71 2023-07-04 MC3D Grant Staten 6m 24s 390ms6:24.39 2023-07-04 MC3D Grant Staten 7m 34s 180ms7:34.18 2023-07-03 MC3D Markk 15m 45s 390ms15:45.39 2022-12-02 MC3D","boost":-3},{"location":"leaderboards/other-leaderboards/","title":"Other Leaderboards","text":"

    Magic Cube 4D Hall of Fame - The original Hall of Fame. New submissions are no longer accepted due to reaching 500 solvers.

    Magic Cube 4D Extended Records - Records for many other puzzles found in MC4D.

    Magic Cube 5D Hall of Insanity - List of everyone who has solved 25-75, and also FMC solutions.

    Magic Cube 7D Solvers - List of 6D and 7D solvers in MC7D.

    Magic 120 Cell Solvers - List of 120 cell solvers in M120C or MPU.

    MagicTile Klein Bottle challenge - List of the first 100 people to solve the Klein Bottle Rubik\u2019s Cube from a competition video made by Mathologer.

    Speedsolving Wiki List of Unofficial World Records - List of a few speedsolving and FMC records.

    "},{"location":"leaderboards/records/","title":"Current World Records image/svg+xml","text":"View Options Time Format Event Name Time Date Program 3x3x3x3 Single Andrew Farkas (Hactar) 1m 56s 420ms1:56.42 2023-05-13 HSC 3x3x3x3 Ao5 Andrew Farkas (Hactar) 2m 03s 180ms2:03.18 2024-05-23 HSC 3x3x3x3 Blindfolded Sebastian Yuste 1h 21m 17s 680ms1:21:17.68 2022-03-17 MC4D 3x3x3x3 No Filters PKEarth 4m 48s 970ms4:48.97 2024-08-10 HSC 2x2x2x2 Single Bilal Mourad 16s 310ms16.31 2024-12-13 HSC 2x2x2x2 Ao5 Bilal Mourad 22s 390ms22.39 2024-12-26 HSC 2x2x2x2 Blindfolded Kevin Liu 10m 12s 770ms10:12.77 2024-10-12 MC4D 4x4x4x4 Single Andrew Farkas (Hactar) 7m 52s 610ms7:52.61 2024-04-24 HSC 5x5x5x5 Single Andrew Farkas (Hactar) 19m 35s 750ms19:35.75 2024-05-16 HSC 6x6x6x6 Single Grant Staten 56m 03s 940ms56:03.94 2023-06-02 HSC 7x7x7x7 Single Grant Staten 1h 58m 21s 230ms1:58:21.23 2023-07-18 HSC 1x3x3x3 Single Tymon Fro 57s 460ms57.46 2023-11-11 MPU 2x2x2x3 Single Rowan Fortier 2m 58s 900ms2:58.90 2024-06-20 MPU 2x2x3x3 Single Emanuele Battistin 22m 12s 690ms22:12.69 2024-06-12 MPU Physical 2x2x2x2 Single The Cube Dude 31s 580ms31.58 2024-12-24 - Physical 2x2x2x2 Ao5 The Cube Dude 59s 890ms59.89 2024-12-27 - Physical 2x2x2x2 One-Handed Tymon Fro 2m 56s 980ms2:56.98 2023-10-08 - Physical 2x2x2x2 Blindfolded Asa Kaplan 47m 14s 000ms47:14.00 2022-08-08 - Physical 3x3x3x3 Single The Cube Dude 56m 22s 110ms56:22.11 2024-07-28 - Virtual Physical 3x3x3x3 Single Grant Staten 9m 58s 950ms9:58.95 2023-05-03 AKKEI-SIM 3-Layer Simplex Single Grant Staten 1m 17s 700ms1:17.70 2023-03-15 MC4D 3x3x3x3x3 Single Rowan Fortier 1h 52m 29s 730ms1:52:29.73 2024-07-01 MC7D 2x2x2x2x2 Single Tymon Fro 48m 22s 310ms48:22.31 2024-07-02 MPU 4x4x4x4x4 Single Luna Harran 15h 20m 24s 490ms15:20:24.49 2024-04-26 MC7D Hemimegaminx Single Milo Jacquet 1m 31s 670ms1:31.67 2024-03-31 MT Canon-Cut Klein Quartic Single Ethan Davis 22m 32s 810ms22:32.81 2024-09-23 MT 3x3x3 in 2D projection with 1D Vision Single Grant Staten 3m 22s 200ms3:22.20 2023-07-05 MC3D","boost":-1},{"location":"leaderboards/rules/","title":"Hypercubing Leaderboard Submission Rules","text":"

    These rules are not written in legalese. Please don\u2019t try to exploit them. Ask in the Hypercubers Discord if you have any questions, comments, or concerns about these rules.

    These rules may be flexible, especially for new categories. For example, if you\u2019re speedsolving a really big puzzle that no one\u2019s done before, it might be reasonable to allow macros for it.

    "},{"location":"leaderboards/rules/#general-rules","title":"General rules","text":"
    • These programs are recommended for speedsolving. If using a program not listed here, you should ask on the Discord first.
    • The puzzle must be solved from a full scramble.
      • No twists may be applied between the puzzle being scrambled and the timer starting, or else the solve counts as a DNF. Full-puzzle rotations, including gyros, are allowed.
      • When the timer is stopped the puzzle must be solved, or else the solve counts as a DNF.
      • Twists may be applied after the timer is stopped, but these moves are discounted.
    • Spectators may commentate, but must not help the solver.
    • Algorithm reference sheets are strongly discouraged.1
    • Reversing the scramble is not allowed; the solution must be novel.
    "},{"location":"leaderboards/rules/#speedsolving-rules","title":"Speedsolving rules","text":"

    These rules apply generally to speedsolving submissions unless the category has specific rules.

    • Macros are not allowed.
      • Exception: macros consisting of a single move STM are allowed, provided that they are created during the solve or demonstrated in the video before the solve starts.
    • Piece filters are allowed.
    • All speedsolves require video evidence, such as a screen recording.
      • The video must include scrambling the puzzle. When using Hyperspeedcube, the statusbar must be visible with the text \u201cScrambled fully.\u201d
      • A timer with precision of 0.01 seconds or better must be visible on the screen during the solve. Timers built into the puzzle software are preferred.
      • Inspection time must not exceed 60 seconds. In other words, the solve timer must be started within 60 seconds of the puzzle being completely scrambled.
    • If using keyboard controls:
      • If possible, a keybinds reference must be visible during the solve.2
      • The keybinds should not be hyperoptimized for specific algorithms.3
    • Log file is not required, but you should still save your PBs!
    • Using software to compute a solution to the puzzle state is not allowed.
    "},{"location":"leaderboards/rules/#blindsolving-rules","title":"Blindsolving rules","text":"
    • Macros are allowed during solving.
    • Piece filters/highlightings are allowed during memorization but not during solving.
    • All blindsolves require video evidence of the solver that clearly shows the screen and that they are not cheating by using some external reference.
    • The solver does not need to be blindfolded, but the colors on the puzzle must be invisible during the solve.
    • Using software to compute a solution to the puzzle state is not allowed.
    • Besides those rules, typical blindsolving rules apply:
      • Final time = memorization time + solving time.
      • Writing anything down isn\u2019t allowed.
    "},{"location":"leaderboards/rules/#real-life-puzzle-rules","title":"Real-life puzzle rules","text":"
    • All real-life puzzle solves must have video evidence that shows the puzzle fully in frame throughout the solve and shows the entire scrambling and solving process as well as the timer.
    • Scrambles must be generated by a computer program that produces a random state or at least this many random moves:
      • Physical 24: see below
      • Hemimegaminx: 30 moves
    "},{"location":"leaderboards/rules/#physical-24-rules","title":"Physical 24 rules","text":"

    See Canonical Moves for more details.

    • All rules for real-life puzzles apply.
    • Allowed moves for physical 2x2x2x2 events are limited to 2 categories, the canonical moveset originally made by the hypercubing mailing list as well as a few additional moves:
      • Canonical moves:
        • All R and L cell rotations
        • U/D y2
        • F/B z2
        • I/O x* (a.k.a. axial twist)
        • Any of many equivalent gyro algorithms
      • Additional moves:
        • U/D x2/z2
        • F/B x2/y2
        • I y2/z2
    • For 24, scrambles must be generated by this page or any equivalent scramble generator.
      • Hand scrambles are no longer allowed, as they have an increased chance to have an easier solution than random state scrambles.
      • It\u2019s ok if you mess up the computer generated scramble a little bit. It should still be way more random than any hand scramble.
    • Any full puzzle rotation that doesn\u2019t change the state (including the Gyro) may be applied to the puzzle during inspection; other moves may not.
    "},{"location":"leaderboards/rules/#fewest-moves-solving-rules","title":"Fewest-moves solving rules","text":"
    • Log files are required for submission.
    • Video evidence is not required.
    • Collaboration is allowed; each collaborator must be given the option to be credited for the solve, and should be credited if their contributions were significant.
    • Moves are counted using STM.
    • Use of computer software to generate a partial or full solution is allowed, but must be noted in the submission.
    "},{"location":"leaderboards/rules/#principles-when-writing-new-speedsolving-software","title":"Principles when writing new speedsolving software","text":"

    If you\u2019re thinking of writing a new program for speedsolving or suggesting a new feature for Hyperspeedcube, here are some principles we try to follow:

    • No more than one move per keypress.
    • If your program has keybinds (especially customizable ones), add some reference that shows what keys are being pressed and what they\u2019re doing.
    • New features should hopefully make solving strategy more interesting. Some examples:
      • Macros on the 34 encourage commutator spam, which is very boring compared to layer-by-layer or F2L methods, so they are not allowed for speedsolves.
      • Piece filters reduce time spent looking for pieces, which is boring to do and boring to watch. There\u2019s a trade-off there: piece filters encourage a stricter solving order, which is less interesting in a way, but this also enables 3-Block, a very interesting method.
    • New features should ideally have some justification based on 3D speedsolving. For example, multiple keybind sets is analogous to different grips on a 3D puzzle.4

    1. We can\u2019t stop you from using an algorithm reference, but it\u2019s not in the spirit of the competition and it\u2019ll probably slow you down.\u00a0\u21a9

    2. If the program doesn\u2019t have a built-in keybinds reference (such as akkei\u2019s physical 34 simulator) then you probably don\u2019t need one.\u00a0\u21a9

    3. In particular, you should not be able to execute a meaningful algorithm by pressing a sequence of keys in a line on the keyboard. Keybinds should be laid out in a way that makes sense moreso than a way that is useful for an algorithm. A keybind set optimized for general RKT is fine. If you have a question about this, ask on the Discord.\u00a0\u21a9

    4. You might hold a 33 differently to execute <M,U> moves more efficiently compared to <R,U,F> moves. Similarly, a 4D creature might have a specialized grip on the 34 for RKT compared to general moves.\u00a0\u21a9

    "},{"location":"leaderboards/solvers/adam/","title":"Adam Marcellus Kelly","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/adam/#rankings","title":"Rankings","text":"Event Rank Time Date Program 3x3x3x3 Single 9 3m 28s 220ms3:28.22 2023-08-15 HSC 3x3x3x3 Ao5 4 3m 58s 960ms3:58.96 2023-08-15 HSC 2x2x2x2 Single image/svg+xml 3 23s 680ms23.68 2024-08-07 HSC 2x2x2x2 Ao5 image/svg+xml 3 28s 080ms28.08 2024-08-28 HSC","boost":1},{"location":"leaderboards/solvers/adam/#history","title":"History","text":"4D Hypercubes 3x3x3x32x2x2x2 SingleAo5 Rank Time Date Program 9 3m 28s 220ms3:28.22 2023-08-15 HSC 3m 49s 300ms3:49.30 2023-01-12 HSC 4m 34s 620ms4:34.62 2023-01-10 HSC 5m 44s 860ms5:44.86 2023-01-09 HSC 6m 58s 670ms6:58.67 2023-01-09 HSC Rank Time Date Program 4 3m 58s 960ms3:58.96 2023-08-15 HSC SingleAo5 Rank Time Date Program image/svg+xml 3 23s 680ms23.68 2024-08-07 HSC 25s 790ms25.79 2024-06-29 HSC 26s 370ms26.37 2024-06-29 HSC 26s 490ms26.49 2023-09-23 HSC 39s 400ms39.40 2023-01-16 HSC 43s 540ms43.54 2023-01-16 HSC Rank Time Date Program image/svg+xml 3 28s 080ms28.08 2024-08-28 HSC 31s 310ms31.31 2024-07-16 HSC 33s 040ms33.04 2024-07-01 HSC","boost":1},{"location":"leaderboards/solvers/akkei/","title":"Akkei","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/akkei/#rankings","title":"Rankings","text":"Event Rank Time Date Program Hemimegaminx Single 5 18m 47s 120ms18:47.12 2024-03-21 -","boost":1},{"location":"leaderboards/solvers/akkei/#history","title":"History","text":"MagicTile Hemimegaminx Single Rank Time Date Program 5 18m 47s 120ms18:47.12 2024-03-21 -","boost":1},{"location":"leaderboards/solvers/alvin/","title":"Alvin","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/alvin/#rankings","title":"Rankings","text":"Event Rank Time Date Program 3x3x3x3 Single 26 12m 13s 260ms12:13.26 2022-11-24 HSC","boost":1},{"location":"leaderboards/solvers/alvin/#history","title":"History","text":"4D Hypercubes 3x3x3x3 Single Rank Time Date Program 26 12m 13s 260ms12:13.26 2022-11-24 HSC","boost":1},{"location":"leaderboards/solvers/asa/","title":"Asa Kaplan","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/asa/#rankings","title":"Rankings","text":"Event Rank Time Date Program Physical 2x2x2x2 Blindfolded image/svg+xml 1 47m 14s 000ms47:14.00 2022-08-08 -","boost":1},{"location":"leaderboards/solvers/asa/#history","title":"History","text":"4D Physical Physical 2x2x2x2 Blindfolded Rank Time Date image/svg+xml 1 47m 14s 000ms47:14.00 2022-08-08","boost":1},{"location":"leaderboards/solvers/battistin/","title":"Emanuele Battistin","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/battistin/#rankings","title":"Rankings","text":"Event Rank Time Date Program 3x3x3x3 Single 15 5m 29s 780ms5:29.78 2024-04-06 HSC 3x3x3x3 Ao5 7 7m 42s 360ms7:42.36 2024-03-24 HSC 2x2x2x2 Single 12 1m 25s 170ms1:25.17 2024-06-01 HSC 4x4x4x4 Single 9 59m 55s 810ms59:55.81 2023-11-10 HSC 2x2x2x3 Single image/svg+xml 3 5m 52s 720ms5:52.72 2024-05-26 MPU 2x2x3x3 Single image/svg+xml 1 22m 12s 690ms22:12.69 2024-06-12 MPU Hemimegaminx Single 4 5m 22s 970ms5:22.97 2024-03-26 MT Canon-Cut Klein Quartic Single 4 1h 33m 54s 020ms1:33:54.02 2024-06-20 MT","boost":1},{"location":"leaderboards/solvers/battistin/#history","title":"History","text":"4DMagicTile HypercubesHypercuboids 3x3x3x32x2x2x24x4x4x4 SingleAo5 Rank Time Date Program 15 5m 29s 780ms5:29.78 2024-04-06 HSC 5m 50s 990ms5:50.99 2024-03-29 HSC 6m 19s 690ms6:19.69 2023-11-06 HSC 7m 24s 840ms7:24.84 2023-11-05 HSC 7m 54s 790ms7:54.79 2023-11-04 HSC 8m 18s 630ms8:18.63 2023-11-02 HSC 8m 38s 990ms8:38.99 2023-10-31 HSC 9m 54s 040ms9:54.04 2023-10-29 HSC Rank Time Date Program 7 7m 42s 360ms7:42.36 2024-03-24 HSC Single Rank Time Date Program 12 1m 25s 170ms1:25.17 2024-06-01 HSC Single Rank Time Date Program 9 59m 55s 810ms59:55.81 2023-11-10 HSC 2x2x2x32x2x3x3 Single Rank Time Date Program image/svg+xml 3 5m 52s 720ms5:52.72 2024-05-26 MPU Single Rank Time Date Program image/svg+xml 1 22m 12s 690ms22:12.69 2024-06-12 MPU HemimegaminxCanon-Cut Klein Quartic Single Rank Time Date Program 4 5m 22s 970ms5:22.97 2024-03-26 MT Single Rank Time Date Program 4 1h 33m 54s 020ms1:33:54.02 2024-06-20 MT","boost":1},{"location":"leaderboards/solvers/bilal/","title":"Bilal Mourad","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/bilal/#rankings","title":"Rankings","text":"Event Rank Time Date Program 3x3x3x3 Single 10 3m 32s 530ms3:32.53 2024-09-26 HSC 3x3x3x3 Ao5 5 4m 36s 200ms4:36.20 2024-09-10 HSC 2x2x2x2 Single image/svg+xml 1 16s 310ms16.31 2024-12-13 HSC 2x2x2x2 Ao5 image/svg+xml 1 22s 390ms22.39 2024-12-26 HSC 4x4x4x4 Single 11 1h 41m 52s 480ms1:41:52.48 2024-07-15 HSC","boost":1},{"location":"leaderboards/solvers/bilal/#history","title":"History","text":"4D Hypercubes 3x3x3x32x2x2x24x4x4x4 SingleAo5 Rank Time Date Program 10 3m 32s 530ms3:32.53 2024-09-26 HSC 3m 43s 430ms3:43.43 2024-09-19 HSC 3m 54s 560ms3:54.56 2024-09-18 HSC 4m 12s 150ms4:12.15 2024-09-17 HSC 4m 14s 990ms4:14.99 2024-09-10 HSC 4m 18s 790ms4:18.79 2024-08-23 HSC 4m 57s 700ms4:57.70 2024-08-18 HSC 5m 48s 740ms5:48.74 2024-07-16 HSC 5m 49s 920ms5:49.92 2024-07-15 HSC 6m 07s 350ms6:07.35 2024-07-15 HSC 6m 14s 240ms6:14.24 2024-07-15 HSC 7m 47s 190ms7:47.19 2024-07-13 HSC 8m 11s 590ms8:11.59 2024-07-10 HSC 8m 36s 530ms8:36.53 2024-06-26 HSC 14m 04s 680ms14:04.68 2024-06-11 HSC Rank Time Date Program 5 4m 36s 200ms4:36.20 2024-09-10 HSC 4m 46s 600ms4:46.60 2024-08-23 HSC 6m 39s 410ms6:39.41 2024-07-15 HSC 8m 57s 530ms8:57.53 2024-07-10 HSC 10m 50s 950ms10:50.95 2024-06-26 HSC SingleAo5 Rank Time Date Program image/svg+xml 1 16s 310ms16.31 2024-12-13 HSC 18s 210ms18.21 2024-12-13 HSC 18s 790ms18.79 2024-12-03 HSC 19s 670ms19.67 2024-11-19 HSC 20s 100ms20.10 2024-10-19 HSC 21s 430ms21.43 2024-09-19 HSC 22s 120ms22.12 2024-09-18 HSC 22s 150ms22.15 2024-09-11 HSC 22s 370ms22.37 2024-09-05 HSC 23s 600ms23.60 2024-08-28 HSC 24s 230ms24.23 2024-08-11 HSC 26s 030ms26.03 2024-08-06 HSC 26s 640ms26.64 2024-08-04 HSC 27s 430ms27.43 2024-07-28 HSC 27s 610ms27.61 2024-07-22 HSC 28s 770ms28.77 2024-07-12 HSC 31s 450ms31.45 2024-07-08 HSC 32s 670ms32.67 2024-06-25 HSC 34s 330ms34.33 2024-06-21 HSC 37s 650ms37.65 2024-06-20 HSC 39s 510ms39.51 2024-06-18 HSC 40s 170ms40.17 2024-06-10 HSC 43s 500ms43.50 2024-06-07 HSC 47s 870ms47.87 2024-06-04 HSC 59s 630ms59.63 2024-06-03 HSC 1m 00s 780ms1:00.78 2024-06-02 HSC 1m 09s 220ms1:09.22 2024-06-02 HSC 1m 14s 810ms1:14.81 2024-05-31 HSC 2m 15s 060ms2:15.06 2024-05-31 HSC Rank Time Date Program image/svg+xml 1 22s 390ms22.39 2024-12-26 HSC 22s 730ms22.73 2024-12-21 HSC 23s 410ms23.41 2024-12-21 HSC 23s 450ms23.45 2024-12-17 HSC 23s 530ms23.53 2024-12-16 HSC 23s 710ms23.71 2024-12-11 HSC 24s 430ms24.43 2024-12-09 HSC 24s 810ms24.81 2024-11-15 HSC 25s 860ms25.86 2024-09-17 HSC 28s 420ms28.42 2024-09-17 HSC 28s 480ms28.48 2024-09-07 HSC 28s 760ms28.76 2024-09-06 HSC 29s 280ms29.28 2024-09-06 HSC 29s 310ms29.31 2024-08-29 HSC 29s 670ms29.67 2024-08-03 HSC 32s 270ms32.27 2024-08-01 HSC 33s 040ms33.04 2024-08-01 HSC 33s 330ms33.33 2024-07-28 HSC 33s 640ms33.64 2024-07-22 HSC 34s 270ms34.27 2024-07-16 HSC 34s 810ms34.81 2024-07-16 HSC 35s 770ms35.77 2024-07-15 HSC 38s 460ms38.46 2024-07-14 HSC 38s 780ms38.78 2024-07-10 HSC 39s 820ms39.82 2024-06-25 HSC 41s 890ms41.89 2024-06-22 HSC 44s 640ms44.64 2024-06-21 HSC 46s 210ms46.21 2024-06-18 HSC 48s 120ms48.12 2024-06-14 HSC 48s 510ms48.51 2024-06-14 HSC 48s 920ms48.92 2024-06-11 HSC 50s 720ms50.72 2024-06-08 HSC 55s 590ms55.59 2024-06-07 HSC 59s 290ms59.29 2024-06-05 HSC 1m 01s 930ms1:01.93 2024-06-05 HSC 1m 05s 370ms1:05.37 2024-06-04 HSC 1m 09s 430ms1:09.43 2024-06-04 HSC 1m 10s 990ms1:10.99 2024-06-03 HSC 1m 13s 310ms1:13.31 2024-06-03 HSC 1m 14s 340ms1:14.34 2024-06-03 HSC 1m 15s 690ms1:15.69 2024-06-02 HSC 1m 19s 540ms1:19.54 2024-06-02 HSC 1m 26s 900ms1:26.90 2024-05-31 HSC Single Rank Time Date Program 11 1h 41m 52s 480ms1:41:52.48 2024-07-15 HSC","boost":1},{"location":"leaderboards/solvers/connor/","title":"Connor Lindsay","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/connor/#rankings","title":"Rankings","text":"Event Rank Time Date Program Physical 2x2x2x2 Single 8 2m 26s 000ms2:26.00 2019-08-11 -","boost":1},{"location":"leaderboards/solvers/connor/#history","title":"History","text":"4D Physical Physical 2x2x2x2 Single Rank Time Date 8 2m 26s 000ms2:26.00 2019-08-11","boost":1},{"location":"leaderboards/solvers/cubedude/","title":"The Cube Dude","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/cubedude/#rankings","title":"Rankings","text":"Event Rank Time Date Program Physical 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HSC","boost":1},{"location":"leaderboards/solvers/tetrian/","title":"Tetrian","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/tetrian/#rankings","title":"Rankings","text":"Event Rank Time Date Program 3x3x3x3 Single 25 10m 11s 870ms10:11.87 2019-05-04 MC4D 3x3x3x3 No Filters image/svg+xml 3 10m 11s 870ms10:11.87 2019-05-04 MC4D 2x2x2x2 Single 11 1m 00s 800ms1:00.80 2021-07-24 MC4D 4x4x4x4 Single 10 1h 12m 07s 650ms1:12:07.65 2019-05-19 MC4D 3-Layer Simplex Single image/svg+xml 3 1m 43s 440ms1:43.44 2021-07-24 MC4D","boost":1},{"location":"leaderboards/solvers/tetrian/#history","title":"History","text":"4D HypercubesOther 3x3x3x32x2x2x24x4x4x4 SingleNo Filters Rank Time Date Program 25 10m 11s 870ms10:11.87 2019-05-04 MC4D Rank Time Date Program image/svg+xml 3 10m 11s 870ms10:11.87 2019-05-04 MC4D Single Rank Time Date Program 11 1m 00s 800ms1:00.80 2021-07-24 MC4D Single Rank Time Date Program 10 1h 12m 07s 650ms1:12:07.65 2019-05-19 MC4D 3-Layer Simplex 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2024-07-18 MC7D","boost":1},{"location":"leaderboards/solvers/vin/#history","title":"History","text":"4D5D Hypercubes 3x3x3x3 Single Rank Time Date Program 29 16m 33s 940ms16:33.94 2024-06-09 HSC 3x3x3x3x3 Single Rank Time Date Program image/svg+xml 2 2h 36m 47s 850ms2:36:47.85 2024-07-18 MC7D 2h 45m 09s 530ms2:45:09.53 2024-07-16 MC7D 3h 20m 00s 680ms3:20:00.68 2024-07-16 MC7D 4h 00m 40s 450ms4:00:40.45 2024-07-01 MC7D 4h 23m 58s 910ms4:23:58.91 2024-06-27 MC7D 6h 40m 34s 760ms6:40:34.76 2024-06-26 MC7D","boost":1},{"location":"leaderboards/solvers/void/","title":"The Void","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/void/#rankings","title":"Rankings","text":"Event Rank Time Date Program 3x3x3x3 Single 30 17m 19s 920ms17:19.92 2023-06-28 HSC","boost":1},{"location":"leaderboards/solvers/void/#history","title":"History","text":"4D Hypercubes 3x3x3x3 Single Rank Time Date Program 17m 19s 920ms17:19.92 2023-06-28 HSC 30 17m 19s 920ms17:19.92 2023-06-28 HSC 20m 12s 920ms20:12.92 2023-06-11 HSC","boost":1},{"location":"leaderboards/solvers/weill/","title":"Freya","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/weill/#rankings","title":"Rankings","text":"Event Rank Time Date Program 3x3x3x3 Single 32 24m 46s 950ms24:46.95 2022-12-02 HSC","boost":1},{"location":"leaderboards/solvers/weill/#history","title":"History","text":"4D Hypercubes 3x3x3x3 Single Rank Time Date Program 32 24m 46s 950ms24:46.95 2022-12-02 HSC","boost":1},{"location":"leaderboards/solvers/yuste/","title":"Sebastian Yuste","text":"View Options Time Format","boost":1},{"location":"leaderboards/solvers/yuste/#rankings","title":"Rankings","text":"Event Rank Time Date Program 3x3x3x3 Blindfolded image/svg+xml 1 1h 21m 17s 680ms1:21:17.68 2022-03-17 MC4D","boost":1},{"location":"leaderboards/solvers/yuste/#history","title":"History","text":"4D Hypercubes 3x3x3x3 Blindfolded Rank Time Date Program image/svg+xml 1 1h 21m 17s 680ms1:21:17.68 2022-03-17 MC4D","boost":1},{"location":"methods/","title":"Methods","text":"

    This page lists solving methods by puzzle.

    It\u2019s a good idea to learn Notation before trying to learn any of these methods

    "},{"location":"methods/#physical-2222","title":"Physical 2x2x2x2","text":"

    While you can use virtual 24 methods on the physical 24, it isn\u2019t recommended because you may have to use an excessive amount of gyros. These methods are specifically designed for the unique restrictive turning of the physical 24 in mind.

    • Luna\u2019s Method
    • Rowan\u2019s Method
    • Grant\u2019s OBC Method
    • P4L (PBLBC)
    • GRCL Method
    "},{"location":"methods/#2222","title":"2x2x2x2","text":"
    • 4Tega
    • Cell By Cell
    "},{"location":"methods/#3333","title":"3x3x3x3","text":"
    • CFOP
    • 3-Block
    • Octachoroux
    "},{"location":"methods/#4444","title":"4x4x4x4+","text":"
    • Reduction
    • 3Block
    "},{"location":"methods/big-cube-reduction/","title":"Big Cube Reduction","text":"

    44 with paired 1c centers

    Just like 3D reduction of big cubes, higher dimensional reduction works by pairing up groups of pieces with more and more colors until the nd is reduced into an (n-1)d. Pairing up pieces is done in a very similar way to the 3D method; If you can solve 33, 43, and 34, then you can solve 44.

    "},{"location":"methods/dimensional-reduction/","title":"Dimensional Reduction","text":"

    Dimensional reduced 34

    Dimensional reduction is a technique to reduce an nd puzzle into an nd-1 puzzle. It does this by pairing up n-piece long bars (e.g. to reduce 34 into 33, pair up 3-piece long bars).

    To get to the dimensionally reduced state from a solved 34, do only turns that don\u2019t change the colours of the I/O axis (moves that look like big 33 moves).

    Does RKT PLC count as Dimensional Reduction?

    RKT PLC does not count as dimensional reduction because certain states of the last cell are not reachable without incurring some form of RKT debt.

    Below is Andrey\u2019s log file of a 34 solve done with dimensional reduction. It\u2019s essentially a CFOP solve, but with some advanced pseudo-slotting and commutators. The solve is similar to doing Pseudo F2L (where the pairs can be in any slot), and then doing OLC/PLC (except solving the PLC into the scrambled state of the rest of the puzzle) and then finishing by solving the reduced puzzle that remains.

    Andrey_34.log Andrey_34.log
    MagicCube4D 3 0 315 {4,3,3} 3\n-0.18955569589822457 0.43991704061351233 -0.877805009971115 -2.7145128148370095E-16\n-2.839993659326744E-9 1.4847274022149884E-10 6.876845398745129E-10 -1.0\n-0.14794206390078768 -0.8966035341133775 -0.41739100175276544 8.357431997622695E-19\n0.970660488472316 -0.05074544307867573 -0.2350385417871584 -2.925836319069701E-9\n*\n104,1,1 12,-1,2 82,1,4 72,1,4 101,-1,1 58,1,4 169,1,4 136,1,2 174,-1,1 158,-1,2\n100,-1,2 44,-1,2 156,1,4 201,-1,1 141,-1,4 40,1,4 208,1,2 162,1,2 143,1,2 171,1,1\n208,-1,1 144,1,4 182,-1,1 56,-1,2 186,1,4 126,1,1 200,-1,2 120,-1,2 156,1,1 194,1,2\n166,-1,2 85,1,1 6,-1,1 143,-1,1 120,1,2 182,-1,2 147,1,4 203,1,1 71,-1,2 192,-1,2\n73,1,2 5,-1,2 159,-1,1 124,-1,2 5,1,2 184,1,4 122,1,1 213,-1,1 m| 102,-1,1\n24,1,1 49,-1,1 20,1,1 131,-1,1 24,1,1 186,1,1 157,1,1 22,1,1 49,-1,1 211,-1,4\n209,1,1 75,-1,1 197,1,1 75,1,1 212,1,4 131,-1,1 194,1,1 131,1,1 195,-1,4 102,-1,1\n211,1,1 102,1,1 214,-1,4 184,1,1 214,-1,1 184,-1,1 214,1,4 78,-1,1 210,1,1 78,1,1\n210,-1,4 209,-1,1 76,1,1 205,1,1 76,-1,1 212,-1,4 78,-1,1 199,1,1 78,1,1 191,-1,4\n202,1,1 52,-1,1 204,1,4 132,1,1 210,1,1 210,1,1 132,-1,1 15,1,1 128,1,1 74,-1,1\n106,-1,1 214,1,4 131,1,1 195,-1,1 131,-1,1 214,-1,4 214,-1,1 187,-1,1 196,-1,4 158,-1,1\n213,1,1 158,1,1 196,1,4 155,1,1 211,1,4 78,1,1 210,-1,1 78,-1,1 22,1,1 214,1,1\n77,1,1 212,1,1 77,-1,1 206,1,1 185,-1,1 212,-1,1 185,1,1 77,1,1 212,-1,1 77,-1,1\n190,1,1 6,-1,1 48,1,1 209,-1,1 48,-1,1 190,1,4 182,1,1 160,-1,1 52,-1,1 24,1,1\n157,1,1 213,1,4 157,-1,1 198,1,1 157,1,1 213,-1,4 157,-1,1 213,1,4 74,1,1 187,-1,1\n189,-1,4 105,1,1 211,1,4 105,-1,1 214,1,1 105,1,1 211,-1,4 105,-1,1 189,1,4 187,1,1\n105,1,1 211,1,4 105,-1,1 214,1,1 105,1,1 211,-1,4 105,-1,1 3,1,1 210,1,1 130,-1,1\n212,-1,4 130,1,1 199,1,1 130,-1,1 212,1,4 130,1,1 192,1,4 74,1,1 193,-1,4 198,1,1\n76,-1,1 210,-1,4 76,1,1 201,1,1 76,-1,1 210,1,4 76,1,1 193,1,4 101,1,1 187,-1,1\n187,-1,1 200,1,4 48,-1,1 209,-1,4 48,1,1 191,1,1 48,-1,1 209,1,4 48,1,1 200,1,4\n51,-1,1 199,1,4 51,1,1 213,-1,1 51,-1,1 199,1,4 51,1,1 182,-1,1 78,1,1 202,1,4\n78,-1,1 210,1,1 210,1,1 78,1,1 202,1,4 78,-1,1 128,1,2 160,1,1 128,1,2 160,-1,1\n182,-1,2 182,-1,2 132,-1,1 202,1,4 132,1,1 196,1,1 132,-1,1 202,1,4 132,1,1 211,1,1\n211,1,1 47,-1,1 201,1,1 101,1,1 74,1,1 201,1,1 128,-1,1 155,-1,1 200,1,1 47,1,1\n155,1,1 200,1,1 52,-1,1 202,1,1 74,-1,1 128,-1,1 202,1,1 182,-1,1 160,1,1 208,1,1\n182,-1,1 155,-1,1 208,1,1 155,1,2 206,1,1 182,1,1 101,1,1 206,1,1 47,1,1 74,-1,1\n207,1,1 182,1,1 128,1,1 207,1,1 187,-1,1 197,1,1 47,1,1 74,1,1 47,-1,1 74,-1,1\n47,1,1 74,1,1 197,1,1 133,-1,1 155,-1,1 155,-1,1 133,-1,1 52,1,1 133,1,1 52,-1,1\n205,1,1 74,-1,1 182,-1,1 74,1,1 182,1,1 74,-1,1 182,-1,1 205,1,1 133,-1,1 203,1,1\n155,-1,1 101,-1,1 155,1,1 101,1,1 155,-1,1 101,-1,1 203,1,1 52,-1,1 79,1,1 52,1,1\n52,1,1 208,1,1 182,-1,1 155,-1,1 182,1,1 155,1,1 182,-1,1 155,-1,1 208,1,1 155,-1,1\n74,-1,1 74,-1,1 182,1,1 74,-1,1 182,-1,1 101,1,1 182,1,1 101,-1,1 182,1,1 101,-1,1\n101,-1,1 74,1,1 74,1,1 47,-1,1 128,1,1 155,-1,1 155,-1,1 47,-1,1 47,-1,1 79,-1,1\n79,-1,1 47,-1,2 47,-1,2 106,1,1 74,-1,1 74,-1,1 47,1,1 133,1,1 133,1,1 79,1,1\n79,1,1 101,1,2 101,1,2 187,-1,1 47,1,1 47,1,1 101,1,1 101,1,1 47,1,1 47,1,1\n101,1,1 101,1,1 47,1,1 47,1,1 74,-1,1 74,-1,1 47,-1,2 74,1,1 74,1,1 47,1,2\n74,1,1 74,1,1 133,1,1 133,1,1.\n

    The move count is extremely low, which is pretty cool.

    "},{"location":"methods/general-cfop/","title":"Generalized CFOP","text":"

    This article is a generator for the steps of a dimension-generalized CFOP method. For CFOP on 34, see CFOP.

    CFOP on: 35

    "},{"location":"methods/general-cfop/#cross","title":"Cross","text":""},{"location":"methods/general-cfop/#f2l","title":"F2L","text":""},{"location":"methods/general-cfop/#oll","title":"OLL","text":""},{"location":"methods/general-cfop/#pll","title":"PLL","text":""},{"location":"methods/ll-strats/","title":"Last Layer Strategies","text":"

    This page is a work in progress.

    This page documents some common last layer strategies for hypercubes (variants of these techniques will work on other puzzles too).

    "},{"location":"methods/ll-strats/#ollpll","title":"OLL/PLL","text":"

    First all of the pieces are oriented in increasing order by number of stickers.

    To orient 2c pieces, use EOLL algorithms from 2-look OLL.

    To orient the 3c pieces, use RKT to set up the E slice of the last cell into a configuration that looks like a possible OCLL case. Then use OCLL algorithms to solve that case.

    Avoiding monotwist/flip

    In higher dimensions, it\u2019s possible to have a single 3c+ piece twisted in place. To avoid this, make sure that your last OCLL algorithm setup will solve all of the remaining pieces of that piece type. For example if you have 5 left to orient, don\u2019t set up an alg that orients 4 because then you will be left over with 1. Instead, you can do an alg that orients 3, and then an alg that orients 2.

    To orient the 4c pieces, use RKT to set up OCLL cases, but this time you have to perform the OCLL alg with RKT. Again, it\u2019s possible to have just 1 4C left to orient, so be smart about your last setup and algorithm.

    \u0651For 4c pices in 2^4, because of a lack of a need to preserve 2 and 3cs, something that can be done is grouping 4 unoriented corners together in a 1x1x2x2 block; then all 4 of these pieces can be oriented with just a U OLL case (F R U R\u2019 U\u2019 F\u2019) with big moves. Something similar can be done with the T and L OLL shapes as well. This way of orienting pieces is useful because not only are you solving more pieces at a time, but they don\u2019t take many moves to do so compared to doing the algs in RKT. If after reaching the OLL step, there are 5 4cs that are misoriented, it may be preferable do something else instead (solve a group of 3, and then 2), as you would want to avoid a monoflip.

    To orient 5c+ pieces, you have to use RKT to setup OCLL cases, and then do the algorithm using double/triple/etc RKT.

    For PLL, first permute the 2c pieces using U-perms or other EPLL algorithms. Then use RKT to permute the last layer like a lower dimensional puzzle using RKT.

    "},{"location":"methods/ll-strats/#direct-ll","title":"Direct LL","text":"

    In 4D+, you can use F2L isolations to help you orient and permute blocks of pieces at the same time.

    "},{"location":"methods/ll-strats/#partial-direct-ll","title":"Partial Direct LL","text":"

    If you use direct LL techniques to just solve the F2L of the LL recursively until you\u2019re left with a \u201csquare\u201d of pieces that need orienting and permuting. This group of pieces is the same amount as in 3D, except they can be oriented wrong in many different ways. From here, you can use algorithms to orient them, and then just normal PLL algorithms to permute them.

    "},{"location":"methods/2x2x2x2/4tega/","title":"4tega","text":"

    4tega is the 4-dimensional equivalent of the Ortega Method on 23. There are 2 main variants, as there is no perfect way to generalize this method to 4d.

    "},{"location":"methods/2x2x2x2/4tega/#variant-1","title":"Variant 1","text":""},{"location":"methods/2x2x2x2/4tega/#orient-both-cells","title":"Orient Both Cells","text":"

    Start by orienting a single side, using either colour from that side or its opposite side. This can easily be done without RKT by using blockbuilding. Next, use 24 OLC strategies to orient the other cell. Because the first cell isn\u2019t solved, you don\u2019t have to worry about messing it up, allowing you to save moves.

    "},{"location":"methods/2x2x2x2/4tega/#separate-colors","title":"Separate colors","text":"

    Separate the colors of the cells that are oriented, leaving you with 2 23 solves.

    "},{"location":"methods/2x2x2x2/4tega/#permute-both-cells","title":"Permute Both Cells","text":"

    Solve the first cell like a 23 without RKT, then solve the other cell like a 23, but using RKT.

    "},{"location":"methods/2x2x2x2/4tega/#variant-2","title":"Variant 2","text":""},{"location":"methods/2x2x2x2/4tega/#orient-both-cells_1","title":"Orient Both Cells","text":"

    Same as in Variant 1.

    "},{"location":"methods/2x2x2x2/4tega/#separate-colors_1","title":"Separate colors","text":"

    Same as in Variant 1.

    "},{"location":"methods/2x2x2x2/4tega/#orient-both-layers-of-both-cells","title":"Orient Both Layers of Both Cells","text":"

    Use RKT to orient both layers of both cells, just like OBL in 3D Ortega. This can be done using the OCLL algorithms.

    "},{"location":"methods/2x2x2x2/4tega/#p4lpblbc","title":"P4L/PBLBC","text":"

    Permtue all 4 layers of both cells at once using algorithms. Some algorithms can be found here.

    "},{"location":"methods/2x2x2x2/cell-by-cell/","title":"Cell by Cell","text":"

    Cell by Cell is the 4D equivalent of Layer by Layer.

    "},{"location":"methods/2x2x2x2/cell-by-cell/#1-solve-the-first-cell","title":"1. Solve the first cell","text":"

    Completely solve one of the sides of the puzzle. You can do this by blockbuilding, or by getting all the same color one on side and then using RKT to permute the pieces around until they are solved.

    "},{"location":"methods/2x2x2x2/cell-by-cell/#2-solve-the-last-cell","title":"2. Solve the last cell","text":""},{"location":"methods/2x2x2x2/cell-by-cell/#a-orient-the-last-cell","title":"a. Orient the last cell","text":"

    Orient the last cell by using RKT to set up OCLL cases, and then using RKT to perform those algorithms.

    "},{"location":"methods/2x2x2x2/cell-by-cell/#b-permute-the-last-cell","title":"b. Permute the last cell","text":"

    Permute the last cell like a 23 using RKT. You may run into RKT parity in this step, or you can avoid it by using certain last layer algorithms.

    "},{"location":"methods/2x2x2x2-physical/grant/","title":"Grant\u2019s OBC Method for physical 2x2x2x2","text":"

    Grant\u2019s OBC Method uses only 3 gyros, making it much faster than Rowan\u2019s OBC Method. It works by secretly setting up to an H OCLL case while orienting both cells.

    "},{"location":"methods/2x2x2x2-physical/grant/#steps","title":"Steps","text":""},{"location":"methods/2x2x2x2-physical/grant/#inspection","title":"Inspection","text":"

    Before starting the solve, inspect the puzzle for an opposite colour pair that you can easily get exactly 8 stickers from that pair to be on the y-axis. They can be in any spot, as long as they are pointing up or down.

    "},{"location":"methods/2x2x2x2-physical/grant/#getting-8-oriented-to-y-axis-then-gyro","title":"Getting 8 Oriented to Y-axis, then gyro","text":"

    This is done in just a few simple moves, and should be fully planned during inspection for speedsolving. This step is very difficult to get used to, but gets easier over time. Once it looks like the image above, perform the gyro algorithm such that those 8 stickers end up on the x-axis (the \u201cpipped\u201d spots).

    "},{"location":"methods/2x2x2x2-physical/grant/#orient-the-remaining-8-to-y-axis","title":"Orient the remaining 8 to y-axis","text":"

    Now that 8 corners have been oriented, you can do any moves (besides a gyro), and not affect their orientation. Pair up the remaining 8 so that they are all on the same cell, and oriented to the y-axis.

    "},{"location":"methods/2x2x2x2-physical/grant/#finish-orientation","title":"Finish orientation","text":"

    Rotate the cell with the pieces you just paired up to the w-axis (facing in/out). Now perform the gyro algorithm, and cancel into the 5 move H OCLL case, and then gyro again.

    "},{"location":"methods/2x2x2x2-physical/grcl/","title":"GRCL Method","text":"

    The Grant Rowan Connor Luna method, or GRCL (pronounced gurkle) is a combination of the best parts of several methods, including Grant\u2019s OBC method, Rowan\u2019s OBC method, Luna\u2019s cell blockbuilding technique, and Connor Lindsay\u2019s P4L algorithms. The name GRCL was proposed by Hypercubers Discord member Jasper on July 18 2022.

    "},{"location":"methods/2x2x2x2-physical/grcl/#steps","title":"Steps","text":"
    • Orient Both Cells using either Grant\u2019s method or Rowan\u2019s method (whatever looks better in that particular scramble)
    • Separate the oriented colours of both cells while orienting 2 layers on those cells by holding the puzzle vertically and using Luna\u2019s method
    • Permuate both layers of both cell using Rowan\u2019s updated version of Connor\u2019s P4L algorithms
    "},{"location":"methods/2x2x2x2-physical/luna/","title":"Luna\u2019s Method for 2x2x2x2","text":"

    A method that Luna developed in 2017-18 making heavy use of an IU move sequence.

    "},{"location":"methods/2x2x2x2-physical/luna/#steps","title":"Steps","text":"
    • Orient the D cell with opposite colours
    • Orient the U cell making use of the IU move
    • Permute a single cell via blockbuilding
    • Permute the last cell using RKT
    "},{"location":"methods/2x2x2x2-physical/luna/#resources","title":"Resources","text":"
    • Luna\u2019s video
    "},{"location":"methods/2x2x2x2-physical/p4l/","title":"P4L","text":""},{"location":"methods/2x2x2x2-physical/p4l/#history","title":"History","text":"

    PBLBC/P4L/PAL is an advanced algorithm set for speedsolving Melinda\u2019s Physical 2x2x2x2. It is used after orientation and separation of 2 opposite cells, followed by (or simultaneously) orienting both layers on both of those cells. This leaves you with two 2x2x2 PBL cases, one on each cell. The original idea for this came from Connor Lindsay, who created most of the algorithms in his algorithm sheet.

    2 years later, Rowan tried optimizing the algorithms, and compiling them in a new document, which can be found here.

    "},{"location":"methods/2x2x2x2-physical/p4l/#algorithms","title":"Algorithms","text":"

    Each layer can be in 3 different permutations: solved, adjacent swap, and opposite swap. This gives us 81 cases, but some of these are impossible. Cases can also be rearranged via Case Manipulation, further reducing the number of cases to 29. Practically however, only a few algorithms are used because all the bad cases are really slow to execute.

    "},{"location":"methods/2x2x2x2-physical/p4l/#big-3d-cases","title":"Big 3D Cases","text":"

    Big 3D cases are the cases where you have 2 pairs of the same layers, such as OOAA. In such a case, you can just use the normal 23 PBL algorithm for that case (but it must be a version with the moveset <R2,L2,F2,B2,U*,D*> only, due to the restrictions in turning the puzzle).

    "},{"location":"methods/2x2x2x2-physical/rowan/","title":"Rowan\u2019s OBC Method for 2x2x2x2","text":"

    Rowan developed this method for Orienting Both Cells while trying to solve the physical 2x2x2x2 for the first time in a YouTube video.

    "},{"location":"methods/2x2x2x2-physical/rowan/#steps","title":"Steps","text":""},{"location":"methods/2x2x2x2-physical/rowan/#inspection","title":"Inspection","text":"

    Before starting the solve, inspect the puzzle for an opposite colour pair that has 4 or fewer stickers oriented to the Left/Right axis (x-axis). They can be made of any colour in the colour pair, and can be in any spot on the puzzle.

    4 red/orange corners oriented to L/R

    "},{"location":"methods/2x2x2x2-physical/rowan/#getting-8-oriented-to-y-axis","title":"Getting 8 Oriented to Y-axis","text":"

    Hold the puzzle vertically, and use block building to orient 8 pieces on a cell to the y-axis using the same colour pair that has 4 or less from inspection. This step is intuitive and does not require any RKT.

    8 red/orange corners oriented to U/D

    "},{"location":"methods/2x2x2x2-physical/rowan/#getting-12-oriented-to-y-axis","title":"Getting 12 Oriented to Y-axis","text":"

    Now use RKT to orient a layer of 4 stickers on the other cell, leaving you with a \u201clast layer\u201d of 4 unoriented pieces on one of the cells.

    12 red/orange corners oriented to U/D

    "},{"location":"methods/2x2x2x2-physical/rowan/#fixing-the-last-layer","title":"Fixing the Last Layer","text":"

    Gyro the puzzle such that the 12 pieces that were oriented end up on the x-axis. Now you should have 4 (or fewer) stickers from that colour pair oriented in other directions besides the x-axis. All you need to do is use RKT to make it into a layer that looks like a possible OCLL case, not worrying about messing up the other 12 corners.

    an H OCLL case set up on LU

    From here, gyro back. Now you will have a normal OCLL case that you can solve using RKT. After doing that algorithm, you should have all 16 corners oriented to the y-axis. Finally, you just gyro to put them all on the x-axis, and OBC is solved.

    "},{"location":"methods/2x2x2x2-physical/rowan/#analysis","title":"Analysis","text":"

    This technique takes more moves than other known OBC methods, and it uses up to 4 gyros. Besides the skip case where everything is already oriented, the best possible case for this technique uses 3 gyros, making it slower than other techniques. It also requires knowing the OCLL algorithms, which other techniques don\u2019t.

    "},{"location":"methods/3x3x3x3/3block/","title":"3-Block","text":"

    3-Block is a method designed for quickly speedsolving the 34. Many speedsolving world records have been set using this method. It was primarily invented by Luna and HactarCE, and has been described as \u201cZZ without EO\u201d. 3-Block is most analogous to the 33 method FreeFOP (which omits a single cross piece in order to pair pieces easier). Compared to 4D CFOP, 3-Block uses ~20% fewer moves.

    Prerequisites

    • Knowledge of how the 34 moves
    • Notation
    • RKT
    • 3D CFOP with 2-look OLL & PLL
    HSC Piece filters (34) Hactar\u2019s filters (pink cross)Rowan\u2019s filters (white cross)Rowan\u2019s new filters (pink cross)Triplets instead of pairs 
    - preset_name: 4-cross\n  visible_pieces: 000200080a5010284280\n- preset_name: Mid (back)\n  visible_pieces: 000200084a5090294284\n- preset_name: Mid\n  visible_pieces: 000200094a5294294294\n- preset_name: Left (cross)\n  visible_pieces: 000200094a5294296294\n- preset_name: Left\n  visible_pieces: 0002000d4a5a943962d4\n- preset_name: BL-a\n  visible_pieces: 0002000d4a7a943963d4\n- preset_name: BL\n  visible_pieces: 0002000d6a7ad4b963d6\n- preset_name: FL-a\n  visible_pieces: 0002000d6b7ad4bd63d6\n- preset_name: FL\n  visible_pieces: 0002008d6b7bd6bd6bd6\n- preset_name: Right (cross)\n  visible_pieces: 0002008d6b7bd6bdebd6\n- preset_name: Right\n  visible_pieces: 0002008d7b7bf6fdebd7\n- preset_name: BR-a\n  visible_pieces: 0002008d7bfbf6fdefd7\n- preset_name: BR\n  visible_pieces: 0002008dfbfbf7ffefdf\n- preset_name: FR-a\n  visible_pieces: 0002008dfffbf7ffffdf\n- preset_name: FR\n  visible_pieces: 0002008fffffffffffff\n- preset_name: OLC 2c\n  visible_pieces: \"01471400000000000000\"\n- preset_name: OLC 3c\n  visible_pieces: aaa8aa20000000000000\n- preset_name: OLC 4c\n  visible_pieces: \"54104150000000000000\"\n- preset_name: OLC\n  visible_pieces: ffffff70000000000000\n- preset_name: PLC 2c\n  visible_pieces: 014714080a5010004000\n- preset_name: PLC cross\n  visible_pieces: \"01479e20000000000000\"\n- preset_name: PLC F2L\n  visible_pieces: 01efff70000000000000\n- preset_name: PLC\n  visible_pieces: ffffff70000000000000\n- preset_name: End\n  visible_pieces: ffffffffffffffffffff\n
    - preset_name: 4/6Cross\n  visible_pieces: 000204080a5830004080\n- preset_name: Middle 1\n  visible_pieces: 00070e080a5830004080\n- preset_name: Middle 2\n  visible_pieces: 00070e080a583000e0c1\n- preset_name: Left Block 1\n  visible_pieces: 00478e080a5a3008e0d1\n- preset_name: Left Block 2\n  visible_pieces: 00478e080e5e3008e0d1\n- preset_name: Left Block 3\n  visible_pieces: 00c78f080e5e3008e0d1\n- preset_name: Left Block 4\n  visible_pieces: 00c78f080e5e3008f0f1\n- preset_name: Left Block 5\n  visible_pieces: 00c78f080f5f3008f0f1\n- preset_name: Left Block 6\n  visible_pieces: 00e7cf080f5f3008f0f1\n- preset_name: Left Block 7\n  visible_pieces: 00e7cf080f5f300cf8f1\n- preset_name: Right Block 1\n  visible_pieces: 00e7df280f5fb00cfaf5\n- preset_name: Right Block 2\n  visible_pieces: 00e7df280fdfb10cfaf5\n- preset_name: Right Block 3\n  visible_pieces: 00e7ff680fdfb10cfaf5\n- preset_name: Right Block 4\n  visible_pieces: 00e7ff680fdfb10cfefd\n- preset_name: Right Block 5\n  visible_pieces: 00e7ff680ffff10cfefd\n- preset_name: Right Block 6\n  visible_pieces: 00efff780ffff10cfefd\n- preset_name: Right Block 7\n  visible_pieces: 00efff780ffff10cffff\n- preset_name: 2c OLC\n  visible_pieces: 01efff7d5ffff12cffff\n- preset_name: 3c OLC\n  visible_pieces: abeffffffffff57dffff\n- preset_name: 4c OLC\n  visible_pieces: 54ffff700ffffb8effff\n- preset_name: 2c PLC\n  visible_pieces: 01efff7d5ffff12cffff\n- preset_name: RKT PLC Cross\n  visible_pieces: ab00000d500000204000\n- preset_name: RKT PLC F2L\n  visible_pieces: ff10008ff00000204000\n- preset_name: RKT PLC LL\n  visible_pieces: ff10008ff0000ef30000\n
    - preset_name: 4-cross\n  visible_pieces: 000200080a5010284280\n- preset_name: odM\n  visible_pieces: 000200094a5014294280\n- preset_name: ouM\n  visible_pieces: 000200094a5294294294\n- preset_name: OLES\n  visible_pieces: 000200094a5294296294\n- preset_name: oLS\n  visible_pieces: 0002000d4a5a943962d4\n- preset_name: oBLE\n  visible_pieces: 0002000d4a7a943963d4\n- preset_name: oDBL\n  visible_pieces: 0002000d6a7a94b963d4\n- preset_name: oUBL\n  visible_pieces: 0002000d6a7ad4b963d6\n- preset_name: oFLE\n  visible_pieces: 0002000d6b7ad4bd63d6\n- preset_name: oDFL\n  visible_pieces: 0002008d6b7ad6bd63d6\n- preset_name: oUFL\n  visible_pieces: 0002008d6b7bd6bd6bd6\n- preset_name: ORES\n  visible_pieces: 0002008d6b7bd6bdebd6\n- preset_name: oRS\n  visible_pieces: 0002008d7b7bf6fdebd7\n- preset_name: oBRE\n  visible_pieces: 0002008d7bfbf6fdefd7\n- preset_name: oDBR\n  visible_pieces: 0002008dfbfbf6ffefd7\n- preset_name: oUBR\n  visible_pieces: 0002008dfbfbf7ffefdf\n- preset_name: oFRE\n  visible_pieces: 0002008dfffbf7ffffdf\n- preset_name: oDFR\n  visible_pieces: 0002008ffffbffffffdf\n- preset_name: oUFR\n  visible_pieces: 0002008fffffffffffff\n- preset_name: OLC 2c\n  visible_pieces: \"01471400000000000000\"\n- preset_name: OLC 3c\n  visible_pieces: aaa8aa20000000000000\n- preset_name: OLC 4c\n  visible_pieces: \"54104150000000000000\"\n- preset_name: PLC 2c\n  visible_pieces: 014714080a5010004000\n- preset_name: PLC cross\n  visible_pieces: \"01479e20000000000000\"\n- preset_name: PLC F2L\n  visible_pieces: 01efff70000000000000\n- preset_name: PLC\n  visible_pieces: ffffff70000000000000\n- preset_name: End\n  visible_pieces: ffffffffffffffffffff\n
    - preset_name: 4-cross\n  visible_pieces: 000200080a5010284280\n- preset_name: odM\n  visible_pieces: 000200094a5014294280\n- preset_name: ouM\n  visible_pieces: 000200094a5294294294\n- preset_name: OLES\n  visible_pieces: 000200094a5294296294\n- preset_name: oLS\n  visible_pieces: 0002000d4a5a943962d4\n- preset_name: triplet 1\n  visible_pieces: 0002000d6a7ad43962d4\n- preset_name: triplet 2\n  visible_pieces: 0002000d6a7ad4b963d6\n- preset_name: triplet 3\n  visible_pieces: 0002008d6b7bd4b963d6\n- preset_name: triplet 4\n  visible_pieces: 0002008d6b7bd6bd6bd6\n- preset_name: ORES\n  visible_pieces: 0002008d6b7bd6bdebd6\n- preset_name: oRS\n  visible_pieces: 0002008d7b7bf6fdebd7\n- preset_name: triplet 5\n  visible_pieces: 0002008dfbfbf7fdebd7\n- preset_name: triplet 6\n  visible_pieces: 0002008dfbfbf7ffefdf\n- preset_name: triplet 7\n  visible_pieces: 0002008ffffff7ffefdf\n- preset_name: triplet 8\n  visible_pieces: 0002008fffffffffffff\n- preset_name: OLC 2c\n  visible_pieces: \"01471400000000000000\"\n- preset_name: OLC 3c\n  visible_pieces: aaa8aa20000000000000\n- preset_name: OLC 4c\n  visible_pieces: \"54104150000000000000\"\n- preset_name: PLC 2c\n  visible_pieces: 014714080a5010004000\n- preset_name: PLC cross\n  visible_pieces: \"01479e20000000000000\"\n- preset_name: PLC F2L\n  visible_pieces: 01efff70000000000000\n- preset_name: PLC\n  visible_pieces: ffffff70000000000000\n- preset_name: End\n  visible_pieces: ffffffffffffffffffff\n
    HSC Piece filters (44, 54, 64, 74)

    These are all Hactar\u2019s filters, which use pink cross.

    44546474 
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\"000000000000000006606066060660066060660606600660666666660660\"\n- preset_name: Belt 1/4\n  visible_pieces: \"000000000000000006666066060660066660660606600666666666660660\"\n- preset_name: Belt 2/4\n  visible_pieces: \"000000000000000006666066060666066660660606660666666666660666\"\n- preset_name: Belt 3/4\n  visible_pieces: \"000000000000000066666066060666666660660606666666666666660666\"\n- preset_name: Belt\n  visible_pieces: \"000000000000000066666066066666666660660666666666666666666666\"\n- preset_name: L center\n  visible_pieces: \"000000000000000066666566566666666665665666666666666666666666\"\n- preset_name: L 1/3\n  visible_pieces: \"000000000000000066666566566666666665665666666666677667766666\"\n- preset_name: L 2/3\n  visible_pieces: \"000000000000000067766566566666677665665666666776677667766666\"\n- preset_name: L 3/3\n  visible_pieces: \"000000000000000067766566566776677665665667766776677667766776\"\n- preset_name: BL 1/3\n  visible_pieces: \"000000000000000067766576576776677665765767766776677767776776\"\n- preset_name: BL 2/3\n  visible_pieces: \"000000000000000067776576576776677765765767766777677767776776\"\n- preset_name: BL 3/3\n  visible_pieces: \"000000000000000067776576576777677765765767776777677767776777\"\n- preset_name: FL 1/3\n  visible_pieces: \"000000000000000067777577576777677775775767776777777777776777\"\n- preset_name: FL 2/3\n  visible_pieces: \"000000000000000077777577576777777775775767777777777777776777\"\n- preset_name: FL 3/3\n  visible_pieces: \"000000000000000077777577577777777775775777777777777777777777\"\n- preset_name: R center\n  visible_pieces: 000000000000000077777f77f7777777777f77f777777777777777777777\n- preset_name: R 1/3\n  visible_pieces: 000000000000000077777f77f7777777777f77f7777777777ff77ff77777\n- preset_name: R 2/3\n  visible_pieces: 00000000000000007ff77f77f777777ff77f77f777777ff77ff77ff77777\n- preset_name: R 3/3\n  visible_pieces: 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\"066006660666066000000000000000000000000000000000000000000000\"\n- preset_name: RLC 4/6\n  visible_pieces: \"066007760776066000000000000000000000000000000000000000000000\"\n- preset_name: RLC 6/6\n  visible_pieces: 06606ff66ff6066000000000000000000000000000000000000000000000\n- preset_name: ELC 2/12\n  visible_pieces: \"000600000000000600000000000000000000000000000000000000000000\"\n- preset_name: ELC 4/12\n  visible_pieces: \"600600000000600600000000000000000000000000000000000000000000\"\n- preset_name: ELC 6/12\n  visible_pieces: \"611600000000611600000000000000000000000000000000000000000000\"\n- preset_name: ELC 8/12\n  visible_pieces: \"611610011001611600000000000000000000000000000000000000000000\"\n- preset_name: ELC 10/12\n  visible_pieces: \"699610011001699600000000000000000000000000000000000000000000\"\n- preset_name: ELC 12/12\n  visible_pieces: \"699690099009699600000000000000000000000000000000000000000000\"\n- preset_name: OLC\n  visible_pieces: ffffffffffffffff00000000000000000000000000000000000000000000\n- preset_name: PLC 2c\n  visible_pieces: 06606ff66ff6066006606f66f6066006606f66f606600000066006600000\n- preset_name: PLC cross\n  visible_pieces: 06606ff66ff66ff600000000000000000000000000000000000000000000\n- preset_name: PLC F2L\n  visible_pieces: 0660ffffffffffff00000000000000000000000000000000000000000000\n- preset_name: PLC\n  visible_pieces: ffffffffffffffff00000000000000000000000000000000000000000000\n- preset_name: Done\n  visible_pieces: ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff\n
    - preset_name: U center\n  visible_pieces: 0000000000000004000000000000000000000000000000000007ec100080000000481200000c9370000000000000000000007ec100000000000000002000000000000000\n- preset_name: D center\n  visible_pieces: 00000000000000040000000000000000837e0000000000000007ec100ec9300000481200000c93700837e0000000000000007ec100000000000000002000000000000000\n- preset_name: B center\n  visible_pieces: 00000000000000040000000000000000837e0000c100c100c107ec100ec9300007481700070c93700837e0000c100c100c107ec100000000000000002000000000000000\n- preset_name: F center\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100000000000000002000000000000000\n- preset_name: O center\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100000000c93700837e0007ec10000000\n- preset_name: 4-cross D center\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100080000c93700837e0007ec10000000\n- preset_name: 4-cross D\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100ec9300c93700837e0007ec10000000\n- preset_name: 4-cross B center\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100ec9300c93700837e8007ec10000000\n- preset_name: 4-cross B\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100ec9300c937e0837ec107ec93000000\n- preset_name: 4-cross U center\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100ec9300c937e0837ec107ec93008000\n- preset_name: 4-cross U\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100ec9300c937e0837ec107ec930ec930\n- preset_name: 4-cross F center\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100ec9300c937e8837ec107ec930ec930\n- preset_name: 4-cross F\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec930e007e817e0070c93700837e0830c930c930c107ec100ec930ec937ec937ec937ec930ec930\n- preset_name: Mid DB ridge center\n  visible_pieces: 00000000000000040000000000000000837e0830c930c930c107ec100ec932e007e817e0070c93700837e0830c930c930c107ec100ec930ec937ec937ec937ec930ec930\n- preset_name: Mid DB ridge\n  visible_pieces: 00000000000000040000000000000000837ec930c930c930c107ec100ec937e007e817e0070c93700837ec930c930c930c107ec100ec930ec937ec937ec937ec930ec930\n- preset_name: Mid DB\n  visible_pieces: 00000000000000040000000000000000837ec930c930c930c107ec100ec937e007e817e0070c93700837ec930c930c930c107ec100ec937ec937ec937ec937ec930ec930\n- preset_name: Mid UB ridge center\n  visible_pieces: 00000000000000040000000000000000837ec930c930c930c107ec100ec937e007e817e0070c93740837ec930c930c930c107ec100ec937ec937ec937ec937ec930ec930\n- preset_name: Mid UB ridge\n  visible_pieces: 00000000000000040000000000000000837ec930c930c930c107ec930ec937e007e817e0070c937e0837ec930c930c930c107ec930ec937ec937ec937ec937ec930ec930\n- preset_name: Mid UB\n  visible_pieces: 00000000000000040000000000000000837ec930c930c930c107ec930ec937e007e817e0070c937e0837ec930c930c930c107ec930ec937ec937ec937ec937ec930ec937\n- preset_name: Mid DF ridge center\n  visible_pieces: 00000000000000040000000000000000837ec930c930c930c107ec932ec937e007e817e0070c937e0837ec930c930c930c107ec930ec937ec937ec937ec937ec930ec937\n- preset_name: Mid DF ridge\n  visible_pieces: 0000000000000004000000000000000c937ec930c930c930c107ec937ec937e007e817e0070c937ec937ec930c930c930c107ec930ec937ec937ec937ec937ec930ec937\n- preset_name: Mid DF\n  visible_pieces: 0000000000000004000000000000000c937ec930c930c930c107ec937ec937e007e817e0070c937ec937ec930c930c930c107ec937ec937ec937ec937ec937ec930ec937\n- preset_name: Mid UF ridge center\n  visible_pieces: 0000000000000004000000000000000c937ec930c930c930c107ec937ec937e007e817e0074c937ec937ec930c930c930c107ec937ec937ec937ec937ec937ec930ec937\n- preset_name: Mid UF ridge\n  visible_pieces: 0000000000000004000000000000000c937ec930c930c930c937ec937ec937e007e817e007ec937ec937ec930c930c930c937ec937ec937ec937ec937ec937ec930ec937\n- preset_name: Mid UF\n  visible_pieces: 0000000000000004000000000000000c937ec930c930c930c937ec937ec937e007e817e007ec937ec937ec930c930c930c937ec937ec937ec937ec937ec937ec937ec937\n- preset_name: L center\n  visible_pieces: 0000000000000000000000000000000c937ec9bac9bac9bac937ec937ec937ea27ea27ea27ec937ec937ec9bac9bac9bac937ec937ec937ec937ec937ec937ec937ec937\n- preset_name: L cross ridge center\n  visible_pieces: 0000000000000000000000000000000c937ec9bac9bac9bac937ec937ec937ea27ea27ea27ec937ec937ec9bac9bac9bac937ec937ec937ec937ec9b7ec937ec937ec937\n- preset_name: L cross ridge\n  visible_pieces: 0000000000000000000000000000000c937ec9bac9bac9bac937ec937ec937ea27ea27ea27ec937ec937ec9bac9bac9bac937ec937ec937eedb7ecdb7fc9b7fe937ec937\n- preset_name: L D ridge center\n  visible_pieces: 0000000000000000000000000000000c937ec9bac9bac9bac937ec937ee937ea27ea27ea27ec937ec937ec9bac9bac9bac937ec937ec937eedb7ecdb7fc9b7fe937ec937\n- preset_name: L D ridge\n  visible_pieces: 0000000000000000000000000000000cdb7fc9bac9bac9bac937ec937fed37ea27ea27ea27ec937ecdb7fc9bac9bac9bac937ec937ec937eedb7ecdb7fc9b7fe937ec937\n- preset_name: L D\n  visible_pieces: 0000000000000000000000000000000cdb7fc9bac9bac9bac937ec937fed37ea27ea27ea27ec937ecdb7fc9bac9bac9bac937ec937fed37eedb7ecdb7fc9b7fe937ec937\n- preset_name: L U ridge center\n  visible_pieces: 0000000000000000000000000000000cdb7fc9bac9bac9bac937ec937fed37ea27ea27ea27ecd37ecdb7fc9bac9bac9bac937ec937fed37eedb7ecdb7fc9b7fe937ec937\n- preset_name: L U ridge\n  visible_pieces: 0000000000000000000000000000000cdb7fc9bac9bac9bac9b7fe937fed37ea27ea27ea27eedb7ecdb7fc9bac9bac9bac9b7fe937fed37eedb7ecdb7fc9b7fe937ec937\n- preset_name: L U\n  visible_pieces: 0000000000000000000000000000000cdb7fc9bac9bac9bac9b7fe937fed37ea27ea27ea27eedb7ecdb7fc9bac9bac9bac9b7fe937fed37eedb7ecdb7fc9b7fe937fed37\n- preset_name: L B ridge center\n  visible_pieces: 0000000000000000000000000000000cdb7fc9bac9bac9bac9b7fe937fed37ea27eaa7ea27eedb7ecdb7fc9bac9bac9bac9b7fe937fed37eedb7ecdb7fc9b7fe937fed37\n- preset_name: L B ridge\n  visible_pieces: 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000000000000effffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff70000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\n- preset_name: PLC LL\n  visible_pieces: fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff70000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\n- preset_name: Done\n  visible_pieces: ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff\n
    "},{"location":"methods/3x3x3x3/3block/#summary","title":"Summary","text":"
    1. 4-Cross \u2014 Solve 4 out of the 6 cross pieces
    2. Middle Block \u2014 Solve 4 2c3c pairs betwen the 4 cross pieces
    3. Left Block \u2014 Blockbuild or pair pieces to solve the Left Cell
    4. Right Block \u2014 Blockbuild or pair pieces to solve the Right Cell
    5. OLL-4 - Orient LC 2c, 3c, and 4c pieces using 3D algorithms
    6. PLL-4 - Permute LC 2c, 3c, and 4c pieces using 3D techniques
    "},{"location":"methods/3x3x3x3/3block/#steps","title":"Steps","text":""},{"location":"methods/3x3x3x3/3block/#4-cross","title":"4-Cross","text":"

    Solve the four 2c cross pieces in a ring in the M slice. Leave the L and R cross pieces unsolved.

    "},{"location":"methods/3x3x3x3/3block/#middle-block","title":"Middle Block","text":"

    Tip

    For the entirety of F2L, the last layer is held on the I cell, as opposed to CFOP where it is held on the U cell. This allows us to see more information at once.

    Create and insert 4 F2L-a (2c3c) pairs into the 4/6 cross. This will solve 2/3 of the M slice. Because the Left and Right cells don\u2019t have their cross pieces, you can use them to aid with building and inserting the pairs.

    "},{"location":"methods/3x3x3x3/3block/#left-block","title":"Left Block","text":"

    Solve the Left cell. This is done in 3 blocks, hence the name of the method. The first block consists of the cross edge, followed by two 2c3c F2L-a pairs that are opposite of each other. This solves the middle column of the left cell. The final two blocks consist of a 2c3c pair, and two 3c4c pairs.

    "},{"location":"methods/3x3x3x3/3block/#right-block","title":"Right Block","text":"

    Solve the Right cell. This is also done by breaking it up into the 3 blocks, except now you don\u2019t have an empty opposite cell to aid you in making pairs. You could just solve the cross edge, and then finish the whole solve using CFOP style F2L and Last Cell. You can also do it the 3 blocks way, except having less freedom means that it is slightly trickier to set up the correct cases.

    "},{"location":"methods/3x3x3x3/3block/#last-layer","title":"Last Layer","text":"

    This is done in the exact same way as CFOP.

    "},{"location":"methods/3x3x3x3/3block/#big-cubes","title":"Big cubes","text":"

    Pair up only the pieces you need during the step of 3-block you\u2019re on. For example: after solving 4 centers, pair up 4 cross ridges. For last cell, you can either just orient everything, and then do an RKT 43 solve, or you can pair everything and end up with an RKT 33 solve.

    "},{"location":"methods/3x3x3x3/cfop/","title":"CFOP","text":"

    3D CFOP can be easily implemented on the 4D Rubik\u2019s Cube. This method was also called Sheerin-Zhao Method (Hybrid) V1, named after the people who discovered its higher dimensional equivalence.

    Prerequisites

    • Knowledge of how the 4D puzzle moves
    • Notation
    • RKT
    • 3D CFOP with 2-look OLL & PLL
    "},{"location":"methods/3x3x3x3/cfop/#summary","title":"Summary","text":"
    1. Cross \u2014 Build a cross by solving six 2c pieces on the O or D cell
    2. F2L-a \u2014 Join 12 2c+3c pairs together and insert them into the first two layers
    3. F2L-b \u2014 Join 8 3c+4c pairs together and insert them into the first two layers
    4. OLL-4 \u2014 Orient LC 2c, 3c, and 4c pieces using 3D EOLL and OCLL
    5. PLL-4 \u2014 Permute 2c pieces of the LL using EPLL, then solve the rest like a 33
    "},{"location":"methods/3x3x3x3/cfop/#steps","title":"Steps","text":""},{"location":"methods/3x3x3x3/cfop/#cross","title":"Cross","text":"

    Using the same techniques from 3D, intuitively place the cross 2c pieces such that they lie between their centers, correctly oriented. After this step, the cross should be kept on the D layer.

    "},{"location":"methods/3x3x3x3/cfop/#f2l-a","title":"F2L-a","text":"

    Find a pair of 2C and 3C pieces. Find a way to bring them onto the P slice using easy intuitve setups. Now, you should be able to pair them up using \u201cnormal\u201d looking 3x3x3 moves.

    Repeat this until you have solved all 12 2c3c pairs of F2L-a.

    Misoriented pairs

    It is possible to insert a pair into its slot, but rotated in place. Make sure to line it up so that it looks like a normal 3x3x3 case before inserting the pair. Oftentimes, this just means doing a Ux2 or Uz2 move beforehand.

    "},{"location":"methods/3x3x3x3/cfop/#f2l-b","title":"F2L-b","text":"

    Find any 3C piece that doesn\u2019t have a U cell colour. Find its respective 4C piece.

    • If they both have the same coloured sticker on the U cell, use RKT on the U cell to pair them up.
    • If the 4C is stuck in a slot in the D cell, bring the edge over the slot such that its colour on the I cell matches the 4C\u2019s colour on the I cell. Then use RKT to pair them up.
    • If the 3C is stuck in a slot in the middle layer, bring the 4C on top of it until its colour on the I cell matches the 3C\u2019s colour on the I cell. Then use RKT to pair them up.
    • If none of the above cases occured, then you kind of just have to fiddle around with it or pick a different pair to solve.

    Repeat for all 8 3c4c pairs of F2L-b.

    "},{"location":"methods/3x3x3x3/cfop/#oll-4","title":"OLL-4","text":""},{"location":"methods/3x3x3x3/cfop/#2c-oll-4","title":"2c OLL-4","text":"

    Use EOLL algorithms from 2-look OLL to orient the 2C pieces. This can always be done in 2 EOLL algorithms (or less).

    "},{"location":"methods/3x3x3x3/cfop/#3c-oll-4","title":"3c OLL-4","text":"

    Use RKT on the last cell to set up the slice layers of the last cell into configurations that look like possible OCLL cases. Then use the OCLL algorithms to solve that case. This can always be done in 3 OCLL algorithms (or less)

    3c monotwist

    It\u2019s possible to have just 1 3c piece twisted in place. To avoid this, make sure that your last OCLL algorithm will solve all of the 3c pieces. For example if you have 5 left, you can\u2019t set it up into an H OCLL case, because that will solve 4/5, leaving you with 1 left. Instead, you can set it up into a Sune case, which would then leave you with 2 (which you can solve using a T or U case OCLL algorithm).

    "},{"location":"methods/3x3x3x3/cfop/#4c-oll-4","title":"4c OLL-4","text":"

    Use RKT on the last cell to set up the 4c pieces into possible OCLL cases. Rotate the last cell to U, such that your OCLL case is in the IU plane, then execute that algorithm with RKT on I.

    4c monoflip

    It\u2019s possible to have just 1 4c piece flipped in place. To avoid this, make sure that your last OCLL algorithm will solve all of the 4c pieces. For example if you have 5 left, you can\u2019t set it up into an H OCLL case, because that will solve 4/5, leaving you with 1 left. Instead, you can set it up into a Sune case, which would then leave you with 2 (which you can solve using a T or U case OCLL algorithm).

    "},{"location":"methods/3x3x3x3/cfop/#pll-4","title":"PLL-4","text":""},{"location":"methods/3x3x3x3/cfop/#2c-pll-4","title":"2c PLL-4","text":"

    It is always possible to solve 2c permutation using just U-perms, but many cases have faster methods. There are seven cases for 2c PLL-4:

    • Solved
    • Adjacent swap - Do a 90-degree twist of I to turn this case into a 3-cycle, then use a 3D U-perm algorithm to solve it, such as M2 U M U2 M' U M2 (7 STM) to cycle IL \u2192 IF \u2192 IR.
    • Opposite swap - Do a 90-degree twist of I to turn this case into a pair of adjacent swaps, then use a 3D Z-perm algorithm to solve it, such as M2 U M2 U M' U2 M2 U2 M' (9 STM) to swap IF \u2194 IR and IB \u2194 IL.
    • Adjacent 3-cycle (clockwise) - Use [[RD, ID], {1-2}LO] = (RD ID RU IU) RO (IF RF IB RB) (RI) (9 STM + 1 move RKT debt) to cycle IF \u2192 IU \u2192 IR.
    • Adjacent 3-cycle (counterclockwise) - Use [[RF, IF], {1-2}LI] = (RF IF RB IB) RI (ID RD IU RU) (RO) (9 STM + 1 move RKT debt) to cycle IR \u2192 IU \u2192 IF.
    • Double adjacent swap (right-handed) - Use [[RU, IU] IR2 RU, IDR] = (RU IU RD IFR RU) IDR (RD IFR RU ID RD) (IUR) (11 STM + 1 move offset) to swap IF \u2194 IU and IR \u2194 IB.
    • Double adjacent swap (left-handed) - Use [[RU, IU] IR2 RU, IUR] = (RU IU RD IFR RU) IUR (RD IFR RU ID RD) (IDR) (11 STM + 1 move offset) to swap IF \u2194 ID and IR \u2194 IB.

    This is currently no known reliable way to recognize which case you have.

    "},{"location":"methods/3x3x3x3/cfop/#pll-3","title":"PLL-3","text":"

    From here, you use RKT to solve the rest of the puzzle like a whole 33. The CFOP method is recommended for this because you arrive at this step inspectionless, meaning that in a speedsolve, you don\u2019t really have the time to count Edge Orientation, or plan a Roux First Block. Finding 4 cross pieces is pretty easy inspectionless.

    RKT parity

    If the \u201ctop face\u201d of the LL is 180 degrees off from the rest of the puzzle, you have to use a special 4D algorithm to correct this. See RKT for algorithms.

    RKT parity avoidance

    You can avoid RKT parity by using 2-look PLL. When you get to PLL, correct any RKT debt you have. Then put the solved LL corner in the UIFR spot. Now you can do whichever A-perm you have (clockwise or anticlockwise). Finally, just solve the 3c with EPLL algorithms.

    "},{"location":"methods/3x3x3x3/octachoroux/","title":"Octachoroux","text":"

    Octachoroux

    • Algorithms: 3D Roux + RKT Parity
    • Move count: way too many

    This method is Rowan Fortier\u2019s attempt to bring the Roux method to the 3x3x3x3.

    "},{"location":"methods/3x3x3x3/octachoroux/#summary","title":"Summary","text":"
    1. 1st Block \u2014 Solve a 1x2x3x3 block using blockbuilding techniques.
    2. 2nd Block \u2014 Solve a 1x2x3x3 block on the other side of the puzzle.
    3. CMLC \u2014 Orient and permute the corners of the U cell.
    4. L/R \u2014 Solve the Left and Right cells.
    5. M slice - Permute the M slice.
    "},{"location":"methods/3x3x3x3/octachoroux/#steps","title":"Steps","text":""},{"location":"methods/3x3x3x3/octachoroux/#1st-block","title":"1st Block","text":"

    Start with the \u201ccross piece\u201d of the 1st block. Then build four 2c3c and four 3c4c pairs around it, to complete the 1x2x3x3 block. No RKT is needed for this, as none of the other pieces have been solved yet.

    "},{"location":"methods/3x3x3x3/octachoroux/#2nd-block","title":"2nd Block","text":"

    Use the free M slice to help you build the pairs to make the 2nd 1x2x3x3 block on the opposite side of the puzzle from the 1st block.

    "},{"location":"methods/3x3x3x3/octachoroux/#cmlc","title":"CMLC","text":"

    Use the same strategies from CFOP, but only for the 4C pieces. This step feels like solving the last cell of a 24, except you have to use 34 algorithms so that you don\u2019t mess up other pieces.

    "},{"location":"methods/3x3x3x3/octachoroux/#lr","title":"L/R","text":"

    Insert the UR and UL 2c pieces using setup moves and EPLL algorithms. Now set up the 3c pieces that need to go to L/R into the IDF spot with the L/R colour on the I cell and the U colour on the D cell. Then move the spot where that L/R edge needs to go above that edge and insert that piece using the RKT algorithm of M D2 M\u2019 D2 (2RO\u2019 IF\u2019 RO2 IF 2RO IF\u2019 RO2 IF). Repeat this for all of the edges (pretty tedious).

    "},{"location":"methods/3x3x3x3/octachoroux/#m-slice","title":"M slice","text":"

    Now all that\u2019s left to be solve is the M slice of the puzzle. Notice how the M slice is more like a 33 than the PLC of CFOP. The centers are already permuted too! However, this step is much more painful than normal RKT PLC due to the strange parities that can happen.

    tricky situations

    • A 3c piece can look \u201cmirrored\u201d in place.
    • A single 3c piece can be rotated wrong.
    "},{"location":"no/","title":"No","text":""},{"location":"no/#_1","title":"No","text":"

    NO

    "},{"location":"puzzles/","title":"Puzzles","text":"

    In hypercubing, we generalize the notion of a twisty puzzle beyond 3D space. The best introduction to hypercubing is the standard 3x3x3x3 4D Rubik\u2019s cube.

    See the navigation sidebar for a list of puzzles that have wiki pages.

    "},{"location":"puzzles/2x2x2x2/","title":"2x2x2x2","text":"

    2x2x2x2

    Shape: Tesseract

    Pieces: 16

    "},{"location":"puzzles/2x2x2x2/#history","title":"History","text":"

    The 2x2x2x2 (normally referred to as the 24) is a 4-dimensional twisty puzzle in the shape of a hypercube that is cut in half on each of the 4 axes.

    The first official 24 solve on the Hall of Fame was done by Jay Berkenbilt in 2004 using MC4D, although it was probably solved several times before that.

    "},{"location":"puzzles/2x2x2x2/#permutations","title":"Permutations","text":"

    The 16 4cs of the 2x2x2x2 can be placed in any even permutation. In order to identify positions that are a whole-puzzle rotation apart from each other, we fix one 4c, leaving 15 4cs that can be permuted. Each 4c can be placed in \\(\\frac{4!}{2}\\) orientations, except the last, which can be placed in only \\(4\\) orientations. Thus, the number of permutations of the puzzle is \\(\\frac{15!}{2} \\cdot \\left(\\frac{4!}{2}\\right)^{14} \\cdot 4 \\approx 3.36 \\cdot 10^{27}.\\)

    "},{"location":"puzzles/2x2x2x2/#physical-version","title":"Physical version","text":"

    Since 2013, Melinda Green has been refining her physical 2x2x2x2. See her project home page for more details.

    "},{"location":"puzzles/2x2x2x2/#see-also","title":"See also","text":"
    • Leaderboard section for 2x2x2x2
    "},{"location":"puzzles/3x3x3/","title":"3x3x3","text":"

    There\u2019s not much we can say about the traditional Rubik\u2019s cube that hasn\u2019t already been said on Wikipedia or the Speedsolving.com Wiki.

    "},{"location":"puzzles/3x3x3/#333-with-1d-vision","title":"3x3x3 with 1D vision","text":"

    What if you were a 2D being trying to solve a 3D Rubik\u2019s Cube? This is directly analagous to us 3D beings trying to solve a 4D cube, and MagicCube3D has a setting to view the 3x3x3 with 1D vision. The first known solver was Markk in December 2022.

    "},{"location":"puzzles/3x3x3x3/","title":"3x3x3x3","text":"

    3x3x3x3

    Shape: Tesseract

    Pieces: 8 1c, 24 2c, 32 3c, 16 4c

    The 3x3x3x3 (normally referred to as the 34) is a 4-dimensional twisty puzzle in the shape of a hypercube that has 2 cuts along each of the 4 axes. It is a direct higher dimensional analogy of the 3x3x3 Rubik\u2019s Cube.

    The 34 is recommended as one\u2019s first introduction to solving higher dimensional puzzles.

    "},{"location":"puzzles/3x3x3x3/#pieces","title":"Pieces","text":"

    The 34 has 81 hypercubies, of which 72 are movable. It has 8 1c, 24 2c, 32 3c, and 16 4c pieces. The 3c pieces can be rotated in 6 different ways, and the 4c pieces in 12 different ways.

    "},{"location":"puzzles/3x3x3x3/#turning","title":"Turning","text":"

    Each turn of the 34 is a rotation of one of its cubic cells (which can be oriented in any of 24 orientations of a cube). A normal move disturbs 8 4c, 12 3c, and 4 2c pieces.

    "},{"location":"puzzles/3x3x3x3/#history","title":"History","text":"

    The 34 has always been the main focus of the community. The first solve was done by Don Hatch in ~1988 using MC4D. After that, the Hall of Fame gradually began to increase in size. It eventually closed on December 7th, 2022 due to it reaching 500 solvers.

    "},{"location":"puzzles/3x3x3x3/#permutations","title":"Permutations","text":"

    The 16 4cs of the 3x3x3x3 can be placed in any even permutation. Each 4c can be placed in \\(\\frac{4!}{2}\\) orientations, except the last, which can be placed in only \\(4\\) orientations, due to the existence of monoflip. The 32 3cs can be in any permutation, and can have \\(3!\\) orientations, except the last, which can only have \\(3\\). The 24 2cs can be in any permutation, and can have \\(2!\\) orientations, except the last, which is determined by the others. However, the permutation parity of the 2cs and 3cs are linked. Thus, the number of permutations of the puzzle is \\(\\left[\\frac{16!}{2} \\cdot \\left(\\frac{4!}{2}\\right)^{15} \\cdot 4\\right] \\cdot \\left[32! \\cdot 3!^{31} \\cdot 3\\right] \\cdot \\left[\\frac{24!}{2} \\cdot 2!^{23}\\right] \\approx 1.76 \\cdot 10^{120}.\\)

    "},{"location":"puzzles/3x3x3x3/#speedsolving","title":"Speedsolving","text":"

    (See the leaderboards for the current records)

    There were a few hypercube speedsolving competitions during the early days of hypercubing, with the controversy at the time being if they were to allow macros or not. One of the first competitions happened in 2010, with the fastest time being 15:57 by Mateusz Burnicki (using prepared macros) Some contestants recieved a t-shirt with an image of the 34 in MC4D on it.

    In mid 2017, The speedsolving scene was single handedly revived by Tetrian22, lowering his best from 37:07 to 10:11 in just under 10 months.

    Starting in November 2022, a 34 speedsolving frenzy began due to the popularity of the recently created Hyperspeedcube program, and its keyboard controls. The record bounced between Hactar and Grant as it smashed through all the barriers, finally lowering to sub-2 minutes in May 2023.

    "},{"location":"puzzles/3x3x3x3x3/","title":"3x3x3x3x3","text":"

    3x3x3x3x3

    Shape: Penteract

    Pieces: 10 1c, 40 2c, 80 3c, 80 4c, 32 5c

    The 3x3x3x3x3 or 35 is a 5-dimensional twisty puzzle in the shape of a penteract that has 2 cuts along each of the 5 axes. The 35 is recommended as one\u2019s first introduction to solving 5D puzzles.

    "},{"location":"puzzles/3x3x3x3x3/#pieces","title":"Pieces","text":"

    The 35 has 243 hypercubies, of which 232 are movable. It has 10 1c, 40 2c, 80 3c, 80 4c, and 32 5c pieces. 1cs are called centers, 2cs ridges, 3cs peaks, 4cs edges, and 5cs corners.

    "},{"location":"puzzles/3x3x3x3x3/#turning","title":"Turning","text":"

    Each turn of the 35 is a rotation of one of its hypercube cells. A hypercube can be reoriented in 192 different ways; a lot of ways just to turn a single side! However in most programs, only some of these moves are able to be done with 2 clicks.

    "},{"location":"puzzles/3x3x3x3x3/#programs","title":"Programs","text":"

    There are 3 programs that currently have implementations of the 35: MPU, MC5D, and MC7D. When HSC2 releases, it will be the 35 program of choice with its amazing piece filters and keybind settings. Until then, the 3 programs all come with their different pros and cons.

    "},{"location":"puzzles/3x3x3x3x3/#mpu","title":"MPU","text":"

    35 in MPU

    Magic Puzzle Ultimate objectively has the nicest view of the 35 out of all the programs. The projection is as symmetrical as possible, treating all dimensions equally. It lacks piece filters such that you can only toggle all pieces with a certain number of colours on or off, which makes finding pieces extremely time consuming. To turn the sides you have to click on 3 different 5c pieces.

    "},{"location":"puzzles/3x3x3x3x3/#mc5d","title":"MC5D","text":"

    35 in MC5D

    This program was designed specifically for 5D puzzles. The pros are that it has a feature to locate a certain piece, as well as an extensive macro system. However, the puzzle is projected in an interesting way that uses the 4D inside to outside projection twice. This makes many of the stickers overlap. Additionally it uses Euler angles for rotation, meaning that sometimes you get stuck at the poles when reorienting the puzzle with the mouse. There are some buttons in the bottom-right of the UI to rotate the puzzle to make up for this.

    "},{"location":"puzzles/3x3x3x3x3/#mc7d","title":"MC7D","text":"

    35 in MC7D

    Magic Cube 7D was made by Andrey Astrelin, and has a different way of projecting higher dimensions. The dimensions higher than 4 are represented with stickers that are smaller than the others. Many of them are also duplicates of each other because they are attached to the other stickers of that piece. While not being the nicest to look at, it does have very useable piece filters and nice 2-click controls, making it a prime candidate to use for one\u2019s first 35 solve.

    "},{"location":"puzzles/3x3x3x3x3/#permutations","title":"Permutations","text":"

    The 32 5cs of the 3x3x3x3x3 can be placed in any even permutation, and each 5c can be placed in \\(\\frac{5!}{2}\\) orientations, including the last one. The 80 4cs can be in any even permutation, and can each have \\(4!\\) orientations, except the last which can only have \\(\\frac{4!}{2}\\) orientations. The 80 3cs can be in any permutation, and can each have \\(3!\\) orientations, except the last which can only have \\(\\frac{3!}{2}\\) orientations. Finally, the 40 2cs can be in any permutation with the same parity as that of the 3cs, and each one can have 2 orientations, except the last, whose orientation is determined by the rest of the 2cs.1 Thus, the number of permutations of the puzzle is \\(\\left[\\frac{32!}{2} \\cdot \\left(\\frac{5!}{2}\\right)^{32}\\right] \\cdot \\left[\\frac{80!}{2} \\cdot \\frac{4!^{80}}{2}\\right] \\cdot \\left[80! \\cdot \\frac{3!^{80}}{2}\\right] \\cdot \\left[\\frac{40!}{2} \\cdot 2^{39}\\right] \\approx 7.02 \\cdot 10^{560}.\\)

    "},{"location":"puzzles/3x3x3x3x3/#speedsolving","title":"Speedsolving","text":"

    (See the leaderboards for the current records)

    A full solve of the 35 had not even been recorded until 2024-03-21 with Rowan Fortier\u2019s first solve, uploading all 15 hours of footage to YouTube. Sadly this did not count as the first speedsolve because it was done over multiple days in different sessions. But a few days later, Luna managed to do it in one sitting, achieving a time of 3h 39m, automatically getting the first world record.

    1. http://www.gravitation3d.com/magiccube5d/permutations.html \u21a9

    "},{"location":"puzzles/4d-skewb/","title":"4D Skewb","text":"

    There are many puzzles that could be called a \u20184D skewb\u2019. Each one has some characteristics similar to the skewb. By constructing the skewb in a certain way, and replacing 3D objects of the definition with similar 4D objects, you can construct a 4D puzzle that has some things in common with the skewb.

    "},{"location":"puzzles/4d-skewb/#cube-like","title":"Cube-like","text":""},{"location":"puzzles/4d-skewb/#vertex-turning-hypercube","title":"Vertex turning hypercube","text":"

    Vertex turning hypercube

    The skewb is a vertex-turning cube with the cuts passing through the origin. If you replace the 3D cube in the definition with the 4D hypercube, you get the half-cut vertex-turning hypercube. This puzzle has 32 edge pieces, 64 X-centers, and 8 center pieces. Both the X-centers and the centers are 1c pieces.

    In three dimensions, a plane perpendicular to the vertex axis passes through 1 corner, then 3, then 3, then 1 corner. This causes a plane passing through the origin to go between the two layers of 3 corners, which is why the skewb has corners. In four dimensions, a hyperplane perpendicular to the vertex axis passes through 1, 4, 6, 4, then 1 corner. Therefore, a hyperplane passing through the origin passes directly through 6 vertices, and there are no corner pieces on this 4D puzzle. Similarly, the pattern of cuts on one cell of the puzzle looks like a dino cube, not a skewb.

    "},{"location":"puzzles/4d-skewb/#skewb-lookalike","title":"Skewb lookalike","text":"

    Skewb lookalike

    You can place a cut to pass between the layers of 4 and 6 corners. If you do this, you get two cuts per axis and three layers per axis. This puzzle has 16 vertex pieces, 64 X-centers, and 24 ridge pieces. Like the skewb, the vertex pieces are separated into two orbits, but this puzzle has a middle layer on each axis. The cut pattern on the cells of this puzzle resembles a 3D skewb with additional trivial tips.

    By rectifying the hypercube, the trivial tip cuts on the cut patterns of the cells are removed, meaning the cells look like cuboctahedral skewbs.

    "},{"location":"puzzles/4d-skewb/#demi-vertex-turning-hypercube","title":"Demi vertex turning hypercube","text":"

    Demi vertex turning hypercube

    On the cube, the vertices can be separated into two tetrahedral subsets (the orbits of the skewb\u2019s corners). Each cut corresponds to one corner in the tetrahedron. On the hypercube, a similar partitioning of the vertices results in two sets of 8, each at the vertices of the 16-cell, or dually, the facets of a differently-oriented hypercube. If you only cut along vertex axes corresponding to these cuts, you will get a puzzle with only 4 cuts, half as many as the vertex turning hypercube. This puzzle has 8 corners and 8 centers. Unlike on the skewb, only half of the vertices of the hypercube have a corner piece, and all the corner pieces are in one orbit.

    "},{"location":"puzzles/4d-skewb/#demi-skewb-lookalike","title":"Demi skewb lookalike","text":"

    Demi skewb lookalike

    If you only use the axes along half the vertices as above, but you use the cut depths of the skewb lookalike, it results in a puzzle that still has three layers per axis, but only half as many axes. The cut pattern on the cells of this puzzle also resembles a 3D skewb, but with only 4 trivial tips per cell. The puzzle has 8 corner pieces, 8 dual corner pieces of a different shape, 32 X-centers, and 24 ridge pieces.

    As before, the hypercube can be rectified, which removes the trivial tips from the cut patterns of the cells.

    Extra turns

    In the two demi puzzles, the 8 vertices are arranged like the cells of a hypercube. This means they are actually shapemods of hypercubic puzzles. As such, they have an additional shapeshifting move in which you turn one layer by 90\u00b0. There is no analogous move on the skewb. In particular, the demi vertex turning hypercube is a shapemod of a 2x2x2x2, and the demi skewb lookalike is a shapemod of a 3x3x3x3, but with the 8 corner pieces that would be at the center of the cells missing.

    Because the 16-cell axis system, or the vertex turning hypercube system, is composed of two disjoint hypercube axis systems, these puzzles also have additional shapeshifting moves in which you turn a layer 90\u00b0. Unlike those in the previous section, though, performing one of these turns would bandage the puzzle. If you unbandaged these puzzles, you would get additional cuts parallel to the cells, resulting in the axis system of a 24-cell.

    "},{"location":"puzzles/4d-skewb/#simplex-like","title":"Simplex-like","text":"

    The skewb can be constructed in a different way: take four axes corresponding to a tetrahedron, and cut perpendicular to each axis through the origin. The tetrahedron is the 3D simplex, so if you replace it with the 5-cell, the 4D simplex, you get a new puzzle in 4D. This puzzle has several types of pieces, but since the shape of the puzzle does not yet exist, we can\u2019t name them after their position. Thus, we will name the pieces by how many layers they are turned by. For the skewb, there are 4 pieces that are in one layer, 6 pieces that are in two layers, and 4 pieces that are in three layers. We will call these 1g, 2g, and 3g pieces. This 4D puzzle will have 5 1g pieces, 10 2g pieces, 10 3g pieces, and 5 4g pieces. The skewb has a symmetry (90\u00b0 rotation of the puzzle) that swaps 1g and 3g pieces and sends 2g to 2g pieces, and like it, this 4D puzzle has a symmetry that swaps 1g and 4g pieces and swaps 2g and 3g pieces. Like the skewb, the cuts on this puzzle can each be offset in a consistent direction while preserving the functionality of the puzzle.

    Constructing the skewb like this, we can describe how to construct its cubic shape. On each face of the cube, there is one 2g piece in its center. Thus, given our abstractly constructed skewb, we can construct a plane perpendicular to each 2g piece\u2019s axis of symmetry. The shape bounded by all 6 of these planes is a cube.

    "},{"location":"puzzles/4d-skewb/#2g-carved-simplex","title":"2g carved simplex","text":"

    2g carved simplex

    Like in 3D, you can put a hyperplane perpendicular to the symmetry axis of each 2g piece. Since there are 10 2g pieces, this creates a polychoron with 10 triangular-bipyramidal cells called the joined 5-cell. On this puzzle, the 1g pieces are 4c at the tetrahedral vertices, the 2g pieces are 1c at the cell centers, the 3g pieces are 3c at the triangular edges, and the 4g pieces are 6c at the triangular-bipyramidal vertices. Unlike on the skewb, where the 1g and 3g pieces look alike, here, the 1g and 4g pieces are distinguishable by their number of colors, and so are the 2g and 3g pieces.

    "},{"location":"puzzles/4d-skewb/#2g-3g-carved-simplex","title":"2g-3g carved simplex","text":"

    2g-3g carved simplex

    You can also put hyperplanes perpendicular to the symmetry axis of both the 2g and 3g pieces. This gives the puzzle a 20-celled shape called the bijungato-10-cell. On this puzzle, the 1g and 4g pieces are both 4c at tetrahedral vertices, and the 2g and 3g pieces are both 1c at the cell centers. Unlike the previous shape but like the skewb, full symmetry of the puzzle is reflected in the shape.

    Each cell of the puzzle either has two 1g and one 2g pieces, or two 4g and one 3g pieces. These two orbits of cells are only adjacent across the 2g-3g boundary. Because of this, this puzzle has multiple solved states, where each cell is a solid color, but the color schemes of the two orbits are rotated relative to each other.

    "},{"location":"puzzles/4d-skewb/#2g-3g-carved-simplex-hemi-colors","title":"2g-3g carved simplex, hemi colors","text":"

    The puzzle can be recolored with 10 colors by giving opposite cells the same color. By asserting that the solved state has to have the two same-colored cells opposite to each other, it restricts the relative orientations of the two orbits to only one configuration, leading to there being only one solved state. Because of the geometry of the puzzle, a 1g-2g cell is opposite a 4g-3g cell, so the pieces on these cells are not swappable and coloring these two cells the same color leads to no ambiguity in the position.

    "},{"location":"puzzles/4d-skewb/#2g-3g-ridge-carved-simplex","title":"2g-3g ridge carved simplex","text":"

    2g-3g ridge carved simplex

    There are 30 square ridges between the 2g and 3g pieces. By constructing a shape bounded by the hyperplanes perpendicular to the symmetry axes of these ridges, you get a 30-cell shape with disphenoidal cells called the bi-10-cell. The 1g and 4g pieces are 12c at triakis-tetrahedral vertices, and the 2g and 3g pieces are 3c at triangular ridges. This shape of the puzzle also reflects the symmetry that swaps 1g and 4g and swaps 2g and 3g.

    "},{"location":"puzzles/4d-skewb/#prism","title":"Prism","text":"

    Prism

    By starting with a skewb, you can extend it into the fourth dimension by taking the cartesian product of it with a line segment. This extends the cube to a cubic prism, and the skewb cut planes to hyperplanes all perpendicular to the original cube. This creates a skewb prism, which has exactly the same pieces as the skewb but with shapes that are the prisms of the originals. It allows all the moves of the skewb, and an additional set of moves that involve flipping the new dimension, which appear as reflection moves on the original skewb. By adding more cuts parallel to the original skewb, you can create multi-layer skewb prisms.

    "},{"location":"puzzles/4d-skewb/#24-cell-like","title":"24-cell-like","text":""},{"location":"puzzles/4d-skewb/#24-cell-skewb-diamond-lookalike","title":"24-cell skewb diamond lookalike","text":"

    24-cell skewb diamond lookalike

    The 24-cell\u2019s cells are octahedra, which have the same symmetry as the skewb, and its dual the skewb diamond. Construct cutting planes parallel to the cells which pass through the centers of the adjacent cells. This creates a puzzle with 24 vertex pieces, 96 ridge pieces, and 144 X-centers. This puzzle is not half-cut, but each cell\u2019s cut pattern looks like a skewb diamond with trivial tips. It is possible to apply RKT to a cell of this puzzle, in which case it acts like a skewb diamond.

    By rectifying the 24-cell, the trivial tip cuts on the cut patterns of the cells are removed, meaning the cells look like cuboctahedral skewbs.

    "},{"location":"puzzles/4d-skewb/#24-cell-skewb-diamond-lookalike-16-cell-cuts","title":"24-cell skewb diamond lookalike, 16-cell cuts","text":"

    24-cell skewb diamond lookalike, 16-cell cuts

    The 24-cell has a subset of cells that correspond to the cells of the 16-cell. If you only keep those cuts, you get a puzzle similar to the one above, but where 8 of the cells no longer appear to have trivial tips in their cut patterns. This puzzle has 16 large corner pieces, 8 small corner pieces, 96 ridge pieces, and 144 X-centers. Like before, RKT can be applied.

    "},{"location":"puzzles/4d-skewb/#24-cell-skewb-diamond-lookalike-8-cell-cuts","title":"24-cell skewb diamond lookalike, 8-cell cuts","text":"

    24-cell skewb diamond lookalike, 8-cell cuts

    The 24-cell also has a subset of cells that correspond to the cells of the 8-cell, or hypercube. If you only keep those cuts, you get another puzzle similar to the one above, but where 16 of the cells do not appear to have trivial tips in their cut patterns, and 8 of the cells appear to have only the trivial tip cuts. This puzzle has 8 large corner pieces, 16 small corner pieces, 32 ridge pieces, and 8 center pieces. Unlike before, RKT cannot be applied to the skewb diamond cells. This puzzle is a shapemod of the 3x3x3x3 without corner pieces.

    "},{"location":"puzzles/hemimegaminx/","title":"Hemimegaminx","text":"

    Hemimegaminx

    Shape: Hemi-dodecahedron

    Pieces: 6 1c, 15 2c, 10 3c

    The hemimegaminx is a twisty puzzle in the shape of a hemi-dodecahedron. In MagicTile, it inhabits the projective plane.

    The puzzle can be constructed by identifying opposite faces on a megaminx. On this realization of the puzzle, opposite faces will turn in opposite directions. In order to be a true hemimegaminx, opposite faces should also be colored the same, so the puzzle will have 6 colors total.

    "},{"location":"puzzles/hemimegaminx/#permutations","title":"Permutations","text":"

    The 10 3cs of the hemimegaminx can be placed in any even permutation. Each 4c can be placed in \\(3!\\) orientations, except the last, which can be placed in only \\(3\\) orientations. The 15 2cs can be placed in any even permutation, and each one has \\(2\\) orientations, except the last, whose orientation is determined by the other pieces. Thus, the number of permutations of the puzzle is \\(\\left[\\frac{10!}{2} \\cdot 3!^{9} \\cdot 3\\right] \\cdot \\left[\\frac{15!}{2} \\cdot 2^{14}\\right] \\approx 5.87 \\cdot 10^{29}.\\)

    "},{"location":"puzzles/hemimegaminx/#physical-version","title":"Physical version","text":"

    In 2019, after several prototypes, Oskar van Deventer built the first working hemimegaminx. In 2024, Akkei created another physical hemimegaminx and did the first known timed solve on it.

    "},{"location":"puzzles/hypercuboids/","title":"Hypercuboids","text":""},{"location":"puzzles/hypercuboids/#introduction","title":"Introduction","text":"

    A hypercuboid is the multi-dimensional version of a cuboid. In a general context, we define a hypercuboid as an \\(n\\)-dimensional puzzle denoted by \\(a_1 \\times a_2 \\times \\dots \\times a_n\\). For the sake of clarity and consistency, we will use \\(a_1,a_2, \\dots a_n\\) as non-decreasing values.

    "},{"location":"puzzles/hypercuboids/#structure","title":"Structure","text":"

    A hypercuboid, as defined, is composed of \\(2n\\) cells, each of which is \\((n-1)\\)-dimensional.

    Given \\(k \\geq 0\\) and \\(n \\geq 1\\), the elementary symmetric polynomial \\(e_k(x_1, x_2, \\dots, x_n) = \\sum_{Y} \\prod_{y\\in Y} y\\), where \\(Y\\) ranges over subsets of \\(\\{x_1, \\dots, x_k\\}\\) where \\(|Y| = k\\). In other words, it is the sum of all terms, each of which are product of distinct \\(x_i\\) taken \\(k\\) at a time.

    • For example: \\(e_1(x_1,x_2,x_3,x_4,x_5) = x_1 + x_2 + x_3 + x_4 +x_5\\), i.e. the sum of terms of 1 element, chosen in \\(x_1, x_2, x_3, x_4, x_5\\).
    • Another example: \\(e_2(x_1,x_2,x_3,x_4) = x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4\\), i.e., the sum of the products of all possible unordered pairs made with \\(x_1, x_2, x_3, x_4\\).

    Note that \\(e_k(x_1, \\dots, x_n)\\) has \\(\\binom{n}{k}\\) terms. From this, we can also see that \\(e_0(x_1, \\dots, x_n)=1\\).

    Using the above notation, the \\(a_1 \\times a_2 \\times \\dots \\times a_n\\) hypercuboid, where \\(a_i > 1\\), has \\(2^k\\cdot e_{n-k}(a_1-2, \\dots, a_n-2)\\) pieces which are \\(k\\)-colored.

    For example, consider the \\(2 \\times 3 \\times 5 \\times 7\\) hypercuboid.

    • For 1-colored pieces we have: \\(2^1 \\cdot e_3(0,1,3,5)=2^1\\cdot (0\\cdot1\\cdot3 + 0\\cdot 3 \\cdot 5 + 1 \\cdot 3 \\cdot 5 + 0\\cdot 1 \\cdot5 )=\\) \\(=2 \\cdot (0+0+15+0)=30\\) pieces.

    • For 2-colored pieces we have: \\(2^2 \\cdot e_2(0,1,3,5)=2^2\\cdot ( 0\\cdot 1+ 0\\cdot 3 +0 \\cdot 5 + 1\\cdot3 +1\\cdot 5 + 3\\cdot 5 )=\\) \\(=4 \\cdot (0+0+0+3+5+15)=92\\) pieces.

    • For 3-colored pieces we have: \\(2^3 \\cdot e_1(0,1,3,5)=2^3\\cdot (0+1+3+5 )=\\) \\(=8 \\cdot 9=72\\) pieces.

    • For 4-colored pieces we have: \\(2^4 \\cdot e_0(0,1,3,5)=2^4\\cdot 1= 16\\) pieces.

    If \\(a_1, \\dots, a_m\\) are all equal to \\(1\\) and \\(a_{m+1} > 1\\), the cuboid is a floppy cuboid. In this case, the number of \\(k\\)-colored pieces is equal to the number of \\((k-2m)\\)-colored pieces on the \\(a_{m+1} \\times \\dots \\times a_n\\) hypercuboid, when \\(k \\geq 2m\\). This includes the case where \\(k = 2m\\), in which case the floppy cuboid has pieces corresponding to the 0-colored pieces of the lower dimensional hypercuboid.

    "},{"location":"puzzles/hypercuboids/#4d-hypercuboids","title":"4D Hypercuboids","text":"

    In 4 dimensions, a hypercuboid is denoted as \\(a \\times b \\times c \\times d\\). \\(a \\times b \\times c \\times d\\) is composed of 8 cells: 2 \\((a \\times b \\times c)\\)-cells, 2 \\((a \\times b \\times d)\\)-cells, 2 \\((b \\times c \\times d)\\)-cells and 2 \\((a \\times c \\times d)\\)-cells. In the following sections, we will denote some of these cells using the classic 3-dimensional puzzle names, in particular:

    • \u201ctower cell\u201d will indicate a \\(2 \\times 2 \\times 3\\)-cell;
    • \u201cdomino cell\u201d will indicate a \\(2 \\times 3 \\times 3\\)-cell;
    • \u201d \\(n\\) -cubic cell\u201d will indicate a \\(n \\times n \\times n\\)-cell.
    "},{"location":"puzzles/hypercuboids/#general-solving-strategies","title":"General solving strategies","text":"
    • Hypercuboids in the form \\(1 \\times a \\times b \\times c\\) can be solved by first orienting the \\(a \\times b \\times c\\)-cells, then solving the puzzle like a 3-dimensional \\(a \\times b \\times c\\).
    • Hypercuboids in the form \\(2 \\times a \\times b \\times c\\) can be solved first by solving the \\(a \\times b \\times c\\)-cells and then solving the opposite, eventually adapting the solution for new possible cases.
    • If 2 dimensions have the same values, the puzzle can be seen as a duoprism.
    • If 3 dimensions have the same values, i.e. there is a couple of \\(n\\)-cubic cells, RKT can be used on these cells.
    • If 4 dimensions have the same values, we have a hypercube.
    "},{"location":"puzzles/hypercuboids/#some-notable-4d-hypercuboids","title":"Some notable 4D hypercuboids","text":"

    In some cases an idea of a possible solution method provided by Ema will be present but not spoiled.

    "},{"location":"puzzles/hypercuboids/#1333","title":"1x3x3x3","text":"Puzzle 4c pieces 3c pieces 2c pieces 1c pieces 1x1x3x3 16 24 12 2 Solve idea (click to reveal)
    • Orient both cubic cells.
    • Solve 3^3 cube, paying attention to corner orientation.
    "},{"location":"puzzles/hypercuboids/#2223","title":"2x2x2x3","text":"Puzzle 4c pieces 3c pieces 2c pieces 1c pieces 2x2x2x3 16 8 0 0 Solve idea (click to reveal)
    • Solve the middle 3-colored pieces of a tower cell (similar to solving a \\(1 \\times 2 \\times 2 \\times 2\\) ).
    • Orient both \\(2\\)-cubic cells at the same time, slicing the solved part for exchanging pieces,being careful to use an even number of slice moves.
    • Use RKT to solve the cubic cells, using the same tower cells as R.
    • Fix tower cell middle layer.
    "},{"location":"puzzles/hypercuboids/#2233","title":"2x2x3x3","text":"Puzzle 4c pieces 3c pieces 2c pieces 1c pieces 2x2x3x3 16 16 4 0 Solve idea (click to reveal)
    • Solve a domino cell.
    • Orient the opposite domino cell, potentially re-solving the first cell.
    • Move pieces on the correct layers of the last cell.
    • Solve last domino cell using 3-dimensional cuboid algorithms an even number of times and conjugating between them.
    "},{"location":"puzzles/hypercuboids/#2333","title":"2x3x3x3","text":"Puzzle 4c pieces 3c pieces 2c pieces 1c pieces 2x3x3x3 16 24 12 2 Solve idea (click to reveal)
    • Orient both 3-cubic cells at the same time.
    • Solve first cubic cell.
    • Solve the second cubic cell using RKT.
    "},{"location":"puzzles/hypercuboids/#2224","title":"2x2x2x4","text":"Puzzle 4c pieces 3c pieces 2c pieces 1c pieces 2x2x2x4 16 16 0 0"},{"location":"puzzles/hypercuboids/#2345","title":"2x3x4x5","text":"Puzzle 4c pieces 3c pieces 2c pieces 1c pieces 2x3x4x5 16 48 44 12

    The smallest 4-dimensional \u201cbrick\u201d hypercuboid.

    "},{"location":"puzzles/hypercuboids/#4d-hypercuboids-in-mpult","title":"4D hypercuboids in MPUlt","text":"

    Here is a way to create your own 4D hypercuboid in MPUlt. The result would not be isometric, but still working.

    Step 1: Recognize the form of your hypercuboid in one of the following

    • \\(a \\times b \\times c \\times d\\),
    • \\(a \\times a \\times b \\times c\\),
    • \\(a \\times a \\times b \\times b\\),
    • \\(a \\times b \\times b \\times b\\),
    • \\(a \\times a \\times a \\times a\\).

    Step 2: Recognize the values of the letters, then substitute the letter with the corresponding string from the following table:

    Value String 2 0.0 3 0.333 -0.333 4 0.5 0.0 -0.5 5 0.6 0.2 -0.2 -0.6 6 0.667 0.333 0.0 -0.333 -0.667 7 0.714 0.429 0.143 -0.143 -0.429 -0.714 8 0.75 0.5 0.25 0.0 -0.25 -0.5 -0.75 9 0.778 0.556 0.333 0.111 -0.111 -0.333 -0.556 -0.778

    So if \\(a=3\\), you need to change \u201cCUT-A\u201d with \u201c0.333 -0.333\u201d in the general puzzle code, and so on.

    Step 3: Insert the created code in \u201cMPUlt_puzzles.txt\u201d file, save and enjoy your puzzle.

    "},{"location":"puzzles/hypercuboids/#case-abcd","title":"Case axbxcxd","text":"

    General code: 

    Puzzle NAME_AXBXCXD\nDim 4\nNAxis 4\nFaces 1,0,0,0 0,1,0,0 0,0,1,0 0,0,0,1\nGroup 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1\nCuts CUT-A\nAxis 0,1,0,0\nTwists 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,0,1,0/0,0,0,1\nCuts CUT-B\nAxis 0,0,1,0\nTwists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0\nCuts CUT-C\nAxis 0,0,0,1\nTwists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0\nCuts CUT-D\n

    "},{"location":"puzzles/hypercuboids/#case-aabc","title":"Case axaxbxc","text":"

    General code: 

    Puzzle NAME_AXAXBXC\nDim 4\nNAxis 3\nFaces 1,0,0,0 0,0,1,0 0,0,0,1\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1\nCuts CUT-C\nAxis 0,0,1,0\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0\nCuts CUT-B\nAxis 0,0,0,1\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0\nCuts CUT-A\n

    "},{"location":"puzzles/hypercuboids/#case-aabb","title":"Case axaxbxb","text":"

    General code: 

    Puzzle NAME_AXAXBXB\nDim 4\nNAxis 2\nFaces 1,0,0,0 0,0,1,0\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1\nAxis 1,0,0,0\nTwists 0,0,1,0/0,0,1,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0,1,1\nCuts CUT-A\nAxis 0,0,1,0\nTwists 1,0,0,0/1,1,0,0 0,0,0,1/1,0,0,0 0,0,0,1/1,1,0,0\nCuts CUT-B\n

    "},{"location":"puzzles/hypercuboids/#case-abbb","title":"Case axbxbxb","text":"

    General code: 

    Puzzle NAME_AXBXBXB\nDim 4\nNAxis 2\nFaces 1,0,0,0 0,0,0,1\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1\nCuts CUT-B\nAxis 0,0,0,1\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0\nCuts CUT-A\n

    "},{"location":"puzzles/hypercuboids/#case-aaaa","title":"Case axaxaxa","text":"

    General code: 

    Puzzle NAME_AXAXAXA\nDim 4\nNAxis 1\nFaces 1,0,0,0\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,-1,0/0,0,0,1 0,2,-1,-1/0,1,1,-2\nCuts CUT-A\n

    "},{"location":"puzzles/hypercuboids/#5d-hypercuboids","title":"5D+ Hypercuboids","text":"

    These hypercuboids haven\u2019t been studied yet, except for some \u201csimpler\u201d versions with lots of \\(1\\)\u2019s.

    "},{"location":"puzzles/nxnxnxn/","title":"NxNxNxN","text":"

    4x4x4x4

    Shape: Tesseract

    NxNxNxN, or N4 is a generic term for a 4-dimensional twisty puzzle in the shape of a hypercube with N layers per axis. It is a direct higher dimensional analogy of the NxNxN Rubik\u2019s Cube.

    This page is concerned with the case where N is greater than 3. The 2x2x2x2 and 3x3x3x3 have their own pages.

    "},{"location":"puzzles/nxnxnxn/#pieces","title":"Pieces","text":"

    The N4 has \\(N^4 - (N-2)^4\\) hypercubies. If N is even, all hypercubies are movable, and if N is odd, all but 8 are movable. It has \\(8(N-2)^3\\) 1c, \\(24(N-2)^2\\) 2c, \\(32(N-2)\\) 3c, and \\(16\\) 4c pieces. These pieces come in many subtypes.

    • 1c
      • Centers: These pieces are at the centers of the facets. When \\(N \\geq 3\\) is odd, 8 of these pieces exist and they are immovable. They are not present when \\(N\\) is even.
      • T-centers: These pieces exist in orbits of 48 between the facet centers and the ridge centers. When \\(N \\geq 5\\) is odd, there are \\(\\frac{N-3}{2}\\) orbits. They are not present when \\(N\\) is even.
      • Y-centers: These pieces exist in orbits of 96 between the facet centers and the edge centers. When \\(N \\geq 5\\) is odd, there are \\(\\frac{N-3}{2}\\) orbits. They are not present when \\(N\\) is even.
      • X-centers: These pieces exist in orbits of 64 between the facet centers and the corners. When \\(N \\geq 5\\) is odd, there are \\(\\frac{N-3}{2}\\) orbits. When \\(N \\geq 4\\) is even, there are \\(\\frac{N-2}{2}\\) orbits.
      • Semi-oblique centers: These pieces exist in orbits of 192. There are several subtypes, each of which have \\(\\frac{(N-3)(N-5)}{4}\\) orbits when \\(N \\geq 7\\) is odd, and \\(\\frac{(N-2)(N-4)}{4}\\) when \\(N \\geq 6\\) is even.
        • TY-centers: These pieces are between the facet centers, ridge centers, and edge centers.
        • TX-centers: These pieces are between the facet centers, ridge centers, and corners.
        • YX-centers: These pieces are between the facet centers, edge centers, and corners.
      • Oblique centers: These pieces exist in orbits of 192 off all hyperplanes of symmetry. They come in two chiralities. When \\(N \\geq 9\\) is odd, there are \\(\\frac{(N-3)(N-5)(N-7)}{8}\\) orbits of each chirality. When \\(N \\geq 8\\) is even, there are \\(\\frac{(N-2)(N-4)(N-6)}{8}\\) orbits of each chirality.
    • 2c
      • Middle ridges: These pieces are at the centers of the ridges. When \\(N \\geq 3\\) is odd, they come in one orbit of 24. They are not present when \\(N\\) is even.
      • T-ridges: These pieces exist in orbits of 96 between the ridge centers and the edge centers. When \\(N \\geq 5\\) is odd, there are \\(\\frac{N-3}{2}\\) orbits. They are not present when \\(N\\) is even.
      • X-ridges: These pieces exist in orbits of 96 between the ridge centers and the corners. When \\(N \\geq 5\\) is odd, there are \\(\\frac{N-3}{2}\\) orbits. When \\(N \\geq 4\\) is even, there are \\(\\frac{N-2}{2}\\) orbits.
      • Oblique ridges: These pieces exist in orbits of 192, but they are not chiral. When \\(N \\geq 7\\) is odd, there are \\(\\frac{(N-3)(N-5)}{4}\\) orbits of each chirality. When \\(N \\geq 6\\) is even, there are \\(\\frac{(N-2)(N-4)}{4}\\) orbits of each chirality.
    • 3c
      • Middle edges: These pieces are at the centers of the edges. When \\(N \\geq 3\\) is odd, they come in one orbit of 32. They are not present when \\(N\\) is even.
      • Wings: These pieces exist in orbits of 64 between the edge centers and the corners. When \\(N \\geq 5\\) is odd, there are \\(\\frac{N-3}{2}\\) orbits. When \\(N \\geq 4\\) is even, there are \\(\\frac{N-3}{2}\\) orbits of each chirality.
    • 4c
      • Corners: These pieces are at the corners. When \\(N \\geq 2\\), they come in one orbit of 16.
    "},{"location":"puzzles/nxnxnxn/#turning","title":"Turning","text":"

    Each turn of the N4 is a rotation of one of its cubic cells, or a slice layer between two parallel cuts. Each layer turned can be oriented in any of 24 orientations of a cube.

    "},{"location":"puzzles/physical/","title":"Physical Puzzles","text":"

    Warning

    It is highly recommended to learn how virtual puzzles work first before trying to understand the physical puzzles. See Why not start with the physical 2x2x2x2?.

    Physical 3x3x3x3

    Physical puzzles refer to higher dimensional puzzles that are physically built in real life (using only 3 dimensions). This involves clever designs, strange symmetry-abusing tricks, and lots and lots of magnets. Often, these puzzles are too impractical to use, defeating their sole purpose. For an explanation of how these puzzles work, see the theory page.

    For a documentary about the history of physical puzzles as of December 2022, see Rowan\u2019s video.

    "},{"location":"puzzles/physical/#physical-puzzles-that-have-been-built","title":"Physical puzzles that have been built","text":"
    • Melinda\u2019s 2x2x2x2
    • Grant\u2019s 2x2x2x3
    • Grant\u2019s 2x2x3x3
    • Grant\u2019s 2x3x3x3
    • Grant\u2019s 1x1x1x1
    • Grant\u2019s 3x3x3x3
    • Dominik\u2019s 4D Pyraminx
    • Tymon\u2019s 1x2x2x2
    • Tymon\u2019s 1x2x2x3
    • Tymon\u2019s 1x2x3x3
    • Tymon\u2019s 1x3x3x3
    "},{"location":"puzzles/physical/#designs-that-havent-been-built-yet","title":"Designs that haven\u2019t been built yet","text":"
    • AxBxCxD (any hypercube or hypercuboid with layers bigger than 1)
    • 1x1xAxB series
    • Markk\u2019s duoprisms
    • {5}x{4} duoprism
    • Akkei\u2019s 2x2x2x2x2
    "},{"location":"puzzles/physical/1x2x2x2/","title":"Tymon\u2019s 1x2x2x2","text":"

    Tymon\u2019s 1x2x2x2

    4D Shape: Hypercuboid

    Physical Shape: Cube

    Pieces: 8 5c

    Magnets: 384

    Completed: 2024 Jan 11

    "},{"location":"puzzles/physical/1x2x2x2/#history","title":"History","text":"

    Tymon F began some initial tests for this puzzle in October 2023. The first 2 pieces were fully assembled and magnetized by December 2023. Finally (conincidentally exactly 1 year after he discovered hypercubing), Tymon completed the 1x2x2x2 hypercuboid on 2024-01-11. After that, he started work on creating the cuboids up to 1x3x3x3.

    "},{"location":"puzzles/physical/1x2x2x2/#moves","title":"Moves","text":"

    Legal moves of a layer include basic 2x2x2 90\u00b0 twists, 180\u00b0 twists in any plane which also rotate pieces in 4d axis, or a combination of both.

    "},{"location":"puzzles/physical/1x2x2x2/#solving","title":"Solving","text":"

    Beside basic stickers (white, yellow, orange, red, green, blue) every piece on the puzzle has pink and purple stickers which indicate in which 4d orientation the piece is. In the solved state every piece must have the same 4d axis orientation. The 1x2x2x2 is not much more difficult than the 2x2x2. It can be solved by: orienting 5c pieces in 4d axis, except \u201cOLL parity\u201d (because of the extra axis, a single 5c can be missoriented) the rest of the puzzle can be solved like a 3d 2x2x2. Tymon made a YouTube video showing an example solve of the puzzle.

    "},{"location":"puzzles/physical/1x2x2x3/","title":"Tymon\u2019s 1x2x2x3","text":"

    Tymon\u2019s 1x2x2x3

    4D Shape: Hypercuboid

    Physical Shape: Cuboid

    Pieces: 8 5c, 4 4c

    Magnets: 512

    Completed: 2024 Jan 15

    "},{"location":"puzzles/physical/1x2x2x3/#history","title":"History","text":"

    A few days after completing the physical 1x2x2x2 Tymon F finished assembling four new 4c pieces that combined with the previous 5c pieces to create a physical analog of the 1x2x2x3 puzzle. This one was another important milestone on the way to create a physical 1x3x3x3 puzzle.

    "},{"location":"puzzles/physical/1x2x2x3/#moves","title":"Moves","text":"

    Legal moves of a layer include basic 2x2x2 90\u00b0 and 180\u00b0 twists that rotate pieces without changing their 4d orientation, 4d 180\u00b0 twists that besides moving pieces in 3d, rotate them in 4d axis, or a combination of both.

    "},{"location":"puzzles/physical/1x2x2x3/#solving","title":"Solving","text":"

    Besides basic stickers (white, yellow, orange, red, green, blue) every piece on the puzzle has pink and purple stickers which indicate in which 4d orientation the piece is. In the solved state every piece must have the same 4d axis orientation. The 1x2x2x3 is a bit more complicated than 2x2x3. It can be solved by: orienting 4c piece in 4d axis, orienting 5c pieces in 4d axis, solving the rest of the puzzle like a 23. Tymon made a YouTube video showing an example solve of the puzzle.

    "},{"location":"puzzles/physical/1x2x3x3/","title":"Tymon\u2019s 1x2x3x3","text":"

    Tymon\u2019s 1x2x3x3

    4D Shape: Hypercuboid

    Physical Shape: Cuboid

    Pieces: 8 5c, 8 4c, 2 3c

    Magnets: 688

    Completed: 2024 Jan 24

    "},{"location":"puzzles/physical/1x2x3x3/#history","title":"History","text":"

    Over a week after completing the physical 1x2x3x3, Tymon F finished assembling another four 4c pieces and two new 3c pieces that combined with the 1x2x3x3 to create the physical 1x2x3x3 puzzle. This was the last puzzle made with the goal of creating a physical 1x3x3x3.

    "},{"location":"puzzles/physical/1x2x3x3/#moves","title":"Moves","text":"

    Legal moves of a layer include normal 2x3x3 90\u00b0 and 180\u00b0 twists that rotate pieces without changing their 4d orientation, 4d 180\u00b0 twists that change the pink/purple stickers, or a combination of both.

    "},{"location":"puzzles/physical/1x2x3x3/#solving","title":"Solving","text":"

    Besides basic stickers (white, yellow, orange, red, green, blue) every piece on the puzzle has pink and purple stickers which indicate in which 4d orientation the piece is. In the solved state every piece must have the same 4d axis orientation. The 1x2x3x3 puzzle is much more complicated than smaller hypercuboids, but it can be solved by: orienting 3c pieces in 4d axis, orienting 4c piece in 4d axis, orienting 5c pieces in 4d axis, solving the rest of the puzzle like 3d 2x3x3. Tymon made a YouTube video showing an example solve of the puzzle.

    "},{"location":"puzzles/physical/1x3x3x3/","title":"Tymon\u2019s 1x3x3x3","text":"

    Tymon\u2019s 1x3x3x3

    4D Shape: Hypercuboid

    Physical Shape: Cube

    Pieces: 8 5c, 12 4c, 6 3c, 1 2c (2 1c)

    Magnets: 936

    Completed: 2024 Jan 26

    "},{"location":"puzzles/physical/1x3x3x3/#history","title":"History","text":"

    Two days after finishing the 1x2x3x3 Tymon F completed his final goal, creating a physical analog of the 1x3x3x3 puzzle. To do this he had to assemble another four 4c pieces, four 3c pieces and one 2c piece that represents two 1c pieces on the virtual puzzle. This ended Tymon\u2019s journey of creating all physical puzzles from the AxBxCx1 family.

    "},{"location":"puzzles/physical/1x3x3x3/#moves","title":"Moves","text":"

    Legal moves of a layer include 90\u00b0 twists, 180\u00b0 twists in any plane, or a combination of both.

    "},{"location":"puzzles/physical/1x3x3x3/#solving","title":"Solving","text":"

    Every piece on the puzzle has pink and purple stickers which indicate which 4d orientation the piece is. In the solved state every piece must have the same 4d axis orientation. 1x3x3x3 is harder than the 3d Rubik\u2019s cube, but it can be solved by: orienting 3c pieces in 4d axis, orienting 4c pieces in 4d axis, orienting 5c pieces in 4d axis, except \u201cOLL parity\u201d (because of the extra axis, a single 5c can be missoriented), then solving the rest of the puzzle like a 3d Rubik\u2019s cube. Tymon made a YouTube video explaining the puzzle as well as showing an example solve.

    "},{"location":"puzzles/physical/1x3x3x3/#gyro","title":"Gyro","text":"

    This puzzle has two 1c pieces that are physicaly represented by one 2c piece (the core). Technically the puzzle is solved only when all of the other pieces have the pink sticker on the outside cell (because pink is outside on the core piece). To be able to solve the puzzle with purple color facing outwards, Tymon created a gyro algorithm that flips all pieces inside out and puts the core outside of the puzzle. This represents that pink/purple axis is flipped and now the puzzle is only solved when all of the pieces are oriented with purple on the ouside. The gyro can be reversed to get back to default projection.

    "},{"location":"puzzles/physical/1x3x3x3/#oll-parity","title":"OLL Parity","text":"

    Becuase of the extra freedom of the 4th dimension a monoflip (single missoriented corner) can occur. It can be solved by doing one of the 4d moves, to set up a valid OLL case or just rotate corners and undoing 4d move. A single corner can also be rotated with a simple commutator.

    "},{"location":"puzzles/physical/1x3x3x3/#alternative-solved-state","title":"Alternative Solved state","text":"

    The original solved state well represents which sticker on a piece is on the inside and which one is on the outside, but some people prefer a solved state with edges rotated 4 dimensionaly. This is because then the overall look of the puzzle is less chaotic and more friendly to non hypercubers.

    "},{"location":"puzzles/physical/2x2x2x2x2/","title":"2x2x2x2x2","text":"

    2x2x2x2x2

    5D Shape: Hypercube

    Physical Shape: \u201cRectangular prism\u201d

    Pieces: 32 5c

    Designed: 2023 Feb 13

    The Physical 2x2x2x2x2 (25) is a valid design that represents its virtual counterpart, designed by Akkei, Grant, and Andreas. Unlike the restricted 2x2x2x2x2, this puzzle would have the full freedom and moves of the virtual puzzle.

    "},{"location":"puzzles/physical/2x2x2x2x2/#history","title":"History","text":"

    In the Hypercubers Discord, Akkei and Andreas were discussing how to represent the symmetries of a 5D 5c piece in 3D. They came upon a certain colouring of the rhombic triacontahedron that worked.

    Grant

    I see

    HHHHHHHHHMMMMMMMMM

    I seeeee

    Akkei quickly finished rendering the full puzzle, with all 32 pieces:

    A few hours later, Grant figured out a simpler way to arrange the stickers that still worked:

    And 2 days later, after an extensive vc discussion, he rendered this different version where the pieces are oriented in a more readable way:

    This works because the rhombic triacontahedron can be viewed as 5 cubes circumscribed within it.

    "},{"location":"puzzles/physical/2x2x2x3/","title":"Grant\u2019s 2x2x2x3","text":"

    Grant\u2019s 2x2x2x3

    4D Shape: Hypercuboid

    Physical Shape: Rectangular prism

    Pieces: 16 4c, 8 3c

    Magnets: 1152

    Completed: 2022 Feb 3

    Grant\u2019s 2x2x2x3 is a physical 4D twisty puzzle that was designed and 3D printed by Grant S. It works almost the same as its virtual counterpart, with the restrictions of only certain moves being canonical, plus the extra gyro algorithms needed to rotate the puzzle 4 dimensionally.

    "},{"location":"puzzles/physical/2x2x2x3/#history","title":"History","text":"

    In December of 2021, Melinda green posted this image to the Hypercubers Discord server, jokingly calling it a 2x2x2x3:

    After Melinda posted that, Luna started sketching out some ideas of possible layouts for an actual 2x2x2x3.

    She invented the design for the 3c pieces by dividing the edges of a cube into 12 sections like this, grouping 4 sets of 3 together symmetrically.

    This is when Grant joined the server, and shared his 3D printed 2x2x2x2. Rowan then challenged him to build Luna\u2019s design for the 2x2x2x3. After a lot of time tweaking the designs, printing them, magnetizing them, and assembling them, the 2x2x2x3 was completed on February 3rd 2022.

    "},{"location":"puzzles/physical/2x2x2x3/#how-does-it-work","title":"How Does it Work?","text":"

    It follows the same basic principles as Melinda\u2019s 2x2x2x2, except gyros get more complicated. Grant made a YouTube video showing off the legal twists, and how it matched the virtual puzzle in MPU exactly.

    "},{"location":"puzzles/physical/2x2x3x3/","title":"Grant\u2019s 2x2x3x3","text":"

    Grant\u2019s 2x2x3x3

    4D Shape: Hypercuboid

    Physical Shape: Rectangular prism

    Pieces: 16 4c, 16 3c, 4 2c

    Magnets: 1728

    Completed: 2022 May 14

    "},{"location":"puzzles/physical/2x2x3x3/#history","title":"History","text":"

    After the invention of grant\u2019s 2x2x2x3, hypercubers were already thinking about what was next. The 2x2x3x3 has an extra middle layer that the 2x2x2x3 doesn\u2019t have, which introduces the 2c piece to the design. Hactar designed the new 2c piece, and then Grant made the first rendering of what a physical 2x2x3x3 would look like.

    It took a while to get the designs just right, but after printing and assembling 4 of the new 2c pieces and 8 more 3c pieces, the puzzle was completed on May 14th, 2022.

    "},{"location":"puzzles/physical/2x2x3x3/#functionality","title":"Functionality","text":"

    In order to prove that it was fully functional, it had to be shown that moves matched the virtual puzzle, and that it was possible to gyro the puzzle into different orientations.

    The gyro starts by removing the middle layer, and then gyroing just like the 2x2x2x3, which involves separating that middle layer and then gyroing like a 2x2x2x2. Next, the middle layer has some 4 dimensional black magic done to it, and then finally the puzzle is reassembled into the gyroed state. After doing this, Grant realized that some of the 2x2x3 cell moves aren\u2019t accessible, meaning that we actually need a 2nd type of gyro! This secondary gyro involves centering a 2x2x3 cell, and then making the middle layer stick out. Now the puzzle was fully functional.

    "},{"location":"puzzles/physical/2x3x3x3/","title":"Grant\u2019s 2x3x3x3","text":"

    Grant\u2019s 2x3x3x3

    4D Shape: Hypercuboid

    Physical Shape: Rectangular prism

    Pieces: 16 4c, 24 3c, 12 2c, 2 1c

    Magnets: 2496

    Completed: 2022 Jul 6

    The next logical step to take after Grant built his 2x2x3x3 was the 2x3x3x3, which once again requires a new piece type: the 1c. But this was no big deal because a 1-colored piece only has 1 color.

    The production went pretty smoothly, and after printing 8 more 3c and 2c pieces, and 2 of the new 1c pieces, the puzzle was completed on July 6th, 2022.

    Now given what we\u2019ve seen before with the previous gyro algorithms, you\u2019d think that the 2x3x3x3 needs an extremely long complicated recursive gyro, and although the design of the puzzle has at least 15 different possible types of gyros, it turns out that it actually doesn\u2019t need one at all! All the moves are accessible from this state, which makes for a really nice solving experience.

    The 4th ever physical 4D puzzle was constructed, bringing us closer and closer to our ultimate goal: the physical 3x3x3x3.

    "},{"location":"puzzles/physical/3x3x3x3/","title":"Grant\u2019s 3x3x3x3","text":"

    Physical 3x3x3x3

    4D Shape: Tesseract

    Physical Shape: Rectangular prism with 8 pieces sticking out

    Pieces: 8 1c, 24 2c, 32 3c, 16 4c

    Magnets: 3,840

    Completed: 2022 July 22

    Permutations: \\(1.7\\times 10^{120}\\)

    "},{"location":"puzzles/physical/3x3x3x3/#history","title":"History","text":"

    With the expansion of Melinda\u2019s 2x2x2x2 through the hypercuboid series, only 1 puzzle remained unmade: the physical 3x3x3x3. A true 4-dimensional analog of the classic 3x3x3 Rubik\u2019s Cube.

    Melinda Green

    I can\u2019t believe progress is being made towards the holy grail. The cost and effort doesn\u2019t matter since one will have to be made if it\u2019s at all possible. (The main reason to be sure it\u2019s right) It also doesn\u2019t matter how clumsy it is to operate, though of course to get it into multiple people\u2019s hands, you\u2019ll want to explore every possible way to improve it.

    At first, it was not clear how to expand the physical 2x3x3x3 into the 3x3x3x3, as it would need extra pieces that would no longer make it a nice cuboid shape. This is because we have one more slice layer than the number of dimensions we\u2019re trying to simulate. For example a 3x3x3 has 3 slice layers, so to represent it in 2D space, some of the pieces have to stick out like this:

    After months and months of effort, Grant completed assembling the puzzle on July 22nd, 2022. To this day, Grant has not scrambled or solved the physical 3x3x3x3 due to it being extremely big, heavy, and awkward to use. The most that was done on it was a checkerboard algorithm, but he did make a video showing off all of the hypercuboids and their legal twists.

    In April 2023, Akkei made a new program that simulated the physical 34, and then Hyperespy became the first person to ever fully solve the physical 34. The actual gyros are pretty painful to do, but luckily the program does them for you :)

    In just under 6 months, the community went from having 1 physical 4d puzzle, to having 5!

    On 2024-07-19, The Cube Dude became the first person ever to solve the physical 34 in real life.

    "},{"location":"puzzles/physical/4d-pyraminx/","title":"Dominik\u2019s 4D Pyraminx","text":"

    Dominik\u2019s 4D Pyraminx

    4D Shape: 5-cell

    Physical Shape: Tetrahedron + small Octahedron

    Pieces: 5 4c, 10 3c

    Magnets: 180 Completed: 2023 Dec 28

    "},{"location":"puzzles/physical/4d-pyraminx/#introduction","title":"Introduction","text":""},{"location":"puzzles/physical/4d-pyraminx/#history","title":"History","text":"

    Based on some ideas for the 4D pyraminx, Dominik first created a paper model in mid-December of 2023. After some suggestions from Melinda Green he created the prototype with magnets which was the first functioning version of this design. In April 2024 he finished an improved 3d printed version of the puzzle.

    "},{"location":"puzzles/physical/4d-pyraminx/#how-does-it-work","title":"How does it work?","text":"

    The puzzle can be moved like a pyraminx but with some additional legal moves. The edges that are in the position of the trivial tips can\u2019t legally be twisted.

    There are a few ways to project a 5-cell into 3d space. In a vertex-first projection the \u201chidden\u201d cell is on the \u201coutside\u201d of the others while in a cell-first projection it is on the \u201cinside\u201d. My design for the 4D pyraminx represents the latter way. In a solved state we have 4 colors on the outside and one on the inside of the puzzle.

    2D visualizations of 3D perspective projections of the first iteration of a pentatope-based fractal: a) vertex-first, b) cell-first, c) face-first, d) edge-first.

    (pictures: projections Source, vertex first pyraminx, cell first pyraminx Source)

    "},{"location":"puzzles/physical/4d-pyraminx/#the-pieces","title":"The Pieces","text":"

    There are ten tetrahedron-shaped 3-colored edge pieces where the fourth face is split into three colors.

    Then there are five octahedron-shaped 4-colored center pieces where four of the faces are split into three colors of the adjacent faces.

    The five 4-colored trivial tips can also be represented by tetrahedrons but with the current design they would need to be a separate arrangement. So this puzzle without the trivial tips represents just the truncated pyramnix.

    "},{"location":"puzzles/physical/4d-pyraminx/#chirality","title":"Chirality","text":"

    For the pieces to fit together in the correct color arrangement they need to have the same chirality. There is a left-handed and a right-handed version of the pieces and the centers and trivial tips need to have the same order of colors. Since the edges are allowed to be in more states the chirality doesn\u2019t matter as much but if the chiralities don\u2019t line up one color will be \u201cfavored\u201d in a certain orientation.

    "},{"location":"puzzles/physical/4d-pyraminx/#moves","title":"Moves","text":"

    There are some legal moves that don\u2019t represent a movement in the 4D puzzle.

    "},{"location":"puzzles/physical/4d-pyraminx/#edge-migration","title":"Edge Migration","text":"

    You can move edges to a corresponding position on the \u201cfree center\u201d and back.

    "},{"location":"puzzles/physical/4d-pyraminx/#reorient-edges","title":"Reorient Edges","text":"

    Edges can be in two states in relation to a center. They can align with the split face towards the center which will be call a \u201ccorrect\u201d state or the split face can be away from the center which will be called an \u201cincorrect\u201d state. In the latter case the edge can legally be oriented in three different ways.

    Next we have moves which do represent a movement in the 4D puzzle. Unlike the normal 3D pyraminx the tips can\u2019t legally be twisted. Other than that the four different moves of turning two layers of the pyraminx is legal. In the 4D version of the puzzle we have 20 different of such slice moves. Here are the four axes of rotations that are allowed for each 2-layered pyramid.

    If you perform such a move on each of the four sides of the tetrahedron you get a total of 16 moves. For the missing four moves we can \u201cequip\u201d the free octahedron with the edges by swapping them onto it and perform the rest of the rotations that way.

    After the rotation you have can move the edges back.

    \u2026

    This is not necessary for a solve, though, since you have access to all the edge pieces through the 16 other moves.

    "},{"location":"puzzles/physical/4d-pyraminx/#legal-states","title":"Legal States","text":"

    To keep the puzzle in a legal state sometimes after certain rotations we need to make edge reorientations. When performing a non-standard slice move an edge that isn\u2019t in a correct state regarding the center that has rotated can after a turn show a split face.

    If it isn\u2019t in a tip position it needs to be reoriented for otherwise the puzzle can get into an illegal or impossible state and also the color alignment becomes unknown for practical purposes when the puzzle is scrambled.

    How does this work? Before the rotation the colors A and B are on the outside.

    After a non-standard rotation either A or B will be on the inside and then there are two possible cases. Firstly, if the edge is \u201ccorrect\u201d then the split face will remain touching the rotated center and the full face with color C will show on the outside.

    The other case is when the edge is \u201cincorrect\u201d. Here the split face will be on the outside and the full color C will be touching the center.

    As a rule you can now rotate the edge in the \u201cdown\u201d direction in regards to the the rotation (where the center would be in the \u201cup\u201d direction) as shown here.

    This way we can simulate a 3-cycle of the faces A, B and C with the 4-sided shape of the tetrahedron.

    "},{"location":"puzzles/physical/4d-pyraminx/#gyro","title":"Gyro","text":"

    While it isn\u2019t necessary for a solve it is possible to reorient the whole puzzle by what\u2019s usually called a gyro. There might be better ways to do it but here is one way that can be done in four steps.

    "},{"location":"puzzles/physical/4d-pyraminx/#step-1","title":"Step 1","text":"

    First you have to align the free octahedron with one of the \u201couter\u201d edges. Regarding the colors it\u2019s best to look at the colors of the adjacent octahedron. In this step you have two cases. Either the edge is already correct the you don\u2019t have to do anything or the edge is incorrect then you have to reorient it towards the free octahedron.

    "},{"location":"puzzles/physical/4d-pyraminx/#step-2","title":"Step 2","text":"

    Now you have to separate one 2-layered pyramid next to the aligned free octahedron. Here you have to watch out for the three edges facing you. In case they are \u201cincorrect\u201d you have to reorient them towards you. Then you can move it together towards the free octahedron.

    "},{"location":"puzzles/physical/4d-pyraminx/#step-3","title":"Step 3","text":"

    Here you have to look at what\u2019s left of the \u201cupper pyramid\u201d. The topmost piece and the \u201cinner\u201d piece have to be reoriented in case they are incorrect. This time the direction you have to reorient them towards is \u201cdown\u201d, which means it\u2019s just like you would do in a normal slice move. One handy thing is that for the \u201cinner\u201d piece you simply have to look on the inside by only taking these three pieces off and if you see a whole face you can just leave the piece there. If the face is 3 colored you can just take it and rotate it together with the other pieces and it will be correct.

    "},{"location":"puzzles/physical/4d-pyraminx/#step-4","title":"Step 4","text":"

    The last step only involves the last edge. Again, if it\u2019s incorrect you first have to reorient it, this time towards the \u201cdown\u201d direction. And then you take the piece and move it \u201cover the Pyraminx\u201d, like this, and simply put it in the right spot.

    That\u2019s all there is to the gyro and here we have all in one movement.

    "},{"location":"puzzles/physical/4d-pyraminx/#comparison-of-moves-with-mc4d","title":"Comparison of Moves with MC4D","text":"Dominik\u2019s Pyraminx MagicCube 4D"},{"location":"puzzles/physical/4d-pyraminx/#example-solve","title":"Example solve","text":"

    Here is a link to a video of an example scramble and solve of the puzzle:

    Also you can watch a tutorial of how to solve it here:

    "},{"location":"puzzles/physical/4d-pyraminx/#notes-for-a-solve","title":"Notes for a solve","text":"

    The 4D pyraminx can end up with a single edge flipped incorrectly unlike the 3D pyraminx. This state can be solved with an algorithm that flips two edges, the right slice move and an additional two edges flip. But if a piece was moved in a wrong way accidentally it can also end up in a state that\u2019s not solvable like this. The reason for this is that there are three states that flipping an edge moves between: (a) solved state, (b) one edge correct and the other two flipped and (c) all edges incorrect.

    Going from (c) to (a) requires two flips so the method above can flip edge A from (b) to (a) or (c) but then edge B will go to state (b). In the following tree diagram it can be seen how all 2 flip moves only lead to looping between unsolved states:

    "},{"location":"puzzles/physical/restricted-2x2x2x2x2/","title":"Restricted 2x2x2x2x2","text":"

    Restricted 2x2x2x2x2

    5D Shape: Penteract

    Physical Shape: Rectangular prism

    Pieces: 32 5c

    Magnets: 768 (without buffer)

    The restricted 2x2x2x2x2 or 25 (also called the physical domino reduced 2x2x2x2x2) is a physical puzzle that represents the state space of a virtual 25 puzzle, with one of the axes only being able to do 180 degree turns. This puzzle is like the 5D equivalent of scrambling a 23 with only <U,D,R2,F2,L2,B2>, which makes the solve way easier than the non-restricted version.

    The first known restricted 25 was made by Grant Staten which he showed off in a YouTube video. Hyperespy also made a version using foldable paper cubes to cover the pieces.

    "},{"location":"puzzles/physical/restricted-2x2x2x2x2/#pieces-and-twists","title":"Pieces and Twists","text":"

    There are 32 5c pieces, but each piece can only be oriented in 12 ways instead of 60 because of the axis restriction. And just like Melinda\u2019s physical 24, this puzzle has certain legal moves that a solver must follow. Each twist of this puzzle is like rotating a tesseract. You can take off half of it and do any reorientation like you would on the physical 24. This includes simple rotations and gyros. The only thing you can\u2019t do is a 90 degree twist of the square layers (this results in a legal puzzle state, but is not analogous to any move on the virtual 25). See Hyperespy\u2019s video for a full overview of the canonical moves and a way to scramble the puzzle quickly.

    "},{"location":"puzzles/physical/restricted-2x2x2x2x2/#how-to-make-one","title":"How to make one","text":"

    Buffer pieces magnet layout

    Order a 24 DIY kit from Melinda Green. Then assemble it with a mirrored colour scheme from your current 24, but with the same magnet layout. For the 5D stickers 192 are needed in total, with half (96) of them being black and the other half grey. This means that each normal physical 24 piece gets 6 stickers on it. A good option is to order a couple of custom sticker sets online such as these.

    Grant used buffer pieces when making his version in order to make it obvious to avoid doing illegal twists. If you want to add buffer pieces, you need 8 monocolored hypercubies with a custom magnet layout. Although there\u2019s only a few different illegal twists that are even possible so it\u2019s not too tricky to avoid altogether.

    "},{"location":"puzzles/physical/2x2x2x2/","title":"Melinda\u2019s 2x2x2x2","text":"

    Melinda\u2019s 2x2x2x2

    4D Shape: Tesseract

    Physical Shape: Rectangular prism

    Pieces: 16 4c

    Magnets: 384

    Permutations: \\(3.3*10^{27}\\)

    See Melinda\u2019s 2x2x2x2 website for her project overview.

    "},{"location":"puzzles/physical/2x2x2x2/#history","title":"History","text":""},{"location":"puzzles/physical/2x2x2x2/#invention","title":"Invention","text":"

    Many hypercubers, including Melinda Green (one of the developers of MC4D) wanted to create a physical 34, but sadly it just seemed too complicated. Eventually, she decided to focus on the 24 due to its simplicity of only having 16 4c pieces.

    Throughout 2013 and 2014, Melinda was in contact with Oskar van Deventer, and together they tried to think of any possible mechanism for a physical 24. Originally they were determined to find a good mechanism that didn\u2019t have to rely on magnets, but after several failed attempts a magnetic mechanism was the only solution. In 2017, Melinda built the first prototype and shared an unlisted YouTube video showing it off to the mailing list members.

    A few months later, Melinda built a 2nd prototype using 3D printing from a company called Shapeways (which just so happened to offer 8 colors, 6 of them being standard Rubik\u2019s Cube colors, plus pink and purple (this is where the standard 4D colour scheme comes from)).

    "},{"location":"puzzles/physical/2x2x2x2/#popularization","title":"Popularization","text":"

    Professor Erno Rubik inspecting a physical 24 at G4G 2018

    At the Gathering For Gardner conference of 2018, Roice Nelson got to show his physical 24 to Erno Rubik. It was reported that his only comment was something along the lines of \u201cnone of the derivative puzzles matter and that only his original invention is important\u201d.

    Also in 2018, the YouCuber CanChrisSolve? made a series of videos where he solved the physical 24, which have now amassed over 100,000 views combined! A month later, a Spanish YouCuber called TheMaoiSha made a video explaining the physical 24 which now sits at over 800,000 views, exposing the world of hypercubing to more and more people.

    Because of all this recognition, many new people bought the puzzle, but it was still pretty expensive because of the costs of 3d printing and the labor of manual assembly. Eventually Melinda moved away from 3D printing in favour of injection molding, which greatly reduced the price while increasing the quality dramatically.

    Where can I buy one?

    Read this section of Melinda\u2019s website.

    "},{"location":"puzzles/physical/2x2x2x2/#how-does-it-work","title":"How Does it Work?","text":"

    Animations showing how the virtual and physical 24 transform into each other

    Each twist and rotation on the virtual 24 is possible to do on the physical puzzle, albeit with certain moves requiring some extra setup. See the canonical moves page for a more detailed explanation.

    Also see Melinda\u2019s video on the official canonical moves.

    In December of 2017, Melinda made a video fully explaining the puzzle, including its canonical moves, scrambling, and more.

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/","title":"Physical 2x2x2x2 Canonical Moves & Notation","text":"

    Info

    Watch Melinda Green\u2019s video for a quick overview of the canonical moves

    The Canonical Moveset is a set of moves agreed upon by the community for solving the physical 24. To get accepted into the official Hall of Fame, you must follow the canonical moves exactly. Make sure that you have already solved a virtual 24, and are familiar with how the pieces of the physical 24 correspond. Also make sure that you know your 3D rotations (x y z) very well.

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#canonical-moves","title":"Canonical Moves","text":""},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#simple-rotations","title":"Simple Rotations","text":"

    Simple Rotations

    A lot of the whole puzzle reorientations can be reached without the need for the gyro algorithm. These are called simple rotations, and consist of rotaing the L and R cells together in opposing directions (as to not change the state of the puzzle, only its orientation).

    zy yz yw wy zw wz

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#cell-twists","title":"Cell Twists","text":"

    The left and right cells of the puzzle can be twisted into any reorientation of a cube. In other words, there are 23 different twists of the L/R cells:

    Left cell twists: Ly Ly' Ly2 Lx2 Lz2 Lx2,y Lx2,y' Lx Lx,y Lx,y' Lx,y2 Lx' Lx',y Lx',y' Lx',y2 Lz Lz,y Lz,y' Lz,y2 Lz' Lz',y Lz',y' Lz',y2

    Right cell twists: Ry Ry' Ry2 Rx2 Rz2 Rx2,y Rx2,y' Rx Rx,y Rx,y' Rx,y2 Rx' Rx',y Rx',y' Rx',y2 Rz Rz,y Rz,y' Rz,y2 Rz' Rz',y Rz',y' Rz',y2

    Each move here either starts with an L or an R, followed by the rotations (separated by commas).

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#insideoutside-twists","title":"Inside/Outside Twists","text":"

    Ix2 twist

    The I and O cells are the sides with the next most turning freedom after L and R. This was referred to as an \u201caxial twist\u201d in Melinda\u2019s video. The canonical moves for the I/IO cells are Ix Ix' Ix2 Ox Ox' Ox2

    These twists can be difficult to perform for speedsolving, so most people tend to split it up by temporarily doing 2 illegal 90 degree twists that yield the same result. Be careful though, as this can lead to an illegal state if you accidentally screw up while doing it this way.

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#slab-twists","title":"Slab Twists","text":"

    U2 twist

    The remaining canonical twists involve picking up a 2x2x4 \u201cslab\u201d off the puzzle, and rotating it 180 degrees in the same plane. These slabs you can pick up are the U, F, D, and B cells. Because the slabs can only be turned 180 degrees those moves will just be referred to as U2 F2 D2 B2

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#gyro","title":"Gyro","text":"

    If you try to scramble the puzzle with all the moves above, you will quickly notice that the 2 colours on the x-axis aren\u2019t mixing with the other colours. This is because the simple rotations and slab twists are restricting what we can do because of the symmetry of the physical puzzle. To fix this, we need a series of illegal moves that rotate the puzzle 4-dimensionally in a way that changes the x-axis. This is called the Gyro, and will allow us to access all the rest of the puzzle rotations (xz zx yx xy xw wx)

    There are several different algorithms for this, some of which gyro different axes. Melinda has several videos about different gyro algorithms, the shortest known one being 6 snaps. Below is a common gyro algorithm that several people in the community use.

    • Take the left endcap off and put it on the right so it becomes the right endcap (this brings the puzzle into the inverted state)
    • Ly Ry'
    • Take the right endcap off and put it on the left so it becomes the left endcap (this brings the puzzle back into the normal state)
    • Rx2 B2 D2 Lx2
    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#expanded-canonical-moveset","title":"Expanded Canonical Moveset","text":"

    The expanded canonical moveset is what is accepted for the Hypercubing.xyz leaderboards. These moves all correspond perfectly with twists on the virtual puzzle, but weren\u2019t allowed in the canonical moveset because people agreed not to for some reason.

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#extra-slab-twists","title":"Extra slab twists","text":"

    Several new slab twists are added. For the U cell, this means going from having just Uy2 to now having Uy2 Ux,y2 Ux',y2 Ux Ux' Ux2 Uz2

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#extra-io-cell-twists","title":"Extra I/O cell twists","text":"

    Iz2 Iy2 Oz2 Oy2 are added (although they are very hard to fingertrick and perform quickly).

    "},{"location":"puzzles/physical/2x2x2x2/canonical-moves/#scramble-notation","title":"Scramble notation","text":"

    Scrambles use comma-separated cell twists and the special move #. # represents a Uy2 move followed by a \u201cleft-to-right restack,\u201d which consists of taking the left endcap off and putting it on the right side of the puzzle. This is not a legal move during solves, but does result in a legal puzzle state so it is used during scrambles.

    "},{"location":"puzzles/physical/2x2x2x2/image-generator/","title":"Physical 2x2x2x2 Image Generator","text":"

    Generate an image of a physical 2x2x2x2 by inputting moves using canonical moves notation, or by specifying the color per sticker. You can also do this directly from the URL by adding ?&moves= or ?&stickers= at the end, and using - to separate (and H instead of # for gyro).

    Generate by move input

    Moves that the generator will accept: zy yz xz zx yx xy yw wy xw wx zw wz Ly Ly' Ly2 Lx2 Lz2 Lx2,y Lx2,y' Lx Lx,y Lx,y' Lx,y2 Lx' Lx',y Lx',y' Lx',y2 Lz Lz,y Lz,y' Lz,y2 Lz' Lz',y Lz',y' Lz',y2 Ry Ry' Ry2 Rx2 Rz2 Rx2,y Rx2,y' Rx Rx,y Rx,y' Rx,y2 Rx' Rx',y Rx',y' Rx',y2 Rz Rz,y Rz,y' Rz,y2 Rz' Rz',y Rz',y' Rz',y2 Ix Ix' Ix2 Ox Ox' Ox2 U2 F2 B2 D2 # Input moves: (separated by spaces)

    Generate Image

    Generate by sticker input

    Colors: 

    W = white\nY = yellow\nR = red\nO = orange\nG = green\nB = blue\nM = magenta (purple)\nP = pink\nS = silver (gray)\n
    Piece/Sticker order: LUBO LUBI LUFI LUFO LDBO LDBI LDFI LDFO RUBI RUBO RUFO RUFI RDBI RDBO RDFO RDFI

    Input stickers by piece: (separated by spaces)

    Generate Image

    "},{"location":"puzzles/physical/2x2x2x2/scramble-generator/","title":"Physical 2x2x2x2 Scramble Generator","text":"

    See Scramble notation. In particular, # represents a Uy2 followed by a right-to-left restack.

    The starting orientation is:

    • R = red
    • U = white
    • F = green
    • O = pink

    Scrambles: 5

    Generate Scrambles

    Scrambles

    "},{"location":"software/","title":"Software","text":"

    Since building physical representations of higher-dimensional puzzles is challenging, we use computer software to simulate them.

    "},{"location":"software/#what-program-should-i-download","title":"What program should I download?","text":"

    We recommend Hyperspeedcube to get started.

    If you want macro support (recorded sequences of moves) or other 4D puzzles, Magic Cube 4D is a good option.

    • For 5D+ puzzles: Magic Cube 5D, Magic Cube 7D, and Magic Puzzle Ultimate
    • For puzzles in non-Euclidean geometries: MagicTile
    • For 3D puzzles: pCubes

    For hypercubing on mobile devices, see Android apps or iOS apps.

    "},{"location":"software/#general-cubing-hypercubing","title":"General cubing & hypercubing","text":"What do the icons mean?
    • Mouse controls
    • Keyboard controls
    • Customizable keyboard controls
    • Find/filter piece by color
    • Find/filter pieces by type
    • Timer
    • Macros
    • Move input
    • Custom colors
    • Custom puzzles
    • VR support
    Program Platforms Features Puzzles Hyperspeedcube \u2022 {1-9}{3-4} Magic Cube 4D \u2022 4D via Schl\u00e4fli symbol MagicTile \u2022 2D tilings Magic Puzzle Ultimate \u2022 3D+ doctrinaire Magic Cube 7D \u2022 {3-5}{4-7} Magic Cube 5D \u2022 {2-7}5 Magic Cube 4D VR \u2022 34 Flat Hypercube \u2022 {1-19}{1-10} Gelatinbrain1 \u2022 many cursed things pCubes nearly every 3D puzzle Ultimate Magic Cube platonic 3D Ultimate Magic Cube 2 \u2022 platonic + misc 3D Twizzle Explorer \u2022 many 3D puzzles IsoCubeSim \u2022 AxBxC, N-layer megaminx Geraniums Pot \u2022 rotating-circle puzzles"},{"location":"software/#specific-puzzles","title":"Specific puzzles","text":"Program Platforms Features Puzzle Magic Hyperbolic Tile {6,3,3} (7 quotients) Magic Simplex 5D \u2022 5D simplex (+ recuts) Magic120Cell \u2022 120-cell (+ recolorings) Nan Ma\u2019s 11-cell \u2022 11-cell Twisty Prisms \u2022 certain 3D prisms"},{"location":"software/#1d-and-2d-puzzles","title":"1D and 2D puzzles","text":"
    • k-Card Game
    • Loopover
    • Heav\u2019s Relocation Puzzle
    • Slidysim
    • Luna\u2019s \u201cGreen\u201d (generalization of Lights Out and Green The Board)
    "},{"location":"software/#complex-laminated-puzzles","title":"Complex & laminated puzzles","text":"
    • Complex Loopover
    • Complex Cube
    • Luna\u2019s Complex Puzzles
    • Milo\u2019s laminated puzzles
    • Ema\u2019s Complex Hemicube Puzzle
    "},{"location":"software/#other","title":"Other","text":"
    • Nan Ma\u2019s puzzles (Reflecube, Lollipop, Clockwork Cube, and more)
    • Akkei\u2019s physical 34 program
    • Rayzchen\u2019s 3to4++ (Physical 34)
    • Arnaud Ch\u00e9ritat\u2019s Hungarian Rings
    • Magic Cube 3D
    • MagmaMcFry\u2019s Quaternion Cube
    • TwistyPuzzles by CasualCuber
    • Banderson\u2019s \u201csus non-associative puzzle\u201d
    "},{"location":"software/#android-apps","title":"Android apps","text":"

    Hypercubing on mobile devices is not recommended; however, if this is your only option, these apps are available for Android:

    • Magic Cube 4D (Raynefork) (recommended)
    • Magic Cube 4D (superceded by Raynefork)
    • The Tesseract Puzzle
    • 4D Hypercube Puzzle
    "},{"location":"software/#ios-apps","title":"iOS apps","text":"

    There are no known apps for hypercubing on iOS.

    If you\u2019re able to connect an external mouse & keyboard, the web version of Hyperspeedcube may work on mobile devices but it\u2019s not officially supported.

    "},{"location":"software/#tools","title":"Tools","text":"
    • Orb, for planning puzzle cut depths
    • RocKeT, for finding RKT cancels for 3D algorithms
    • Hypersolve, for generating short 24 solutions and scrambles
    "},{"location":"software/#games","title":"Games","text":"

    This is not a complete list of 4D games. This is specifically 4D software that hypercubers often recommend for understanding 4D geometry better.

    • 4D Blocks
    "},{"location":"software/#visualizations","title":"Visualizations","text":"
    • IMAC2 Rubik4D by Oradimi, melokye, and MMeche

    1. Full name: gelatinbrain\u2019s Virtual Magic Polyhedra (permutationpuzzles)\u00a0\u21a9

    "},{"location":"software/hyperspeedcube/","title":"Hyperspeedcube","text":"

    Hyperspeedcube (HSC) is a modern, beginner-friendly 3D and 4D Rubik\u2019s cube simulator with customizable mouse and keyboard controls and advanced features for speedsolving. It\u2019s been used to break numerous speedsolving records and runs on all major operating systems plus the web. Hyperspeedcube was first released in early 2022 and is developed by Andrew Farkas (a.k.a. HactarCE).

    "},{"location":"software/hyperspeedcube/#downloadinstallation","title":"Download/installation","text":"

    • Download Hyperspeedcube

    • Use Hyperspeedcube online

    HSC does not have an installer. On Windows, open hyperspeedcube_win64.zip and move hyperspeedcube.exe out of to a folder on your computer.

    Features missing from the web version

    • Saving & loading logs to file (can still save/load via clipboard)
    • Awareness of alternate keyboard layouts
    • Antialiasing
    "},{"location":"software/hyperspeedcube/#troubleshooting","title":"Troubleshooting","text":"

    If none of the instructions below help, join the Hypercubers Discord server and start a thread in the #\u2753help forum. Mention in your post that you\u2019ve read the FAQ.

    "},{"location":"software/hyperspeedcube/#windows","title":"Windows","text":"My antivirus thinks Hyperspeedcube is malicious

    Try opening the Windows Security app, and digging around in the settings there to disable it, then try the download again.

    \u201cWindows protected your PC\u201d

    Microsoft Defender SmartScreen prevented an unrecognized app from starting. Running this app might put your PC at risk.

    Click More info and then Run anyway.

    In order to prevent that message from appearing, Hactar would have to spend a lot of money to buy a Microsoft developer license and go through a lot of hassle every time there is a new release of HSC. You\u2019ll only have to click through the warnings once.

    \u201cThe program can\u2019t start because VCRUNTIME140.dll is missing from your computer.\u201d

    You need to install the Microsoft Visual C++ Redistributable. Here is a direct download.

    \u201cHyperspeedcube crashed. A crash report has been saved to \u2026\u201d

    First, make sure your graphics drivers are up-to-date! This is the most common cause of Hyperespeedcube crashes. Here is a video tutorial if you don\u2019t know how to do that.

    If you still get an error message, then follow these steps:

    1. Press +R, type %LocalAppData%, and press Enter
    2. Open the Temp folder
    3. Find the file named in the error message and send it to Hactar either in a GitHub issue or a new thread in the #\u2753help forum on Discord
    "},{"location":"software/hyperspeedcube/#macos","title":"macOS","text":"\u201c\u2018Hyperspeedcube.app\u2019 cannot be opened because the developer cannot be verified.\u201d

    macOS cannot verify that this app is free from malware.

    Go to System Settings \u2192 Privacy & Security, then scroll down to Security. Select App Store and identified developers, then click Open Anyway.

    In order to prevent that message from appearing, Hactar would have to spend a lot of money to buy an Apple developer license and go through a lot of hassle every time there is a new release of HSC.

    "},{"location":"software/hyperspeedcube/#linux","title":"Linux","text":"

    Check that you have up-to-date graphics drivers installed. If that doesn\u2019t help, you\u2019re on your own.

    If there\u2019s any changes you make to the Hyperspeedcube source code to get it working, please open an issue or PR on GitHub. If you\u2019re able to make an package for HSC for your package manager, please let me know either on Discord or GitHub Issues.

    "},{"location":"software/hyperspeedcube/#history","title":"History","text":""},{"location":"software/hyperspeedcube/#10","title":"1.0","text":"

    On April 24th, 2020, Hactar wrote a small program called Keyboard Speedcube that simulated a 33 using keybinds inspired by Ryan Heise\u2019s Rubik\u2019s Cube Simulator. In October 2021, rudimentary 4D support was added and a screenshot was posted on the Hypercubers Discord server. A month later, Rowan Fortier asked Hactar if he could receive an early version of the program, and showcased it in a YouTube video. Shortly before the video\u2019s release, the project was renamed to Hyperspeedcube.

    The first official release was v0.1.0 in January 2022, featuring customizable keybinds and support for the 33 and 34. Over the next year, many community-requested features were added, including mouse controls and more n3 and n4 puzzles. (See the changelog for a complete list of releases and changes.)

    Hyperspeedcube began to see widespread use after the addition of mouse controls and more advanced piece filters in August 2022, kickstarting the hyperspeedsolving revolution. The web version was released in January 2023.

    Shortly after the web version was released, Hactar and Luna began reading the MPU source code, intending to make a tool to ease the process of writing MPU puzzle definitions. Instead, they were able to replicate MPU\u2019s puzzle generation algorithms, and Hactar set to work on a new puzzle simulator using this backend. Over the next year, they adapted the algorithm to use Conformal Geometric Algebra in the hopes of supporting curved cuts, but this proved too challenging in higher dimensions. Together with Milo, they built a puzzle definition system using Lua, since it is a well-known programming language with a simple type system that is easy to embed, sandbox, and extend with custom types.

    The first dev build of the new version, Hyperspeedcube v2.0.0-pre.1, was released on July 16th, 2023, and supported Lua shape definitions but not twists. As of mid-2024, HSC 2.0 is still in active development.

    "},{"location":"software/hyperspeedcube/#20","title":"2.0","text":"

    HSC 2.0 is the next major update in the works, with no set release date as of yet. Below is a list of upcoming features:

    • Build nearly any puzzle in 3D to 7D space
    • Complete overhaul of the graphics engine
    • Built-in timer for speedsolves, including a configurable autosplitter
    • Timeline of progress during solve
    • More piece filter customization

    See Hactar\u2019s website for more details.

    "},{"location":"software/hyperspeedcube/#development-screenshots","title":"Development screenshots","text":""},{"location":"software/hyperspeedcube/#early-shape-generation-and-twisting-prototypes-late-2022","title":"Early shape generation and twisting prototypes (late 2022)","text":""},{"location":"software/hyperspeedcube/#graphics-debugging-early-2024","title":"Graphics debugging (early 2024)","text":""},{"location":"software/hyperspeedcube/#ui-development-mid-2024","title":"UI development (mid 2024)","text":""},{"location":"software/magiccube4d/","title":"Magic Cube 4D","text":"

    Magic Cube 4D (MC4D) was one of the first ever programs capable of simulating higher dimensional twisty puzzles. It was first released in 1988 and was developed by Melinda Green, Don Hatch, Jay Berkenbilt, and Roice Nelson.

    34 puzzle in Magic Cube 4D

    "},{"location":"software/magiccube4d/#downloadinstallation","title":"Download/installation","text":"

    Magic Cube 4D requires Java to be installed. Once you have Java, go to the Superliminal website and click on the Click here to download MagicCube4D button. Save the executable mc4d-4-3-343.jar file somewhere you will remember, as this file is how you open the program.

    "},{"location":"software/magiccube4d/#alternative-versions","title":"Alternative versions","text":""},{"location":"software/magiccube4d/#raynefork","title":"Raynefork","text":"

    Raynefork is Raymond Zhao\u2019s version of MC4D, which has the updated default colour scheme (standard 33 colours plus pink/purple), and some more options in the settings menu. It can be found from its GitHub here.

    "},{"location":"software/magiccube4d/#dons-version","title":"Don\u2019s version","text":"

    Don\u2019s version contains some 2D puzzles, many 3D puzzles, and some interesting 4D puzzles. It also has a menu for 5D and 6D puzzles, however they don\u2019t work when you select them. Don\u2019s version can be downloaded from its GitHub here.

    "},{"location":"software/magiccube4d/#troubleshooting","title":"Troubleshooting","text":"

    Try reading MC4D\u2019s FAQ on the superliminal website. If you\u2019re still having an issue, try asking for help on the Discord server or mailing list.

    my computer wants to open the .jar file with notepad

    This means that you haven\u2019t installed Java properly. Make sure to get the newest version from Java\u2019s website. After the installation, right click on the mc4d-4-3-343.jar file, and click open with Java(TM) Platform SE Binary.

    facecolors.txt doesn\u2019t change my colors
    1. Try closing and reopening the program, and then reloading the puzzle using the menu.
    2. If you\u2019re on windows and you have show file extensions off in settings, then you may have accidentally named the file facecolors.txt.txt. Try renaming it to just facecolors. (Alternatively, enable show file extensions in settings)
    "},{"location":"software/magicpuzzleultimate/","title":"Magic Puzzle Ultimate","text":"

    Magic Puzzle Ultimate (MPU or MPUlt) is a higher-dimensional puzzle simulator developed by Andrey Astrelin that can simulate nearly any symmetric doctrinaire puzzle and supports, including user-defined ones.

    Download MPU

    32-cell puzzle in MPU

    "},{"location":"software/magicpuzzleultimate/#records","title":"Records","text":"

    Some records for the shortest and first solutions of a puzzle are kept on the Superliminal Wiki page.

    "},{"location":"software/magicpuzzleultimate/#virtual-puzzles","title":"Virtual Puzzles","text":"

    While we\u2019re waiting for Hyperspeedcube 2.0 to release, Magic Puzzle Ultimate is a great program to get your feet wet in the world of other 4D puzzles.

    Open the location where your MPUlt folder is. Next, open the MPUlt_puzzles.txt file. This file is a big list of all the puzzles in the program. Now you can copy & paste the puzzles that you want below under their appropriate Block. Just save the file, reopen MPUlt, and the puzzle you added will be in the menu.

    "},{"location":"software/magicpuzzleultimate/#puzzle-definitions","title":"Puzzle Definitions","text":"3D Puzzles 3x3x3 3x3x3
    Puzzle 3x3x3\nDim 3\nNAxis 1\nFaces 1,0,0\nGroup 1,0,0/1,1,0 1,0,0/1,0,1\nAxis 1,0,0\nTwists 0,1,0/0,1,1\nCuts -0.33 0.33\n
    Skewb Skewb
    Puzzle Skewb\nDim 3\nNAxis 1\nFaces 1,0,0\nGroup 1,0,0/1,1,0 1,0,0/1,0,1\nAxis 1,1,1\nTwists 1,-1,0/1,0,-1\nCuts 0\n
    Compy Rainbow Compy Rainbow
    Puzzle Compy_Rainbow\nDim 3\nNAxis 1\nFaces 1,0,0 0.6667,0.6667,0.6667\nGroup 1,0,0/1,1,0 1,0,0/1,0,1\nAxis 1,1,1\nTwists 1,-1,0/1,0,-1\nCuts -0.45 0.45\nFixedMask 2\n
    Cuboctahedron Cuboctahedron
    Puzzle Cuboctahedron\nDim 3\nNAxis 2\nFaces 1,0,0 0.667,0.667,0.667\nGroup 1,0,0/1,1,0 1,0,0/1,0,1\nAxis 1,0,0\nTwists 0,1,0/0,1,1\nCuts 0.5 -0.5\nAxis 1,1,1\nTwists 1,-1,0/1,0,-1\nCuts 0.5 -0.5\n
    4D Puzzles Tesseract Family 2x2x1x1 2x2x1x1
    Puzzle 2x2x1x1\nDim 0\nNAxis 2\nFaces 1,0,0,0 0,0,0,0.5\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,1,1\n
    2x2x2x1 2x2x2x1
    Puzzle 2x2x2x1\nDim 4\nNAxis 2\nFaces 1,0,0,0 0,0,0,0.5\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1\nCuts 0 0\nAxis 0,0,0,1\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0\nCuts\n
    2x2x2x3 2x2x2x3
    Puzzle 2x2x2x3\nDim 4\nNAxis 2\nFaces 1,0,0,0 0,0,0,1.5\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1\nCuts 0\nAxis 0,0,0,1\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0\nCuts 0.5 -0.5\n
    2x2x3x3 2x2x3x3
    Puzzle 2x2x3x3\nDim 4\nNAxis 2\nFaces 1,0,0,0 0,0,1.5,0\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1\nAxis 1,0,0,0\nTwists 0,0,1,0/0,0,1,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0,1,1\nCuts 0\nAxis 0,0,1,0\nTwists 1,0,0,0/1,1,0,0 0,0,0,1/1,0,0,0 0,0,0,1/1,1,0,0\nCuts 0.5 -0.5\n
    2x2x3x4 2x2x3x4
    Puzzle 2x2x3x4\nDim 4\nNAxis 3\nFaces 1,0,0,0 0,0,1.5,0 0,0,0,2\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1\nCuts 0\nAxis 0,0,1,0\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0\nCuts 0.5 -0.5\nAxis 0,0,0,1\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0\nCuts 1 0 -1\n
    2x3x4x5 2x3x4x5
    Puzzle 2x3x4x5\nDim 4\nNAxis 4\nFaces 1,0,0,0 0,1.5,0,0 0,0,2,0 0,0,0,2.5\nGroup 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1\nCuts 0\nAxis 0,1,0,0\nTwists 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,0,1,0/0,0,0,1\nCuts 0.5 -0.5\nAxis 0,0,1,0\nTwists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0\nCuts 1 0 -1\nAxis 0,0,0,1\nTwists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0\nCuts 1.5 0.5 -0.5 -1.5\n
    3x3x3x1 3x3x3x1
    Puzzle 3x3x3x1\nDim 4\nNAxis 2\nFaces 1.5,0,0,0 0,0,0,0.5\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1\nCuts 0.5 -0.5\nAxis 0,0,0,1\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0\nCuts\n
    3x3x3x2 3x3x3x2
    Puzzle 3x3x3x2\nDim 4\nNAxis 2\nFaces 1.5,0,0,0 0,0,0,1\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1\nCuts -0.5 0.5\nAxis 0,0,0,1\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0\nCuts 0\n
    4x4x4x4 4x4x4x4
    Puzzle 4^4\nDim 4\nNAxis 1\nFaces 1,0,0,0\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,-1,0/0,0,0,1 0,2,-1,-1/0,1,1,-2\nCuts 0.5 0 -0.5\n
    Other {4}x{4} 3 {4}x{4} 3
    Puzzle {4}x{4} 3\nDim 4\nNAxis 2\nFaces 1,0,0,0 0,0,1,0\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,0,1,0 0,0,1,0/0,0,1,1\nCuts 0.5 -0.5\nAxis 0,0,1,0\nTwists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1\nCuts 0.5 -0.5\n
    3^4 Skewb 3^4 Skewb
    Puzzle 3^4 Skewb\nDim 4\nNAxis 1\nFaces 1,0,0,0\nSimplified\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1\nAxis 1,1,1,1\nTwists 0,2,-1,-1/0,1,1,-2 1,-1,0,0/0,0,1,-1\nCuts 0\n
    5-5_Duotegum 5-5_Duotegum
    Puzzle 5-5_Duotegum\nDim 4\nNAxis 1\nFaces -1,1,0,0\nGroup 1,0,0,0/0.809016994,0,0.587785252,0 1,1,0,0/0,0,1,1\nAxis -1,1,0,0\nTwists 0,0,1,0/0,0,0,1 1,1,0,0/0,0,1,1 1,1,0,0/0,0,-1,1\nCuts 0.65\n
    16-cell Face Turning 16-cell Face Turning
    Puzzle 16-cell_FT\nDim 4\nNAxis 1\nFaces 1,1,1,1\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1\nAxis 1,1,1,1\nTwists 0,2,-1,-1/0,1,1,-2 1,-1,0,0/0,0,1,-1\nCuts 0.6 -0.6\nFixedMask 2\n
    Chamfered Pentagonal Duoprism Chamfered Pentagonal Duoprism
    Puzzle Chamfered_Pentagonal_Duoprism\nDim 4\nNAxis 2\nFaces -1.41429,0,0,0 1.41429,0,0,0 -1,1,0,0\nGroup 1,0,0,0/0.809016994,0,0.587785252,0 1,1,0,0/0,0,1,1\nAxis -1,0,0,0\nTwists 0,1,0,0/0,0.809016994,0,0.587785252 0,0,1,0/0,0,0,1\nCuts 1.3 -1.23\nAxis -1,1,0,0\nTwists 0,0,1,0/0,0,0,1 1,1,0,0/0,0,1,1 1,1,0,0/0,0,-1,1\nCuts 0.85\n
    Chamefered Tesseract Chamfered Tesseract
    Puzzle Chamfered_Tesseract\nDim 4\nNAxis 2\nFaces 1,0,0,0 0.70710678,0.70710678,0,0\nGroup 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,0,0/0,1,0,1 0,0,1,0/0,0,1,1\nCuts 0.85 -0.85\nAxis 1,1,0,0\nTwists 0,0,1,0/0,0,1,1 1,-1,0,0/0,0,1,0 1,-1,0,0/0,0,1,1\nCuts 0.57 -0.57\n
    Octahedral Prism Octahedral Prism
    Puzzle Octahedral_Prism\nDim 4\nNAxis 2\nFaces 1,0,0,0 0,1,1,1\nGroup 1,0,0,0/0,1,0,0 0,1,0,0/0,1,1,0 0,1,0,0/0,1,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,1,0 0,1,1,0/0,0,1,1 0,1,0,0/0,0,1,1\nCuts 0.5 -0.5\nAxis 0,1,1,1\nTwists 0,1,-1,0/0,1,0,-1 1,0,0,0/0,1,-1,0\nCuts 0.5 -0.5\n
    Snub 24-cell Snub 24-cell
    Puzzle Snub24cell\nDim 4\nNAxis 2\nFaces 1,0,0,0 0.809017,0.809017,0,0 0.9045085,0.6545085,0.25,0\n#Faces 1,0,0,0 0.809017,0.809017,0,0 0.9045085,0.6545085,0.25,0 0.9045085,0.6545085,-0.25,0\nGroup 0,2,-1,-1/0,1,1,-2 0,1,1,2/0,2,-1,1 2,-2,-2,0/1,-1,-1,3\nAxis 1,0,0,0\nTwists 0,2,-1,-1/0,1,1,-2 0,0,1,0/0,0,0,1 0,1,1,2/0,2,-1,1\nCuts 0.9 -0.9\nFixedMask 2\nAxis 0.809017,0.809017,0,0\nTwists 2,-2,-2,0/1,-1,-1,3 1,-1,-1,-3/2,-2,-2,0 0,0,0,1/0,0,1,0\nCuts 0.95 -0.95\nFixedMask 2\n
    Square Antiprism Prism Square Antiprism Prism
    Puzzle Square_Antiprism_Prism\nDim 4\nNAxis 3\nFaces -0.5,0,0,0 0,0,0.42044820,0 0,0.56903559,0.14014940,0\nGroup 0,0,1,0/0,0.38268343,0,0.92387953 0,0,0,1/0,1,0,1 1,0,0,0/0,0,0,1\nAxis 1,0,0,0\nTwists 0,1,0,0/0,1,0,1 0,0,1,0/0,0.38268343,0,0.92387953 0,0,1,0/0,-0.38268343,0,0.92387953\nCuts 0.1666 -0.1666\nAxis 0,0,1,0\nTwists 0,1,0,0/0,1,0,1 1,0,0,0/0,1,0,0 1,0,0,0/0,1,0,1\nCuts 0.1235 -0.1235\nAxis 0,0.56903559,0.14014940,0\nTwists 1,0,0,0/0,0,0,1\nCuts 0.621\n
    Triangular Antitegmatic Icoschoron Triangular Antitegmatic Icosachoron
    Puzzle Triangular-antitegmatic_Icosachoron\nDim 4\nNAxis 1\nFaces 1,0,0,0\nGroup 1,0,0,0/0.5,0.866025404,0,0 0,0.577350269,0.816496581,0/0,0,0.612372436,0.790569415\n#1,0,0,0/0.5,sqrt(3)/2,0,0 0,1/sqrt(3),sqrt(2/3),0/0,0,1/sqrt(6),sqrt(5/6)\nAxis 1,0,0,0\nTwists 0,0.577350269,0.816496581,0/0,0,0.612372436,0.790569415 0,0.790569,-0.559017,0.25/0,0.57735,1.22474,0.912871\nCuts 0.75 -0.75\n
    5D Puzzles Penteract Family 1x1x1x1x2 1x1x1x1x2
    Puzzle 1x1x1x1x2\nDim 5\nNAxis 2\nFaces 0.5,0,0,0,0 0,0,0,0,1\nGroup 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/1,0,0,1,0 1,0,0,0,0/0,0,0,0,1\nAxis 1,0,0,0,0\nTwists 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,0,0,0,1\nCuts\nAxis 0,0,0,0,1\nTwists 1,0,0,0,0/1,1,0,0,0\nCuts 0\n
    1x1x1x2x2 1x1x1x2x2
    Puzzle 1x1x1x2x2\nDim 5\nNAxis 2\nFaces 0.5,0,0,0,0 0,0,0,1,0\nGroup 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1\nAxis 1,0,0,0,0\nTwists 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1\nCuts\nAxis 0,0,0,1,0\nTwists 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/0,0,0,0,1\nCuts 0\n
    1x1x2x2x2 1x1x2x2x2
    Puzzle 1x1x2x2x2\nDim 5\nNAxis 2\nFaces 0.5,0,0,0,0 0,0,1,0,0\nGroup 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0 0,0,1,0,0/0,0,1,0,1\nAxis 1,0,0,0,0\nTwists 0,1,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0\nCuts\nAxis 0,0,1,0,0\nTwists 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1\nCuts 0\n
    1x2x2x2x2 1x2x2x2x2
    Puzzle 1x2x2x2x2\nDim 5\nNAxis 2\nFaces 0.5,0,0,0,0 0,1,0,0,0\nGroup 1,0,0,0,0/0,1,0,0,0 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,1,0,1,0 0,1,0,0,0/0,1,0,0,1\nAxis 1,0,0,0,0\nTwists 0,1,0,0,0/0,1,1,0,0\nCuts\nAxis 0,1,0,0,0\nTwists 1,0,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0\nCuts 0\n
    2x2x2x2x2 2x2x2x2x2
    Puzzle 2^5\nDim 5\nNAxis 1\nFaces 1,0,0,0,0\nGroup 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/1,0,0,1,0 1,0,0,0,0/1,0,0,0,1\nAxis 1,0,0,0,0\nTwists 0,1,0,0,0/0,1,1,0,0\nCuts 0\n
    Other Simplex Prism Simplex Prism
    Puzzle Simplex_Prism\nDim 5\nNAxis 2\nFaces 0,0,0,0,1 1,0,0,0,0\nGroup 1,0,0,0,0 0,0,1,1,0/0,0,1,-1,0 0,2,-1,-1,0/0,1,1,-2,0 0,2,-2,0,0/0,1,1,-1,-2.236068\nAxis 0,0,0,0,1\nTwists 1,0,0,0,0/0,1,-1,0,0 0,0,1,1,0/0,0,1,-1,0 0,2,-1,-1,0/0,1,1,-2,0\nCuts 0\nAxis 1,0,0,0,0\nTwists 0,2,-1,-1,0/0,1,1,-2,0\nCuts 0\n
    {3,3}x{4} {3,3}x{4}
    Puzzle {3,3}x{4}\nDim 5\nNAxis 2\nFaces 1,1,1,0,0 0,0,0,1.73205081,0\nGroup 1,1,0,0,0/1,0,-1,0,0 1,1,0,0,0/0,1,-1,0,0 0,0,0,1,0/0,0,0,1,1\nAxis 1,1,1,0,0\nTwists 0,0,0,1,0/0,0,0,1,1 1,-1,0,0,0/1,0,-1,0,0 1,-1,0,0,0/0,0,0,1,0\nCuts -0.33333\nAxis 0,0,0,1,0\nTwists 1,1,0,0,0/1,0,-1,0,0 0,0,0,0,1/1,-1,0,0,0\nCuts 0\n
    "},{"location":"software/magicpuzzleultimate/#making-your-own-puzzles","title":"Making your own puzzles","text":"

    If you figure out the notation, you can try adding your own puzzles. You define the symmetry group, rotational axes, cutting planes, possible twists and puzzle boundary.

    For some definitions of various hypercuboids, see hypercuboids.

    There is also a general formula for 4D duoprism puzzles, made by Luna: {p}x{q}

    Puzzle {p}x{q}\nDim 4\nNAxis 2\nFaces 1,0,0,0 0,0,1,0\nGroup 1,0,0,0/1,tan(pi/p),0,0 0,0,1,0/0,0,1,tan(pi/q)\nAxis 1,0,0,0\nTwists 0,1,0,0/0,0,0,1 0,0,1,0/0,0,1,tan(pi/q)\nCuts ...\nAxis 0,0,1,0\nTwists 0,0,0,1/0,1,0,0 1,0,0,0/1,tan(pi/p),0,0\nCuts ...\n

    "},{"location":"techniques/","title":"Solving Techniques","text":"

    Hypercubing methods typically rely on a few fundamental techniques. By mastering these, you can even develop your own solutions to new puzzles.

    Learn Notation before trying to learn any of these techniques

    • F2L
    • Commutators
    • RKT
    "},{"location":"techniques/F2L/","title":"F2L","text":"

    First 2 Layers (F2L) is a solving technique for cubes and some other puzzles that involves pairing pieces and inserting those pairs into slots to build up the first 2 layers (of a 3-layered puzzle). The first known description of this technique was by John Conway in 1979, but is most commonly credited to Jessica Fridrich due to the massive popularity her website describing the CFOP method had in the early 1990s.

    The goal of this page is to give you a generalized understanding of higher dimensional F2L, not to hand out algorithms for certain cases. There are so many cases for higher dimensional F2L that it would be nearly impossible to list out all the cases. It\u2019s recommended to have a fairly strong intuitive understanding of 3D F2L before reading this page.

    Dimensional Analogy

    In 3D F2L, whenever you twist a side axis, a 1x1x3 block of pieces (containing the F2L slot) goes to the top layer (essentially a 1D line). In 4 dimensions, 2 adjacent sides intersect at a 2D plane. So when you do a twist you\u2019re bringing a 1x1x3x3 block to the top layer, which contains 3 F2L pairs: that being two F2L-b slots and an F2L-a slot. This makes inserting a single pair trickier, because now you need to use more moves to not disturb the other two subslots.

    Continuing this analogy into 5D, you bring a whole 3D cube\u2019s worth of F2L slots to the top (1x1x3x3x3) that you mustn\u2019t distrub while inserting a single pair. This pattern continues for higher and higher dimensions, meaning that it gets trickier to insert a single pair. Pro solvers of higher dimensional puzzles will wait until they have 3 (or multiples of 3) pairs, then pairs all those pairs into a gigapair, and then insert that gigapair like inserting a lower dimensional pair.

    Terminology

    To talk about F2L in a dimensionally neutral way, we need some more terms. Most importantly, the following:

    • Head: the piece in the pair with the most colours (corner in 3D)
    • Body: the piece in the pair with one less colour (edge in 3D)

    For more terms, see the glossary.

    Regardless of dimension and type of pair, they are broken down into the following categories:

    1. both in top, head facing side axes
    2. both in top, head facing top axis
    3. body in slot, head in top facing side axis
    4. body in slot, head in top facing top axis
    5. head in slot facing side axis, body in top
    6. head in slot facing base axis, body in top
    7. both stuck in slot
    "},{"location":"techniques/F2L/#f2l-4a","title":"F2L-4a","text":"

    F2L-a pairs consist of a 2c and a 3c, exactly the same as in 3D. The solutions feel very similar to 3D cases, but there are a few extra tricks that are possible.

    Insert

    Normal F2L insert: IU RU IU' RU'

    Hide and reveal

    Notice: same color on top.

    Can either use RKT on the I cell to pair it or use hide and reveal (preferred).

    RU IU RU' //pair\nIU2 RU IU' RU' //insert\n
    3-mover

    3-move insert. Still works if the body is at IU, IF, or ID except you\u2019ll just need an extra RKT move.

    Notice how the F sticker of the pair matches the F cell colour. (If it doesn\u2019t, then the pair will be flipped after inserting!)

    Capping

    • Body stuck in slot, solved
    • Head has base color on top
    • Cap head on top of body
    • Push, rotate pair, pull

    Solution: IU2 IF RU IR2 RU'

    "},{"location":"techniques/commutators/","title":"Commutators","text":"

    A commutator (or \u201ccomm\u201d) is a sequence of moves where you do a sequence A, then a sequence B, then the inverse of A, and finally the inverse of B. Usually, you want the intersection of the pieces affected by sequence A and the pieces affected by sequence B to be small but non-empty. In turn, creating a commutator with those sequences makes it easier to solve a puzzle by only affecting a few specific pieces at a time.

    One of the earliest solutions for the 3x3x3x3 was The Ultimate Solution to a 3x3x3x3 by Roice Nelson, which made heavy use of intuitive setup moves and commutators. Commutators were also much easier to use in MC4D due to its Macro feature, which allowed users to record sequences of moves and then replay them later from any angle.

    "},{"location":"techniques/commutators/#notation","title":"Notation","text":"

    Commutator notation extends traditional twisty puzzle notation by adding the following definitions:

    • Commutators are notated [A, B], which expands to A B A' B'.
    • Conjugates are notated [A: B], which expands to A B A'.

    Here are some example commutators that can be executed on a 3x3x3:

    • [R, U] expands to R U R' U'.
    • [M', U2] expands to M' U2 M U2, which cycles 3 edges on the M slice.
    • [D: [M', U2]] expands to D (M' U2 M U2) D', which cycles 3 slightly different edges. Conjugates are often combined with commutators to influence different sets of pieces.
    • [F: [R, U]] expands to [F: R U R' U'], which expands further to F (R U R' U') F'.
    • [D, [R, U]] expands to D (R U R' U') D' (U R U' R'), which cycles 3 corners on the F layer.

    (Parentheses are added for readability but have no special meaning.)

    "},{"location":"techniques/commutators/#strategy","title":"Strategy","text":"

    Once you find commutators for cycling different types of pieces on a puzzle (2c, 3c, etc) then you can almost solve the whole thing! (There may be some situations where you need an algorithm to swap 2 pieces of a type and 2 of a different type.)

    "},{"location":"techniques/rkt/","title":"RKT","text":"

    You must know 34 notation in order for this page to make sense!

    RKT is a technique that lets you treat a single cell of a (cell-turning) higher dimensional puzzle as if it were a lower dimensional puzzle. This is very useful to do moves that damage fewer pieces. For example: RKT lets you treat a side of a 34 just like a 33 cube, meaning that you can use all the 3D algorithms you already know to solve the full 4D puzzle.

    It has been invented independently several times, but was mainly popularized by Raymond Zhao in his article here.

    "},{"location":"techniques/rkt/#naming","title":"Naming","text":"

    Other proposed names:

    • SFM (Single Facet Manipulation)
    • FRM (Facet Redution Method)

    The name RKT comes from the move set <RK,T*> on n4 puzzles (in the old notation, the Inside and Outside cells were called Top and Kata). In the new notation it should be called ROI, but the old name stuck.

    Grant

    Also it\u2019s the part of the solve where you can get ReKT

    "},{"location":"techniques/rkt/#doing-moves-with-rkt-n4","title":"Doing moves with RKT (n4)","text":"

    RKT lets us do any n3 sequence of moves on one side of an n4. The beginner\u2019s way to learn this is by \u201ctranslating\u201d 3D algorithms into RKT (although this is not the best way to think about how RKT works beacuse you can really perform any sequence of moves). Below is a simple 3 step guide to translate your algorithms into RKT.

    Example: Sune algorithm with RKT

    Say you want to do the Sune algorithm R U R' U R U2 R' with RKT. The first step is to try and think of how you can execute the algorithm using only R moves and cube rotations. R U R' U R U2 R' becomes R z R z' R' z R z' R z R2 z' R'. Now we replace R with RO, and cube rotations with I cell rotations: RO IF RO IF' RO' IF RO IF' RO IF RO2 IF' RO'

    1. Rewrite the algorithm to only use cube rotations and R moves
    2. Rewrite cube rotations in this new algorithm (x, y, or z) to their respective I cell rotation (IR, IU, IF)
    3. Replace all R moves with their RO counterpart

    This can definitely be very confusing and disorienting to beginners, as you have to think about 3x3x3 moves with only R moves and rotations. It takes a lot of practice to get comfortable with RKT, but it is an extremely powerful technique.

    The best way to learn RKT before doing your first 34 solve is by manually scrambling the last cell using any RO moves and random I cell rotations until it looks scrambled. Then practice doing PLC by solving it with RKT.

    Tip

    Even though this technique is called RKT, it is not limited to those types of turns. Thus, you can use LO and I- moves instead, or rotate your view and use RU and D- instead of RK and I-.

    Another popular way to do RKT is to use wide O cell turns (instead of normal I cell turns). This has the benefit of keeping the cell you\u2019re doing RKT to in the same orientation the entire time. Instead of rotating a side of the I cell to the RI plane, you rotate all the outside layers over the side that you\u2019re twisting.

    "},{"location":"techniques/rkt/#parity","title":"Parity","text":"

    RKT Parity

    RKT parity is a state you can get to that appears to rotate a single layer of a cell by 180\u00b0. If you try and fix this with RKT, then the outer layers will be off by 180\u00b0. This means that you must use an algorithm (or intuition) to solve it.

    You can use a 3D supercube algorithm that rotates the U center 180\u00b0 ((R U R' U)5, or (L R U2 R' L' U)2).

    Harder to memorize (but much lower in movecount) is this 9-mover RKT parity alg: IU UR IU' IF' UO' IF RF UR RF' UIR, found by Tetrian22.

    • 34 Commutators List by Alvin (RKT parity algs are near the bottom)
    • PLL + RKT parity algs by Eff

    On 24, the algorithm is shorter because it doesn\u2019t have to worry about messing up other pieces besides corners. A commonly used one is R2 B2 R2 U R2 B2 R2 U'RKT.

    On bigger n4 puzzles (where \\(n>3\\)), it can look like a single slice layer of a cell is off by 180\u00b0. An intuitive way to solve this is to do the 24 RKT parity algorithm with wide moves, and then the normal 34 RKT parity algorithm. It can also be avoided by just lining up your slice the same way you line up centers in 3D before finishing last 4 edges (when using freeslice).

    "},{"location":"techniques/rkt/#debt","title":"Debt","text":"

    RKT Debt is when the R cell isn\u2019t aligned with the rest of the puzzle aftering performing an algorithm or sequence of moves with RKT. For example: after executing a T perm algorithm with RKT the R cell will be misaligned by 90\u00b0. RKT Debt always has to be \u201cpaid back\u201d at some point during the solve. During complicated setup moves for fancy inserts, RKT debt can be used as \u201cammo\u201d. That is, undoing the debt in a useful way to help solve the puzzle. An easy way for beginners to \u201cavoid\u201d debt is to just do a move on a layer that you don\u2019t care about messing up at that point in the solve. For example: in a CFOP solve during the final F2L-b pair insertion, you could do any U* move that fixes the debt.

    "},{"location":"techniques/rkt/#cancels","title":"Cancels","text":"

    Sune with RKT: left = normal RKT, right = RKT cancel

    RKT cancelling is a technique that reduces the move count of certain RKT algorithms and triggers by abusing symmetry of rotations. HactarCE made a program called RocKeT to find cancels for 34 algorithms. Often, it just involves inserting some flipping moves at certain points throughout the algorithm.

    Consider R U R' U'. Conventional RKT rotates after every move, but we don\u2019t have to do that. We can build up multiple moves of RKT debt and then cancel them later:

    • RO UO \u2014 do R U using RKT, building up two moves of RKT debt (R U)
    • IF RO' \u2014 do R' using RKT, undoing the debt from U
    • IF2 UO' \u2014 do U' using RKT, undoing the debt from the R

    But we can do even better!

    • RO UO \u2014 do R U using RKT
    • IUR \u2014 swap R with U
    • UO' RO' \u2014 do R' U' using RKT (because we swapped R and U), undoing the debt from R U

    If you squint, you might notice that this is a conjugate [RO UO: IUR]. This corresponds to the fact that R U R' U' can be written as a conjugate where one part is a pure rotation: [R U: z x2]. When executing this algorithm, we apply it to I with the rotation (so it\u2019s equivalent to R U R' U') and to the outer layers without the rotation (so [RU: _] expands to R U U' R', which completely cancels out). This is the fundamental theory behind how to do RKT cancels: by rewriting algorithms as conjugates and commutators with pure rotations.

    Recall how in 3D you can replace a U y' with Dw. We can do the same thing here: replace IUR with {1-2}OUR, which is a rotation of the outer layers instead of the inner layers. Now it just looks like a normal algorithm with a flip thrown in the middle: RO UO {1-2}OUR RO' UO'. So we can notate it a little more simply: R U {1-2}OUR R' U'

    Here\u2019s a more complex example, the Sune algorithm: RO UO RO' {1-2}OUR UO RO {1-2}OUR UO2 RO'. This works because you can rewrite Sune using conjugates with rotations: [R U: [R': z x2] [U: z x2]]. Notice how if you remove the rotations, the whole thing cancels out. (You don\u2019t need to expand the conjugates to see this.)

    "},{"location":"techniques/rkt/#rkt-cancel-algorithms","title":"RKT Cancel Algorithms","text":"

    These won\u2019t help you solve a puzzle for the first time

    You do not need to learn RKT cancel algs to solve any puzzle. RKT can and should be done completely intuitively unless you are already speedsolving at a high level and wish to improve this step.

    • See Commutator notation first.

    • Single letter moves are the \u201cbig 3d\u201d moves. (R = RO, U = UO, etc.)

    • TODO: explain xyz notation and {1-2}O notation in notation page. put a big caveat that its purpose is for execution, and it\u2019s totally arbitrary as far as what\u2019s actually going on with the puzzle.

    44 edge swap parity Image Description RKT Cancel Algorithm UF and UR edges swapped [f' l': [[r' U' l': D2], Iy2]] OCLL Image Name RKT Cancel Algorithm Sune R U R' {1-2}Ozx2 U R {1-2}Ozx2 U2 R' Antisune R U2 {1-2}Ozx2 R' U' {1-2}Ozx2 R U' R' H R U2 R' U' {1-2}Ozx2 R U R' U' {1-2}Ozx2 R U' R' U (R U R' U R U2 R') Iy (R U2 R' U' R U' R') Iy' T (R U R' U R U2 R') Iy' (R U2 R' U' R U' R') Iy L (R U R' U R U2 R') Iy2 (R U2 R' U' R U' R') Iy2 Pi R U2 {1-2}Ozx2 R2 U' R2 U' R2 {1-2}Ozx2 U2 R CPLL Image Name RKT Cancel Algorithm Aa (Ix) R' U {1-2}Oz R' D2 R {1-2}Oz' U' R' D2 {1-2}Oz' R2 Ab (Ix) R2 {1-2}Oz D2 R U {1-2}Oz R' D2 R {1-2}Oz' U' R Na R U' L U2 R' U L' {1-2}Oy2 R U' L U2 R' U L' TODO Nb EPLL Image Name RKT Cancel Algorithm Ub R2 U R U {1-2}Ozx2 R' U' R' U' R' {1-2}Ozx2 U R' Ua R U' {1-2}Ozx2 R U R U R {1-2}Ozx2 U' R' U' R2 Z (R U' R U R U R U' R' U' R2) Iy2 (R2 U R U R' U' R' U' R' U R') Iy2alternate: M2 U' M U2 M' Iy2 M U2 M' U M2 H M2 U' M2 U2 M Iy2 M' U2 M Iy2 M U M2"},{"location":"techniques/rkt/#simultaneous-rkt","title":"Simultaneous RKT","text":"

    RKT can be done by using 2 opposite sides (e.g. RO and LO turns) as well as normal I cell turns. This would allow you to execute <R,U,L> gen algorithms easier, at the cost of having to fix RKT debt on both layers. Another interesting way to do this is in a method like Belt Method. After solving the belt, orienting opposite sides, and separating the colours, the user is left with solving 2 opposite cells that can be solved simultaneously using RKT. If you turn one of the sides with the belt going through it, it essentially does a twist to both cells. One case that can be annoying is if you want to do a 180\u00b0 turn on one side, and a 90\u00b0 turn on the other (you would have to wait until both cells\u2019 debt lined up).

    "},{"location":"techniques/rkt/#higher-dimensional-rkt","title":"Higher Dimensional RKT","text":"

    The term Double/Triple/Quadruple/etc\u2026 RKT is used to refer to using RKT to do RKT in 5D+ puzzles. Because RKT lets us treat a single side of an nd like an nd-1, using RKT on the 35 lets us treat a single 4D cell as a 34. And if you know how to use RKT to treat a 34 like a 33, then you can do Double RKT. This generalizes to any number of dimensions, but the movecount doubles each time, making it impractical. This is why bigger nd puzzles are mostly solved using commutators.

    "},{"location":"techniques/rkt/#rkt-in-3d","title":"RKT in 3D","text":"

    If RKT treats a single layer of an nd puzzle like an nd-1 puzzle, then technically the last step of the 33 Roux method counts as RKT. The last step of Roux is to permute the M-slice like a 32 (with mirroring moves allowed).

    Another thing that \u201cfeels like RKT\u201d is solving 3x3xn cuboids where \\(n>3\\). A typical strategy for these is to solve from the innermost layers to the outermost layers, treating it as several nested 3x3x2 puzzles. Doing an R2 on the whole puzzle does an R2 to each of the subpuzzles.

    "},{"location":"theory/gods-number/","title":"God\u2019s Number","text":"

    God\u2019s number is the minimum number of moves that is sufficient to solve a twisty puzzle from any starting position. For 3x3x3, this has been proven to be 20 HTM (or 26 QTM).

    God\u2019s number for 33 took lots of creative mathematical work and 35 years of CPU time to prove \\(\\sim 4.3 \\times 10^{19}\\) states reachable in 20 moves or less. For comparison, the 24 has \\(\\sim 3.4 \\times 10^{27}\\) states and 43 has \\(\\sim 7.4 \\times 10^{45}\\) states. There isn\u2019t a single nontrivial 4D puzzle for which God\u2019s number is known, let alone remotely possible to compute.

    There are three strategies we can use to estimate it:

    1. Setting a lower bound using the branching factor of move sequences
    2. Setting an upper bound by analyzing the worst-case solution of every stage in a given method
    3. Estimation by measuring move counts produced by a near-optimal solver
    "},{"location":"theory/gods-number/#summary","title":"Summary","text":"Puzzle Lower Bound Upper Bound Estimate 24 15 STM 37 STM ~20 \u00b15 34 51 STM 570 STM ~125 \u00b150"},{"location":"theory/gods-number/#2222","title":"2x2x2x2","text":""},{"location":"theory/gods-number/#lower-bound","title":"Lower bound","text":"

    We can compute a lower bound of 15 in the STM by computing a bound on the number of positions reachable by algorithms of a certain length.

    Derivation

    Moves here will be measured using STM. Our focus here is on positions that can be reached by algorithms of a certain length. Note that on the 24, we consider one piece fixed when counting permutations. There are \\(\\sim 3.4 \\times 10^{27}\\) positions on the 24.

    One position, the solved position, is reachable without making any moves. There are 23 moves on each of the 4 cells that do not affect the fixed piece, so at most 92 one-move positions are reachable by 1 move. For subsequent moves, there are only 69 non-canceling moves since any move on the same axis as the previous move will cancel. Therefore there are at most \\(92 \\times 69^{k-1}\\) reachable by exactly \\(k\\) moves (where \\(k\\ge 1\\)). Let \\(P_n\\) denote the number of positions reachable in \\(n\\) moves or less. Based on the previous analysis,

    \\[P_n \\le 1+\\sum_{k=1}^n 92 \\times 69^{k-1} = \\frac{23}{17} (69^n-1)+1.\\]

    Let \\(n_G\\) be God\u2019s number. By definition, \\(P_{n_G}\\) must be equal to the total number of positions. Combining this with the bound on \\(P_n\\), we find

    \\[P_{n_G} = 3.4\\times 10^{27} \\le \\frac{23}{17} (69^{n_G}-1)+1,\\]

    which implies \\(n_G \\ge 15\\).

    "},{"location":"theory/gods-number/#upper-bound","title":"Upper bound","text":"

    We can compute an upper bound of 37 in the STM by computing the worst-case move count of a known method.

    Derivation

    Hypersolve uses a method that has a worst-case of 39 STM. Furthermore, it has been shown that all cases with move counts more than 37 STM are avoidable. Therefore God\u2019s number is at most 37 STM.

    "},{"location":"theory/gods-number/#estimate","title":"Estimate","text":"

    Hypersolve typically produces solutions in the range of 20-30 STM. Note that this solver does not produce optimal solutions1, but based on this, God\u2019s number for 24 is probably not higher than 20-30.

    "},{"location":"theory/gods-number/#conclusion","title":"Conclusion","text":"

    God\u2019s number for 24 is definitely between 15 and 37 inclusive, and probably around \\(\\sim 20 \\pm 5\\).

    Could this be improved?

    • A better method or lots of compute time might improve slightly on the upper bound. More worst-case scenarios can be checked for avoidability.
    • Unless there is some fundamental breakthrough in our understanding of computation, there\u2019s likely no way to improve on the lower bound. If you\u2019re an expert in quantum computing then perhaps you can devise some clever quantum algorithm to help, but as of 2023 quantum computers haven\u2019t solved a single real-world problem faster than a classical computer so we remain skeptical.
    "},{"location":"theory/gods-number/#3333","title":"3x3x3x3","text":""},{"location":"theory/gods-number/#lower-bound_1","title":"Lower bound","text":"

    We can compute a lower bound of 51 in the STM. This works by showing that algorithms of limited length can generate, at most, only a subset of the possible positions on the 34. We will also demonstrate some known optimizations (often useful for other puzzles), although they do not immediately improve the lower bound in the case of 34.

    Lower Bound, Winning Ways Method

    Turns here are measured in the Slice Turn Metric (STM).

    We can describe turns on the \\(3^4\\) as happening on one of four axes, and with one of three layers. Each layer can be turned in 23 ways, so we have \\(4 \\times 3 \\times 23 = 276\\) turns measured as one move in the STM. After turning a layer, we want subsequent turns to be noncancelling (turning the same layer twice can be written as a single move), so we have \\(276-23=253\\) choices for subsequent turns.

    From here, we can proceed without the Winning Ways improvement, where we find the number of positions reachable by 50 turns or fewer is at most

    \\[1 + 276 \\times \\sum_{k=0}^{49}253^k \\approx 1.57 \\times 10^{120} < \\frac{\\left(24!\\times 2^{24}\\right) \\times \\left(32! \\times 6^{32}\\right) \\times (16! \\times 12^{16})}{48} \\approx 1.78 \\times 10^{120}.\\]

    This shows there are positions on the \\(3^4\\) which require 51 turns or more to solve. In the book Winning Ways for Your Mathematical Plays Vol. 4, the authors optimize this argument (as it applies to the Rubik\u2019s Cube in the half turn metric) by accounting for relations like \\(LR = RL\\). We can apply a similar optimization for the \\(3^4\\).

    We will refer to the set of possible new positions reachable after exactly \\(n\\) turns as \\(T_n\\). \u201cNew\u201d here means that \\(T_n\\) does not include positions that were in \\(T_{n-1}\\). \u201cExactly\u201d means that positions reachable in more than \\(n\\) turns are not in \\(T_n\\). The number of elements in \\(T_n\\) is \\(|T_n|\\).

    By the computation from before, there are 276 distinct puzzle states after a single turn. So \\(|T_1|=276\\). Finding \\(|T_2|\\) is more complicated. First, we need to setup a more detailed way of describing turning axes.

    A primary axis is a line through the core of a puzzle in any of the four cardinal directions. Primary axes have three layers, which we might label \\(-1,\\ 0,\\) and \\(1\\). The white-yellow primary axis\u2019 three primary layers would correspond to the yellow cell (\\(-y\\)), white cell (\\(y\\)), and the slice between the white and yellow cells. A \\textbf{secondary} axis is a line through the center of a facet and any of the pieces in that facet. Importantly, the secondary axis can be described using a single cardinal direction when it is through a ridge piece (the \\(90^\\circ\\) degree cell turns and their doubles). A primary axis, secondary axis, and an angle which we turn by is enough to describe any turn in the STM.

    We have just 253 noncancelling chocies for a second turn. Some of these second turns commute with a first turn, which means we will overcount postions reachable by two algorithms differing by the order of commuting turns. Noncancelling commuting pairs of turns can happen in two ways:

    1. The first and second turns happen in the same primary axis and on different layers (for example, \\(UO\\ \\ \\{2\\}UO' = \\{2\\}UO'\\ \\ UO\\).\\

    2. The first turn has primary and secondary axes \\(a\\) and \\(b\\), while the second turn has primary and secondary axes \\(b\\) and \\(a\\) (for example, \\(OL\\ LO= LO\\ OL\\)).

    In describing commuting cases of the first kind, the second turn is one that happens on the same primary axis as the first and on a different layer. There are two other layers, and we can make one of \\(23\\) turns on that layer. This means that after a single turn is made, \\(2\\times 23 = 46\\) of the following turns may lead to commuting cases of the first kind. However, two of these actually lead to single turn puzzle states (like \\(UO\\ \\{2\\}UO = DO\\)) which were counted in \\(T_1\\) and should not be part of \\(T_2\\). So, only 44 of these turns lead to distinct new puzzle states. This means there are \\(\\frac{44\\times 276}{2}\\) distinct new puzzle states reachable by two commuting moves of this first kind.

    Commuting cases of the second kind can only happen if the first turn\u2019s secondary axis goes through a ridge piece. There are \\(108\\) turns like this. The second turn\u2019s primary/secondary axes are determined by transposing the first turn\u2019s primary/secondary axes, and then can happen in one of three ways on one of three layers. So, we have \\(\\frac{9\\times 108}{2}\\) distinct new puzzle states reachable by two commuting moves of this second kind. NOTE: If the first turn is one of the \\(276-108= 168\\) other turns that cannot be followed by a commuting turn of the second kind, those nine second turns happen on a different primary axis and will neither cancel nor commute.

    Now we can compute an upper bound on \\(|T_2|\\). There are \\(\\frac{44\\times 276}{2}\\) distinct new puzzle states reachable by two commuting moves of the first kind, and \\(\\frac{9\\times 108}{2}\\) distinct new puzzle states reachable by two commuting moves of the second kind. There are \\(276-23-44-9=200\\) noncancelling, noncommuting second turns. In total, we have \\(|T_2| \\leq \\frac{44\\times 276}{2} + \\frac{9\\times 108}{2} + 9\\times 168 + 200\\times 276 = 63,270\\). As mentioned in the \u201cNOTE\u201d, following one of the other 168 first moves (with secondary axis \\textit{not} through a ridge peice) with one of those nine turns suggested by commuting case of the second kind will never cancel or commute, so we add \\(9\\times 168\\) to complement the \u201cleftovers\u201d that were ignored in the \\(\\frac{9\\times 108}{2}\\) term.

    To complete the argument, we can derive a recurrence relation (more accurately a recurrence estimate) that gives an upper bound on the size of \\(|T_{n+2}|\\) in terms of \\(|T_n|\\) and \\(|T_{n+1}|\\).

    In the case where the \\(n+2\\)th turn will not commute or cancel with the \\(n+1\\)th turn, we have at most \\(9\\times 168|T_{n}| + 200\\times |T_{n+1}|\\) possible puzzle states. In the case where the \\(n+2\\)th turn \\textit{will} commute (but still not cancel) with the \\(n+1\\)th turn, we have at most \\(\\left(\\frac{44\\times 253}{2} + \\frac{9\\times 108}{2}\\right)|T_n|\\) possible puzzles states. This gives us the estimate \\(|T_{n+2}| \\leq 200|T_{n+1}| + 7,582|T_n|.\\)

    This is an order 2 linear recurrence estimate with constant coefficients. We can show that

    \\[|T_n| \\leq A\\lambda_1^n + B\\lambda_2^n,\\]

    where

    \\[ \\begin{array}{c c} \\lambda_{1}=100+\\sqrt{17,582}, & \\lambda_{2}=100-\\sqrt{17,582}, \\\\ A=\\displaystyle \\frac{4,035}{7,582}+\\frac{160,704}{3,791}\\sqrt{\\frac{2}{8,791}}, & B=\\displaystyle \\frac{4,035}{7,582}-\\frac{160,704}{3,791}\\sqrt{\\frac{2}{8,791}}. \\\\ \\end{array} \\]

    Finally, the number of turns reachable by 50 turns or fewer is at most

    \\[1+\\sum_{n=1}^{50}|T_n| \\leq 2.52\\times 10^{118} < \\frac{\\left(24!\\times 2^{24}\\right) \\times \\left(32! \\times 6^{32}\\right) \\times (16! \\times 12^{16})}{48} \\approx 1.78 \\times 10^{120}.\\]

    This shows there are positions on the \\(3^4\\) which require 51 turns or more to solve.

    Unfortunately, we did not improve the lower bound here. It is worth noting that in the estimate from the beginning (with few optimizations), the actual value of the sum on the LHS is quite close to the number of scrambles possible on the \\(3^4\\) (within the same order of magnitude). Compare this to our new estimate, where the sum is bounded above by a number two orders of magnitude smaller than the number of scrambles possible on the \\(3^4\\). So as we would have expected, our improvements do reduce the size of the estimate, just not quite enough to bring down the number of turns derived.

    Below is an example of a similar argument used to derive a lower bound of 56 turns in the OBTM. Note that this argument is missing some optimizations that were applied in the STM lower bound derivation.

    OBTM Lower Bound, Winning Ways Method

    Moves here will be measured using something equivalent (as far as this discussion is concerned) to OBTM. Our focus here is on positions that can be reached by algorithms of a certain length. Wide moves contribute to algorithm length in the same way that single cell turns do (wide move = single cell move + cube rotation, where cube rotations count as 0 moves), so we will make a simplification by restricting to single cell turns. There are 23 moves on each of the 8 cells, so 184 one-move algorithms are possible at any given time.

    We know the 34 can be scrambled in over 1.7 novemtrigintillion ways. When solving, we have 184 choices for the first turn, and 161 (non cancelling) choices for the following turns. The number of positions reachable by 54 turns or fewer is at most

    \\[1 + 184 \\times \\sum_{n=0}^{53} 161^n \\approx 1.695 \\times 10^{119} < 1.757 \\times 10^{120}.\\]

    This shows that there exist positions on the 34 that require 55 or more turns to solve.

    In the book Winning Ways for Your Mathematical Plays Vol. 4, the authors optimize this argument (as it applies to the Rubik\u2019s Cube in the half turn metric) by accounting for relations like \\(LR = RL\\). We can apply a similar optimization for the 34.

    We will refer to the number of possible 34 positions after \\(n\\) moves as \\(u_n\\). It is not too hard to convince ourselves that \\(u_1 = 184\\). After turning one cell, moving the same cell again is what we might call a \u201ccancelling move\u201d, since the two moves could have been measured as a single move. So, we have \\(161\\) non cancelling choices for a second move. We should be able to reduce that \\(161\\) further by avoiding double counting positions reachable by two algorithms which differ only by the order of commuting elements.

    There are different ways that a pair of moves can commute on 34. One is by the cell turns affecting completely distinct subsets of pieces, which happens when turning opposite cells, like \\(OL\\ IL = IL\\ OL\\). This is analogous to the commuting cases on the Rubik\u2019s Cube as addressed in Winning Ways. There is at least one more way, which can involve certain adjacent cell moves. \\(OL\\ LO = LO\\ OL\\) is one example. We have not yet reached a description of all commuting cases of this kind. Accounting for these may allow us to further improve the lower bound. For now, we will factor out the commutativities that we can describe, which we will call \u201copposite-commuting\u201d moves.

    First, there are \\(161-23 = 138\\) possible non-cancelling, non-opposite-commuting choices for the second move. We\u2019ll add the opposite-commuting cases separately.

    There are four \u201cflavors\u201d of opposite-commuting cases: I cell moves with O cell moves, R moves with L moves, U moves with D moves, and B moves with F moves. In any of these, choose any one of the 23 moves possible on each cell (order doesn\u2019t matter), so there are \\(23 \\times 23 = 529\\) opposite-commuting cases of each flavor. Then across the four flavors, we have \\(529\\times 4 = 2,116\\) distinct cases after the second move, when it opposite-commuted with the first. So we have that the number of \\(3^4\\) positions after \\(n\\) moves is at most \\(138u_1 + 2,116 = 27,508\\). That is, we can say \\(u_2 \\leq 27,508\\).

    As a sanity check, we could note at this point the \\(27,508\\) positions after two turns is better than the \\(184\\times 161 = 29,624\\) positions we would have estimated without accounting for opposite-commutativity of two successive turns here.

    To complete the argument, we can derive a recurrence relation (more accurately a recurrence estimate) that gives us an upper bound on the size of \\(u_{n+2}\\) in terms of \\(u_n\\) and \\(u_{n+1}\\)

    In the case where the \\(n+2\\)th move will not opposite-commute (or cancel) with the \\(n+1\\)th move, we have \\(138u_{n+1}\\) possible puzzle states. If the \\(n+2\\)th move is going to opposite-commute with the \\(n+1\\)th move, we might have \\(529\\times 4 = 2,116\\) cases to follow up the \\(n\\)th move. However, one of those four flavors of opposite commuting cases would have cancelled with the \\(n\\)th move, so this time we only have \\(529 \\times 3 = 1,587\\) cases to follow the \\(n\\)th move. This gives us the recurrence estimate \\(u_{n+2} \\leq 138u_{n+1} + 1,587u_{n}\\).

    Note that as we have defined it, \\(u_n\\) is the number of positions accessible by exactly \\(n\\) moves. If we sum the \\(u_n\\)s from 1 to \\(N\\), that will gives us (an upper bound on) the number of positions accessible by \\(n\\) moves or fewer.

    In this case, our recurrence estimate is an order 2 linear recurrence with constant coefficients. There exists a method to derive a general formula for \\(u_n\\). In the end, we find

    \\[u_n = C\\lambda_1^n + D\\lambda_2^n,\\]

    where

    \\[ \\begin{array}{c c} \\lambda_{1}=69+46\\sqrt{3}, & \\lambda_{2}=69-46\\sqrt{3}, \\\\ C=\\displaystyle \\frac{1}{4}\\left(-22+13\\sqrt{3}\\right), & D=\\displaystyle \\frac{1}{4}\\left(-22-13\\sqrt{3}\\right). \\\\ \\end{array} \\]

    Finally, the number of positions reachable by 55 turns or fewer is at most

    \\[1\\ +\\ \\sum_{n=1}^{55}u_n \\approx 3.865 \\times 10^{118} < 1.757 \\times 10^{120}.\\]

    This shows that there exist positions on the \\(3^4\\) that require 56 or more turns to solve.

    We should address the other side of the inequality,

    \\[1.757 \\times 10^{120} <1\\ +\\ \\sum_{n=1}^{56}u_n \\approx 5.746 \\times 10^{120}.\\]

    So, the argument as applied here can\u2019t necessarily show that there are positions requiring 57 or more turns to solve.

    "},{"location":"theory/gods-number/#upper-bound_1","title":"Upper bound","text":"

    Hactar computed a generous upper bound of 570 STM using CFOP.

    Step-by-step breakdown
    • cross: 6 pieces x 4 STM per piece = 24 STM
    • F2L-a: 12 pairs x 13 STM per pair = 156 STM
      • 3 STM to put corner on I (with corner first-cell color not on I)
      • 3 STM to put edge on I
      • 3 STM to pair + 2 STM to fix RKT debt
      • 4 STM to insert
    • F2L-b: 8 pairs x 26 STM per pair = 208 STM
      • 6 STM to put corner on I (with corner first-cell color not on I)
      • 6 STM to put edge on I
      • 4 RKT = 7 STM to pair them and fix RKT debt
      • 7 STM to insert
    • ridge OLC: worst case (all ridges bad) is 17 STM
    • edge OLC:
      • solve at least 3/4 pieces in each slice using OCLL: 3 slices x 11 STM = 33 STM
      • worst case is 3 unsolved edges remaining: 5 RKT + 9 STM = 18 STM
        • 2 RKT to arrange any two edges so that they are adjacent in the same slice but do not form a valid 2-corner OLL case
        • 2 RKT to place the last edge in a way that forms a valid 3-corner OLL case
        • 1 RKT to fix RKT debt
        • 9 STM to solve the OLL case (consider OCLLs with at least one corner solved)
    • corner OLC: 58 STM
      • 11 RKT = 23 STM to solve a 2^3 into a state where the D and U layer each contains a valid 3-corner OLC case
        • if there is exactly one remaining corner unsolved, then misorient two corners when solving one of those 3-corner OLC cases.
        • if there are exactly two remaining corners unsolved, then misorient one corner when solving one of the 3-corner OLC cases.
        • arrange the cases so that the 3 unsolved corners left will form a sune
      • 13 STM to solve each OLL case on U/D using RKT cancels (worst case is double antisune)
      • 9 STM to solve the remaining sune case
    • ridge PLC: 14 STM (2x U perm in the worst case)
    • PLC: 42 STM
      • solve a 3^3 using God\u2019s algorithm in 20 moves
      • adjust I cell to solve
    "},{"location":"theory/gods-number/#estimate_1","title":"Estimate","text":"

    We do not have a near-optimal 34 solver. Good FMC speedsolves average 200 STM, so God\u2019s number is probably lower than that.

    "},{"location":"theory/gods-number/#conclusion_1","title":"Conclusion","text":"

    God\u2019s number for 34 is definitely between 56 and 570 inclusive, and probably \\(\\sim 125 \\pm 50\\)[citation needed].

    Could this be improved?

    • A better method could easily improve the upper bound, probably to around 300[citation needed].
    • A better manual computation could give a slightly better lower bound.
    • An automated 34 solver or search program could give a much better estimate.

    1. It does eventually converge on optimal solutions when run for a sufficient amount of time, but this amount of time is impractical for all but the simplest scrambles.\u00a0\u21a9

    "},{"location":"theory/grip_theory/","title":"Grip theory","text":"

    Grip theory is a framework for analyzing many different kinds of twisty puzzles in a geometry-independent way. It was first developed on the TwistyPuzzles forum in 2009 in this thread. Grip theory was originally used to describe doctrinaire puzzles in 3D, but it can be extended to describe bandaged puzzles, jumbling puzzles, higher-dimensional puzzles, and more.

    "},{"location":"theory/grip_theory/#basic-grip-theory","title":"Basic grip theory","text":""},{"location":"theory/grip_theory/#the-333","title":"The 3x3x3","text":"

    Consider a cube in 3-dimensional euclidean space. Rotations of this 3-dimensional space form a group: \\(SO(3)\\). This group is said to be acting on the 3D space. Consider a subset of these rotations that sends the cube to itself. These rotations form a subgroup of order 24, which is isomorphic to the symmetric group \\(S_4\\), although it goes by many other names. To see how this subgroup acts on the cube, we can define the 6 sides of the cube (as subsets of 3D space), and label them e.g. \\(R\\), \\(U\\), \\(F\\), \\(L\\), \\(D\\), and \\(B\\). Because we restricted our attention to rotations that preserve the cube, the image of any side under any rotation will always also be one of the sides. Thus we can define an action of the group on the 6-element set of sides \\(\\{R, U, F, L, D, B\\}\\).

    A piece of the 3x3x3 can be thought of as a subset of the cube, and it is incident with one or more of its faces. Corners are incident with 3 faces, edges with 2, centers with 1, and the core with 0. At any time each piece has 3 important properties:

    • Set of faces with which it was incident when solved. This defines which piece it is, and is reflected in e.g. the stickers that are on the piece.
    • An element of the aforementioned rotation group which sends the piece from its solved state to its current state.
    • Set of faces with which the piece is incident now. This determines whether the piece moves or not when a particular twist is performed.

    These properties are redundant: suppose we call these \\(X, g, Y\\) respectively. Then \\(Y = \\{ x \\cdot g \\mid x \\in X\\}\\), or simply \\(Y = X \\cdot g\\); and likewise and \\(X = Y \\cdot g^{-1}\\).

    When we perform a twist, we choose a face \\(f\\), and a rotation \\(r\\) of the cube that sends the chosen face to itself: \\(f \\cdot r = f\\). Then for each piece one of two things happen:

    • If the piece is currently incident with \\(f\\), it gets acted on by \\(r\\), sending \\((X, g, Y) \\mapsto (X, gr, Y \\cdot r)\\).
    • If the piece is currently not incident with \\(f\\), it remains unchanged.

    For example when we do an R twist, it\u2019s as if the pieces currently incident with the \\(R\\) face underwent an x cube rotation, and other pieces remained static.

    Note that when twisting around \\(f\\), we didn\u2019t change whether any piece is incident with \\(f\\), meaning the twist can be inverted by using the same face \\(f\\) and the rotation \\(r^{-1}\\).

    "},{"location":"theory/grip_theory/#the-grips-of-the-333","title":"The grips of the 3x3x3","text":"

    Instead of working with geometric shapes embedded in 3D space, we can work with abstract sets. Consider a set \\(\\Gamma\\) of 6 abstract objects, called grips:

    \\[\\Gamma = \\{R, U, F, L, D, B\\}.\\]

    Next consider a group acting on this set, generated by two 4-cycles that cycle \\(F, U, B, D\\), and \\(R, F, L, B\\) respectively. More precisely, we mean the subgroup \\(G\\) of \\(S_\\Gamma\\) generated by \\((F,U,B,D)\\) and \\((R,F,L,B)\\). We will call this the grip group. This group and its action are isomorphic to the rotations of the cube and their action on the 6 faces we defined above.

    A piece is identified by a set \\(X \\subseteq \\Gamma\\) \u2013 the set of solved grips. At any time a piece has some attitude \\(g\\), which is an element of the grip group \\(G\\). From that we can also compute the set of active grips, which is defined to be \\(Y = X \\cdot g\\). The grip group acts on the piece like so:

    \\[(X, g, Y) \\cdot r = (X, gr, Y \\cdot r),\\]

    so the active grips follow the piece.

    The attitude of a piece with active grip set \\(Y\\) can be decomposed into two components: the orientation of the piece, which is an element of the stabilizer subgroup \\(\\operatorname{Stab}(Y)\\); and the position, which is the set of active grips \\(Y\\) itself. By the orbit-stabilizer theorem we can identify the orbit \\(X \\cdot G\\) (the possible positions of this piece) with the cosets in \\(G / \\operatorname{Stab}(Y)\\).

    For example, \\((\\{R, U\\}, (U,R)(L,D)(F,B), \\{R, U\\})\\) is an abstract description of the up-right edge in its solved position, but wrong orientation. We can verify its set of active grips:

    \\[\\{R, U\\} \\cdot (U,R)(L,D)(F,B) = \\{R, U\\}.\\]

    Note that the groups \\(\\operatorname{Stab}(X)\\) and \\(\\operatorname{Stab}(X \\cdot g) = \\operatorname{Stab}(Y)\\) are conjugate, and thus isomorphic, and in general we have a fiber bundle:

    \\[\\operatorname{Stab}(X) \\hookrightarrow G \\rightarrow G / \\operatorname{Stab}(X).\\]

    Next, define a twist to be a pair of: a grip \\(f\\) we twist around, or axis, and element of \\(\\operatorname{Stab}(f)\\), the transformation. Equivalently, we can choose an arbitrary transformation \\(r \\in G\\) first, and then an axis \\(f\\) from the set \\(\\Gamma^r\\) of fixed points of \\(r\\). As before, the action of a twist \\((f, r)\\) on a piece is:

    \\[(X, g, Y) \\mapsto \\begin{cases} (X, gr, Y \\cdot r), & \\text{if } f \\in X \\cdot g, \\\\ (X, g, Y), & \\text{if } f \\notin X \\cdot g. \\\\ \\end{cases}\\]

    For instance, applying the twist with axis \\(U\\) and transformation \\((R,F,L,B)\\), which fixes \\(U\\), to the piece from above, we get a new attitude

    \\[(U,R)(L,D)(F,B) * (R,F,L,B) = (R,U,F)(L,D,B),\\]

    and a new active grip set

    \\[\\{U, R\\} \\cdot (R,F,L,B) = \\{U, F\\}.\\]

    The set of all positions our piece can move to are all sets of two grips except \\(\\{R, L\\}\\), \\(\\{U, D\\}\\), and \\(\\{F, B\\}\\). These exactly correspond to the edge positions of the 3x3x3. An edge piece is only affected by a twist if one of its colored faces is on the face corresponding to the axis of the twist, and after the twist, the positions of its colored faces moves. This is exactly the behavior of the active grips. Therefore, the piece above is an edge piece.

    We can similarly construct three more sets of pieces: the corners, with active grips like \\(\\{R, U, F\\}\\), the centers, with active grips like \\(\\{R\\}\\), and the core, with empty active grip set. Note that even though the centers have only one grip, they still have attitude, so the centers on this puzzle are actually super centers.

    "},{"location":"theory/grip_theory/#states","title":"States","text":"

    Above we have chosen some set of pieces \\(P\\), a subset of \\(2^\\Gamma\\). Notably it is an invariant subset under the action of the grip group: \\(P \\cdot G = P\\). The state of the puzzle comprises of the states of all pieces, i.e. some set of triples \\((X \\in P, g \\in G, Y \\in P)\\) such that \\(X \\cdot g = Y\\). This defines a relation between \\(P\\), \\(P\\), and \\(G\\). Due to the redundancy, there are three ways to look at it:

    • A function from \\(X\\) to \\(g\\), called \\(a\\), indicating that the piece \\(X\\) has attitude \\(a(X)\\), and has active grips \\(Y(X) = X \\cdot a(X)\\).
    • A function from \\(Y\\) to \\(g\\), also called \\(g\\), indicating that the piece whose active grips are \\(Y\\) has attitude \\(g(Y)\\) and its solved grips were \\(X(Y) = Y \\cdot g(Y)^{-1}\\).
    • A permutation of \\(X\\)\u2019s into \\(Y\\)\u2019s, \\(\\pi : P \\to P\\), together with a choice of \\(g\\) at each fiber such that \\(\\pi(X) = X \\cdot g\\).

    Denoting the set of such relations \\(G \\wr P\\), the first description allows us to straightforwardly define a group structure on it. If we treat elements of \\(G \\wr P\\) as functions from \\(P\\) to \\(G\\) in the first sense, then given \\(a, b \\in G \\wr P\\), their product \\(ab\\) is defined by:

    \\[ab(X) = a(X) b(X \\cdot a(X)).\\]

    The second and third descriptions allow us to identify \\(G \\wr P\\) as a subgroup of the wreath product \\(G \\wr_P S_P\\) (hence the notation). Indeed, suppose \\(g, h : P \\to G\\) are maps sending active grips to attitudes, and \\(\\pi, \\sigma \\in S_P\\) are permutations of pieces. Defining multiplication as:

    \\[(\\pi, g) * (\\sigma, h) = (\\pi\\sigma, Y \\mapsto g(\\sigma^{-1}(Y)) h(Y)),\\]

    we end up with the group structure of the semidirect product \\(S_P \\rtimes G^P\\), with the action of reindexing, i.e. exactly the wreath product \\(G \\wr_P S_P\\). The group \\(G \\wr P\\) is then the subgroup consisting of pairs \\((\\pi, g)\\) that agree:

    \\[\\forall X \\in P, \\pi(X) = X \\cdot g(\\pi(X)).\\]

    Note that the \\(\\pi\\)\u2019s are limited to those that preseve the orbits \\(P / G\\), else there doesn\u2019t exist an agreeing \\(g\\). The wreath product thus splits into a product of independent groups, one for each orbit. In turn, for a piece with active grips \\(Y\\), the set of agreeing \\(g(Y)\\)\u2019s is some coset of the stabilizer \\(\\operatorname{Stab}(Y)\\) in \\(G / \\operatorname{Stab}(Y)\\). Within an orbit, all these stabilizers are isomorphic, thus we can write \\(G \\wr P\\) as:

    \\[\\prod_{O \\in P / G} \\operatorname{Stab}(Y \\in O) \\wr_O S_O,\\]

    provided we choose a designated element \\(Y\\) in each orbit \\(O\\), and an identification of \\(\\operatorname{Stab}(Y)\\) with its cosets.

    For our 3x3x3, there are four orbits, corresponding to the four types of piece we defined above:

    • 12 edges with stabilizers \\(\\mathbb{Z}_2\\),
    • 8 corners with stabilizers \\(\\mathbb{Z}_3\\),
    • 6 faces with stabilizers \\(\\mathbb{Z}_4\\),
    • and 1 core with stabilizer \\(G\\).

    The group of puzzle states is thus:

    \\[(\\mathbb{Z}_2 \\wr S_{12}) \\times (\\mathbb{Z}_3 \\wr S_8) \\times (\\mathbb{Z}_4 \\wr S_6) \\times (G \\wr S_1).\\]"},{"location":"theory/grip_theory/#twists","title":"Twists","text":"

    A twist acts on a piece, and thus can act on the puzzle state \u2013 a set of pieces. Because the set of states \\(G \\wr P\\) admits a group structure, the action of a twist \\((f, r)\\) is actually equivalent to multiplication on the right by a certain element of \\(G \\wr P\\) \u2013 the state you get if you apply the twist to the solved puzzle:

    \\[\\pi(X) = \\begin{cases} X \\cdot r, & \\text{if } f \\in X, \\\\ X, & \\text{if } f \\notin X; \\\\ \\end{cases}\\] \\[g(Y) = \\begin{cases} r, & \\text{if } f \\in Y, \\\\ 1, & \\text{if } f \\notin Y. \\\\ \\end{cases}\\]

    This defines a group homomorphism \\(\\operatorname{twist}_f : \\operatorname{Stab}(f) \\to G \\wr P\\). To be able to talk about a sequence of twists about different axes, we can define the twist group as the free product (coproduct) of the groups of twists about each axis:

    \\[\\coprod_{f \\in \\Gamma} \\operatorname{Stab}(f),\\]

    which comes with a free homomorphism \\(\\operatorname{twist} : \\coprod\\limits_{f \\in \\Gamma} \\operatorname{Stab}(f) \\to G \\wr P\\), which simply applies multiple twists in sequence.

    Often not all puzzle states are reachable by twists. The image of this homomorphism is exactly the set of puzzle states that are. This is a subgroup of \\(G \\wr P\\) and is called the puzzle group.

    Various reasons why the puzzle group is smaller than \\(G \\wr P\\) are called piece invariants. For example, no twist on a cube ever relocates a center, thus in the puzzle group\u2019s \\(\\mathbb{Z}_4 \\wr S_6\\) component, the \\(S_6\\) permutation is always the identity. For more complicated reasons, in the \\(\\mathbb{Z}_2 \\wr S_{12}\\) component, the sum across all copies of \\(\\mathbb{Z}_2\\) must be \\(0\\) \u2013 this is the edge orientation parity.

    "},{"location":"theory/grip_theory/#complex-puzzles","title":"Complex puzzles","text":"

    We do not need to be limited to the pieces that exist on the physical 3x3x3. We can take pieces corresponding to every possible subset of the grip set, thus \\(P = 2^\\Gamma\\), which has \\(2^6 = 64\\) pieces. The puzzle you get when you include all the pieces of another puzzle in this sense is called the complex version of that puzzle.

    The complex 3x3x3 has 10 types of pieces, which include the core, centers, edges, and corners from the 3x3x3. It also includes the double-axles, with active grips like {R, L}, and the bridges, with active grips like {R, U, L}. The other types of pieces are generated by the anti- automorphism of this puzzle. The anti- automorphism takes every piece with active grip set \\(A\\) to the piece whose active grip set is \\(A^c\\), its complement. Given a twist with axis \\(a\\) and transformation \\(g\\), the composition of two actions of the puzzle: the first action is a rotation of every piece by \\(g^{-1}\\), and the second is the original twist. When viewed on an actual 3x3x3, this automorphism maps moves to wide moves of the opposite faces. The anti- version of a piece is the image of a piece under the anti- automorphism. Thus, we can call the other pieces of the complex 3x3x3 anti-edges, anti-double-axles, anti-centers, and the anti-core. Anti-corners and anti-bridges are isomorphic to the corners and bridges, so they do not need to be recounted.

    "},{"location":"theory/grip_theory/#more-puzzles","title":"More puzzles","text":"

    This definition is immediately generalizable by replacing the set of grips \\(\\Gamma\\) and the grip group \\(G\\). For instance, using 12 grips and the grip group of a dodecahedron, we can create a megaminx, and by including all pieces, we can construct the complex megaminx, which has \\(2^{12} = 4096\\) pieces of 96 types1.

    We can also take 8 grips, R, U, F, O, L, D, B, and I, with the grip group being the 192 symmetries of the hypercube. This allows us to construct the 34. One notable feature of this group that is not present in the previous cases is that when we take two adjacent grips (e.g. R and U), there are non-identity elements of the grip group that fix both of these, even though there are also non-identity elements that only fix one of them. This means that RKT is possible on this puzzle.

    "},{"location":"theory/grip_theory/#more-layers","title":"More layers","text":"

    More layers can be added to a puzzle by adding more grips. For instance, to build a 5x5x5, we need two grips for every grip of the 3x3x3, which we can call 1R and 2R, and similarly for the others. The grip group will act identically on the layer-1 grips and the layer-2 grips, but will not swap them. In this way, either layer of a face of the 5x5x5 can be selected by choosing which grip on that face to use. The complex 5x5x5 can be constructed out of this grip system.

    Odd-layered puzzles can be constructed like this, but even-layered puzzles are less clear. They can be constructed like the odd-layered puzzles with one more layer, but this results in identical grip systems and complex variants. Alternatively, we can use stored grips, but this results in less symmetrical puzzles.

    "},{"location":"theory/grip_theory/#group-automorphisms","title":"Group automorphisms","text":"

    There are more automorphisms that these puzzles can have. An inner automorphism of the grip group, which is a conjugate by an element of the group, can be extended to an automorphism of the puzzle by interpreting the inner automorphism as a whole-puzzle rotation. More interesting is what happens when we use a non-inner automorphism. The one we choose should be realizable as conjugation by a permutation of grips, though not necessarily one in the group. We consider the 12-grip complex megaminx. The grip group of this puzzle has an outer automorphism of reflection. By applying this outer automorphism, we obtain a reflection of the whole puzzle, showing that the puzzle is not chiral, even though some pieces are. More interesting is the other outer automorphism of this group, which manifests as an exotic automorphism that takes adjacent grips on the dodecahedron to non-adjacent but non-opposite grips and vice versa. This automorphism is called the co- automorphism, and it shows that complex puzzles can have a higher degree of symmetry than the geometric puzzle they were based on.

    In general, if we have the grip set \\(\\Gamma\\) with group \\(G\\) acting on it, the normalizer \\(N_{S_\\Gamma}(G)\\) is the subgroup of the automorphism group of \\(G\\) of automorphisms that can be realized as acting on \\(\\Gamma\\). Thus, the cos of the grip system are elements of the group \\(N_{S_\\Gamma}(G)/G\\). This group is a subgroup of \\(\\operatorname{Out}(G)\\), and manifests as additional symmetries of the grip system and the complex puzzle created with it. For example, the cube has two cos: the identity and the reflection. The dodecahedron has four: the identity, the reflection, and the co mentioned above and its inverse. That co applied twice is a reflection, so in a way, it can be considered a square root of the reflection.

    "},{"location":"theory/grip_theory/#advanced-example","title":"Advanced example","text":"

    Suppose we have 7 grips, labeled 001, 010, 011, 100, 101, 110, and 111, which are acted on by \\(GL_3(\\mathbb{F}_2)\\) in the standard way. This group has order 168. The action of the grip group on the grips can be thought of as the Fano plane: a line is a set of 3 grips where one is the XOR-sum of the other two, and the group is all permutations of the grips that map lines to lines. Using this, we can build the complex puzzle and identify its piece types. First, there is clearly the core, with empty active grips, and 7 centers, with only one active grip each (e.g. {001}). Then, among the 15 2-grip pieces, there is only one orbit since the action is 2-transitive, so there is only one piece type, for instance that with active grip set {001, 010}. Among 3-grip pieces, there are only two types: those that contain three grips on a line like {001, 010, 011}, and those that don\u2019t, like {001, 010, 100}. There are 7 of the former piece type and 28 of the latter type. Finally, for 4-, 5-, 6-, and 7- grip pieces, we can use the anti- automorphism to observe that these pieces are exactly the anti- variants of the 0- through 3-grip pieces. Thus, we have classified all the pieces of the complex Fano plane puzzle.

    "},{"location":"theory/grip_theory/#extensions-to-grip-theory","title":"Extensions to grip theory","text":""},{"location":"theory/grip_theory/#infinite-grips","title":"Infinite grips","text":"

    Consider a set of grips \\(\\mathbb{Z}^2\\), whose grip group are the rotations and translations that preserve the lattice. A puzzle constructed like this would have infinitely many twists, and would be unwieldy. A way to simplify this puzzle is to restrict the axes that twists are allowed on. Now, a grip that is not under an allowed twist is called stored. For instance, if twists are only allowed with axes \\((0,0)\\) and \\((1,0)\\), you can make a selection of pieces to make a two-circle puzzle like the Rashkey. However, the complex Rashkey would still have uncountably many pieces, many of which can reach infinitely many attitudes. A solution to this is to only allow pieces with finite active grip sets. If the active grip set of a piece is finite and there are only a finite number of allowed axes, each with finitely many transformations corresponding to a turn, the piece can only reach finitely many attitudes, despite the infinitude of the grip group.

    "},{"location":"theory/grip_theory/#bandaging","title":"Bandaging","text":"

    The 3x3x3 is a doctrinaire puzzle. If a corner {R, U, F} and edge {R, U} are bandaged together, the puzzle ceases to be doctrinaire. This can be implemented into grip theory by adding a new kind of grip to pieces: bandaged grips. Bandaged grips follow the attitude just like active grips. We remove the corner and edge above and replace them with a new piece with active grips {R, U} and bandaged grips {F}. When we apply a twist, if its axis would turn a piece along a bandaged grip, the entire twist is illegal in this state. This exactly mimics the bandaging of the 3x3x3, because by removing F from the set of active grips of the corner and the set of inactive grips of the edge, it prevent us from performing any twist that would only move one of the two pieces.

    "},{"location":"theory/grip_theory/#jumbling","title":"Jumbling","text":"

    We can combine these two notions. Consider the curvy copter (ignore overhang bandaging for now). Clearly, it has at least 12 grips, corresponding to the edges of a cube, with a grip group that contains the rotations of a cube acting on them. However, it should also have jumbling moves, rotations of \\(\\arccos\\frac{1}{3}\\) around one of the 12 allowed axes. If we take the image of all the grips under repeated application of this type of rotation, we end up with a countably infinite dense set of grips with a countably infinite grip group. Each piece of the curvy copter can be compared with every grip to identify its active and bandaged grips. This produces a description of the curvy copter in grip theory. This description can be simplified so there are only finitely many active and bandaged grips per piece.

    "},{"location":"theory/grip_theory/#non-faithful-actions","title":"Non-faithful actions","text":"

    In all previous examples, the action on the grip group has been faithful. However, this does not have to be the case. Consider the rotational symmetry of the cube, the octahedral group. This group is a quotient of the binary octahedral group of order 48, so the binary octahedral group can act on 6 grips via its quotient. The resulting puzzle has the same pieces as the complex 3x3x3, but each piece can have twice as many additudes, and thus twice as many orientations in a given position. Because the binary octahedral group can be interpreted as a group of spinors, the puzzle with the 3x3x3\u2019s pieces but the binary octahedral grip group is the quaternion cube.

    "},{"location":"theory/grip_theory/#multiplex-puzzles","title":"Multiplex puzzles","text":"

    Rather than simply assigning pieces a set of active grips, it is possible to create a set of grips combined with endomorphisms of the transforms associated with each grip. This allows for a puzzle like the 3x3x3 where some pieces perform an R while others perform R2 or R'. A full multiplex puzzle can be built by including a piece for every combination of these grip-endomorphism pairs for a given set of endomorphisms. A simple example is the Triplex Tetrahedron on this page.

    Of particular note are puzzles using automorphisms of the transforms, as these preserve symmetry between different versions of the same grip. These include combination puzzles like Entangled 3x3x3s.

    "},{"location":"theory/grip_theory/#lamination","title":"Lamination","text":"

    Lamination is a restriction and generalization of grip theory. It allows fewer types of pieces than are described by grip theory, but the pieces it describes are closer to those of actual puzzles. It has multiple related interpretations.

    "},{"location":"theory/grip_theory/#group-action-interpretation","title":"Group action interpretation","text":"

    Suppose, as above, there is a grip group that acts on the set of grips. We now select a block system of this action, and each block is called an axis. Now, instead of a piece having a selection of active grips, a piece will have a layer on each axis, where a layer is an assignment of a symbol (WLOG element of \\(\\mathbb{Z}\\)) to each grip in that axis. Then, a twist still has an axis, but now it has a selected layer within that axis, as well as a transformation, an element of the grip group, that pointwise stabilizes the axis. A laminated puzzle is one that, given a selection of axes an selection of layers on each axis, contains a piece for every combination of layers. In this way, it is similar to complex puzzles.

    For instance, to construct a 3x3x3, first start with the 6 grips R, U, F, L, D, and B. Then, let the axes be x = {R, L}, y = {U, D}, and z = {F, B}. We can choose a set of layers for each axis: the layers of x will be the assignments {R\u21a60, L\u21a60}, {R\u21a61, L\u21a60}, and {R\u21a60, L\u21a61}, and similary for y and z. Since there are 3 layers per axis and 3 axes, the laminated 3x3x3 has \\(3^3 = 27\\) pieces, exactly as many as a 3x3x3. In fact, the laminated 3x3x3 is exactly the super 3x3x3.

    "},{"location":"theory/grip_theory/#representation-interpretation","title":"Representation interpretation","text":"

    Now, we still have a grip group, but instead of acting on a set of grips, it acts linearly on a vector space \\(V\\). The axes are a set of linearly independent subspaces that span \\(V\\) (which implies the \\(V\\) is the direct sum of the axes), and this set should be setwise stabilized by the grip group. A layer is a point in a subspace. A piece as before has one layer per axis, but now, since layers are points in orthogonal subspaces, we can also identify a piece with a point in \\(V\\) whose projections onto the subspaces are its layers. Thus, a twist, being a selection of axis \\(a\\), layer \\(\\lambda\\), and transformation \\(g\\), has the effect of selecting all pieces in an affine space passing through \\(\\lambda\\) parallel to the direct sum of all axes that are not \\(a\\), and applying \\(g\\) to them.

    Constructing the 3x3x3 is even more geometrically intuitive than before. Now, we have three axes x, y, z, which correspond to the three coordinate axes of \\(\\mathbb{R}^3\\). The grip group acts on \\(\\mathbb{R}^3\\) as the rotational symmetries of the cube, and the layers of x will be the surfaces \\(x=0\\), \\(x=-1\\), and \\(x=1\\), and similarly for y and z. Like this, the 27 pieces of the laminated 3x3x3 form an arrangement just like that of the actual 3x3x3. However, this approach is not limited to 3 dimensions, and can be used to construct all the puzzles of doctrinaire grip theory.

    "},{"location":"theory/grip_theory/#more-layers_1","title":"More layers","text":"

    Laminated puzzles, as the name suggests, are made of layers, so adding more layers is trivial. Moreover, even-layered puzzles can be constructed as naturally as odd-layered ones. This makes them a more natural description of these puzzles.

    1. https://www.gcsu.edu/sites/files/page-assets/node-808/attachments/powers.pdf \u21a9

    "},{"location":"theory/invariants/","title":"Piece invariants","text":"

    Many puzzles exhibit a phenomenon where the position or orientation of some pieces are determined by the position or orientation of the other pieces. This can often be captured with an invariant, a function of position that does not change when moves are applied, and calculating the invariant can identify whether a position is valid or not. Invariants are useful when computing the total numer of positions of a puzzle.

    The most common kinds of invariants are permutation parity and total orientation.

    "},{"location":"theory/invariants/#permutation-parity","title":"Permutation parity","text":"

    The permutation parity of a permutation is the parity of the number of swaps needed to produce it. It is exhibited well in k-Card Game.

    "},{"location":"theory/invariants/#monoflip","title":"Monoflip","text":"

    Monoflip is a phenomenon where a single corner piece on the 3x3x3x3 can be flipped in place, with two 2-swaps of stickers, while all other corners are solved. Conversely, if 15 corners on the puzzle are solved, the last corner must either be solved or in one of 3 orientations of the monoflip. This effect is unique to 4D hypercubes. In 3D, having 7 solved corners implies the last corner must also be solved, and in 5D and above, the orientation of the last corner is independent of the orientations of the others.

    The monoflip exists because the group of orientations of a single corner, ignoring others, is \\(A_4\\), and its commutator subgroup is the Klein four-group \\(\\mathbb{Z}_2 \\times \\mathbb{Z}_2\\). Each element of this group corresponds to either the identity or a monoflip. Its quotient in \\(A_4\\) is \\(\\mathbb{Z}_3\\), which represents an invariant analogous to total corner twist in 3D. This will allow us to prove that the orientation of the last corner, given that the other 15 are solved, must be identity or monoflip.

    First, we construct a particular set of 8 ridges by choosing two sets of 4 parallel ridges which are mutually orthogonal (for instance, choose two disjoint rings of 4 cells each, and select the ridges which lie between two cells of the same ring). We say these ridges are red. We can take two more sets of 8 ridges the same way both disjoint from the first one, and we call those ridges green and blue. Notice that if we perform a monoflip on a corner colored like this, the arrangement of colors does not change.

    If we pick a particular corner position, we can call its position even. Then, we specify that positions adjacent to even positions are odd, and positions adjacent to odd positions are even. This is well-defined and unique given our original choice. Coordinate-wise, if the coordinates of the corners are in \\(\\{-1,1\\}^4\\), corners whose coordinates have product \\(1\\) can be even and those with product \\(-1\\) can be odd. Then, in any position of the 3x3x3x3, we define the variant of a particular corner based on two factors: the color in the solved position of where the red ridge of the corner is, and whether the position of the corner is even or odd.

    Even Odd Red 0 0 Green 1 2 Blue 2 1

    In effect, in an odd position, the roles of green and blue are swapped. By adding the variant of each corner, we get the invariant of the position.

    We now analyze how the invariant changes under turns. Since quarter turns form a generating set, we only need to consider those. Each quarter turn fixes one color of ridge and swaps the other two. It also takes corners in even positions to odd positions and vice versa, which swaps blue and green (for the purposes of computing the variant). Thus, it effectively either keeps all colors fixed or 3-cycles them. This has the effect of adding \\(k\\) to the variants of even-positioned corners and subtracting \\(k\\) from the odd-positioned corners for some \\(k \\in \\mathbb{Z}_3\\). Thus, the invariant stays fixed. Since the invariant of the solved position is \\(0\\), it is \\(0\\) in any position and so if 15 corners are solved, the last one can only be in a variant-\\(0\\) position, that is, identity or monoflip.

    "},{"location":"theory/physical/","title":"Physical puzzles","text":"

    Higher-dimensional (\u201cvirtual\u201d) puzzles naturally exist in more than 3 dimensions. Physical puzzles are a way to make a 3-dimensional object that functions similarly to this higher-dimensional puzzle. They must be able to perform the moves of the puzzle, so everything that can be done on the virtual puzzle can be done on the physical puzzle. However, most physical puzzles are able to perform other moves that are not allowed on the virtual puzzle, and these must be disallowed by the rules of operating the puzzle. The more rules a puzzle requires, the more complicated operating it becomes, and it becomes less interesting. The most interesting and natural physical puzzles implement as many of the restrictions as possible as part of the mechanism.

    Warning

    The relationship between virtual and physical puzzles is reversed from what it usually is in 3 dimensions! In the normal case, the physical puzzle has a mechanism that restricts the possible moves, and the unrestricted moves become legal, and a virtual simulation of the puzzle should copy the moves from the physical puzzle. However, when constructing a virtual higher-dimensional puzzle, the legal moves are defined abstractly, and the physical puzzle needs to have restrictions applied to it by rules.

    The simplest way to construct a physical puzzle is to create the most unrestricted object possible, and apply all restrictions via the rules. We can make one physical piece for each sticker of the virtual puzzle, and observe on the virtual puzzle the permutation of the stickers when applying one move. When we allow this set of moves on the physical puzzle, we get sticker soup. This is the simplest way to construct a physical puzzle, and since it has no mechanism at all, the rules of operation are just the permutation. This is generally considered uninteresting.

    Since in a virtual puzzle, stickers are part of pieces and cannot be separated, it makes sense to make this a restriction of a physical puzzle as well. In this case, a physical piece is a virtual piece. This applies some design constraints to the puzzle. On the virtual puzzle, there are generally moves that reorient a piece, which has the effect of permuting its stickers. This produces the orientation group of the piece. A physical puzzle built like this must have pieces that can represent that symmetry group.

    "},{"location":"theory/physical/#melindas-24","title":"Melinda\u2019s 24","text":"

    The 24 has 16 pieces, each with 4 stickers. By performing moves on a virtual 24, a piece can be brought back to its position with any even permutation of stickers. (It is not just the monoflip because that assumes the rest of the puzzle is solved, which is not important here.) In three dimensions, a tetrahedron with its four vertices representing the stickers has the same symmetry. (This is not a coincidence; a 24 built on the surface of a hypersphere will have its pieces be spherical tetrahedra with the stickers at the vertices.) Thus, the pieces of Melinda\u2019s 24 have tetrahedral symmetry.

    This is not the only design decision resulting in a usable physical puzzle. Since the pieces of the 24 can be arranged in two cubic layers, the pieces are cubes to allow them to stick together in this fashion. This makes many of the moves simply rotating one whole cube of 8 pieces, and the others rotate 4 pieces from each layer in a simple way. These moves are unable to swap a white or yellow sticker with a sticker of a different color, which means the puzzle simulates a cubic prism puzzle: a 23x21 cuboid, where the first three axes can turn into each other but the last one cannot in the same way that a 3x3x3x2 is restricted.

    The virtual 24, on the other hand, does allow moves that take a white or yellow sticker to a sticker of a different color, and the pieces were designed so that this is possible. It would be possible to observe each move that does that, record which piece goes where, and apply that to the physical 24, but this is unwieldy. Instead, Melinda introduced a gyro, a sequence of actions that does the equivalent of a whole-puzzle rotation on the virtual puzzle, but takes a white sticker to a sticker on a different axis. This means other moves are accessible by performing the gyro and then a simple move. On this puzzle, a gyro is usually about six physical twisting actions, but on other puzzles, they can be more complicated, so finding a puzzle that supports a short gyro is valuable.

    "},{"location":"theory/physical/#restricted-25","title":"Restricted 25","text":"

    The pieces of a virtual 25 have 60 orientations, the even permutations of its 5 stickers. It is difficult to design a piece with sufficient symmetry to represent this, so the restricted 25 sets one axis to be unswappable with the others, i.e. it is a 24x21 in the notation of the previous section. This means the pieces only need to be able to evenly permute 4 of their stickers, which is the same symmetry as the pieces of the physical 24. The pieces from that can thus be reused to build the restricted 25, with one sticker added to each piece. The extra sticker needs to be invariant under all rotations of the piece, so it needs to be applied to every face of the piece.

    The design of the puzzle is simple: two copies of the physical 24, where the legal moves are doing a whole puzzle rotation on one, doing the same twist on both, or swapping 8 pieces from each half by twisting a side of the puzzle. The design requires a buffer layer (pictured above in purple), because if it were not present, doing a 90\u00b0 rotation of the whole puzzle corresponds to three moves on the virtual restricted 25, which is considered illegal in solves.

    "},{"location":"theory/physical/#unrestricted-25","title":"Unrestricted 25","text":"

    It is not impossible to design a piece with 5 stickers that supports all even permutations of its stickers. The symmetry group of this piece is called \\(A_5\\), the alternating group on 5 points, and it is isomorphic to the symmetry of the icosahedron. It is not obvious whether there are actually 5 objects permuted by this, but there are: the 5 tetrahedra of the compound of 5 tetrahedra, or the 5 cubes of the compound of 5 cubes. Any even permutation of the 5 components of either shape can be realized as a rotation of the whole shape.

    The compounds of 5 tetrahedra and 5 cubes1

    Observe that in the compound of 5 cubes, if you take one cube as fixed, the other four cubes each share one pair of opposite vertices with it. In particular, the arrangement of these 4 other cubes has the same symmetry as the 4 stickers of the piece of the physical 24, or the 4 exchangeable stickers of the restricted 25. Thus, by designing a piece based on this object, the true physical 25 can be implemented that shares all the moves of the restricted 25, along with one new gyro move. Because of the complexity of the puzzle, the gyro is very difficult to execute.

    25 gyro, by Akkei

    "},{"location":"theory/physical/#26","title":"26","text":"

    The virtual 26 has pieces, which have 6 stickers each. Each piece must be able to perform all even permutations of all 6 stickers, which results in the group \\(A_6\\) of order 360. A rigid piece in 3 dimensions must have as its symmetry a chiral point group. However, the only discrete chiral point groups in 3 dimensions of order 360 all contain a rotation by \\(\\frac{\\pi}{90}\\), which has order 180. \\(A_6\\) has no elements of order 180, so none of these point groups are isomorphic to \\(A_6\\), and a physical 26 with rigid pieces is impossible. (The lowest dimension to have an object with the requisite symmetry is 5, with the 5-simplex.)

    This does not preclude other designs for a physical 2^6 that have not been developed yet. For instance, a skewb has 6 center pieces, and all even permutations of them are accessible with skewb twists. This means that skewbs can be used as pieces for a physical 26. Whether there is a design for it that does not require manipulating the individual pieces every turn is unknown.

    "},{"location":"theory/physical/#restricted-26","title":"Restricted 26","text":"

    There are several ways to restrict a 26 to make it able to be made physical. One way is to make it a 25x21, which can be made physical out of two copies of the 25 in the same way as the 24x21 was made out of the 24. Another way is the 24x22, pictured above. Each piece of the 24x22 has four 4-stickers and two 2-stickers. The four 4-stickers can have an even permutation applied to them, or the two 2-stickers can be swapped and an odd permutation can be applied to the four 4-stickers. This enlarges the symmetry group of the piece from \\(A_4\\) to \\(S_4\\), the symmetric group on 4 points. Fortunately, octahedral symmetry acts on the four space diagonals of a cube by \\(S_4\\), and one endpoint from each diagonal corresponds to the vertices of a tetrahedron, used for the stickers on the 24. Therefore, the 4-stickers can be placed on opposite corners of a cube. The stickers on the periphery of the pieces correspond to the 2-stickers, which swap whenever the piece is rotated in a way that does not preserve the original tetrahedron, i.e. one that does an odd permutation. It may require one or more buffer layers.

    1. Images created with Stella software.\u00a0\u21a9

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https://hypercubing.xyz/puzzles/3x3x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/3x3x3x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/3x3x3x3x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/4d-skewb/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/hemimegaminx/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/hypercuboids/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/nxnxnxn/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/1x2x2x2/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/1x2x2x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/1x2x3x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/1x3x3x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/2x2x2x2x2/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/2x2x2x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/2x2x3x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/2x3x3x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/3x3x3x3/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/4d-pyraminx/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/restricted-2x2x2x2x2/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/2x2x2x2/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/2x2x2x2/canonical-moves/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/2x2x2x2/image-generator/ + 2025-01-03 + daily + + + https://hypercubing.xyz/puzzles/physical/2x2x2x2/scramble-generator/ + 2025-01-03 + daily + + + https://hypercubing.xyz/software/ + 2025-01-03 + daily + + + https://hypercubing.xyz/software/hyperspeedcube/ + 2025-01-03 + daily + + + https://hypercubing.xyz/software/magiccube4d/ + 2025-01-03 + daily + + + https://hypercubing.xyz/software/magicpuzzleultimate/ + 2025-01-03 + daily + + + https://hypercubing.xyz/techniques/ + 2025-01-03 + daily + + + https://hypercubing.xyz/techniques/F2L/ + 2025-01-03 + daily + + + https://hypercubing.xyz/techniques/commutators/ + 2025-01-03 + daily + + + https://hypercubing.xyz/techniques/rkt/ + 2025-01-03 + daily + + + https://hypercubing.xyz/theory/gods-number/ + 2025-01-03 + daily + + + https://hypercubing.xyz/theory/grip_theory/ + 2025-01-03 + daily + + + https://hypercubing.xyz/theory/invariants/ + 2025-01-03 + daily + + + https://hypercubing.xyz/theory/physical/ + 2025-01-03 + daily + + \ No newline at end of file diff --git a/sitemap.xml.gz b/sitemap.xml.gz new file mode 100644 index 0000000000000000000000000000000000000000..685407ebb37faa22ed475c5618e4a677b08edd2b GIT binary patch literal 1038 zcmV+p1o8VHiwFpSD0gQ9|8r?{Wo=<_E_iKh0M*>hj@vd60N{I`V&S_iue}AD!t2~} z>ajf+6gif-qB!z!Xj^{zQvOLdDH0%;V8%MwD+}5W^msTVC2v2ShwyX+N*?3u;(57R 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    Hyperspeedcube

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    Hyperspeedcube (HSC) is a modern, beginner-friendly 3D and 4D Rubik’s cube simulator with customizable mouse and keyboard controls and advanced features for speedsolving. It’s been used to break numerous speedsolving records and runs on all major operating systems plus the web. Hyperspeedcube was first released in early 2022 and is developed by Andrew Farkas (a.k.a. HactarCE).

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    3×3×3×3 with the far cell mid-twist +3×3×3×3 near the end of F2L-b with many tools and settings windows open +Solved 2×2×2

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    Download/installation Release badge

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    HSC does not have an installer. On Windows, open hyperspeedcube_win64.zip and move hyperspeedcube.exe out of to a folder on your computer.

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    Features missing from the web version

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    • Saving & loading logs to file (can still save/load via clipboard)
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    • Awareness of alternate keyboard layouts
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    • Antialiasing
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    Troubleshooting

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    If none of the instructions below help, join the Hypercubers Discord server and start a thread in the #❓help forum. Mention in your post that you’ve read the FAQ.

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    Windows

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    +My antivirus thinks Hyperspeedcube is malicious +

    Try opening the Windows Security app, and digging around in the settings there to disable it, then try the download again.

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    +“Windows protected your PC” +
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    Microsoft Defender SmartScreen prevented an unrecognized app from starting. Running this app might put your PC at risk.

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    +

    Click More info and then Run anyway.

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    In order to prevent that message from appearing, Hactar would have to spend a lot of money to buy a Microsoft developer license and go through a lot of hassle every time there is a new release of HSC. You’ll only have to click through the warnings once.

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    +“The program can’t start because VCRUNTIME140.dll is missing from your computer.” +

    You need to install the Microsoft Visual C++ Redistributable. Here is a direct download.

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    +“Hyperspeedcube crashed. A crash report has been saved to …” +

    First, make sure your graphics drivers are up-to-date! This is the most common cause of Hyperespeedcube crashes. Here is a video tutorial if you don’t know how to do that.

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    If you still get an error message, then follow these steps:

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    1. Press +R, type %LocalAppData%, and press Enter
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    2. Open the Temp folder
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    3. Find the file named in the error message and send it to Hactar either in a GitHub issue or a new thread in the #❓help forum on Discord
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    macOS

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    +“‘Hyperspeedcube.app’ cannot be opened because the developer cannot be verified.” +
    +

    macOS cannot verify that this app is free from malware.

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    Go to System Settings → Privacy & Security, then scroll down to Security. Select App Store and identified developers, then click Open Anyway.

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    In order to prevent that message from appearing, Hactar would have to spend a lot of money to buy an Apple developer license and go through a lot of hassle every time there is a new release of HSC.

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    Linux

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    Check that you have up-to-date graphics drivers installed. If that doesn’t help, you’re on your own.

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    If there’s any changes you make to the Hyperspeedcube source code to get it working, please open an issue or PR on GitHub. If you’re able to make an package for HSC for your package manager, please let me know either on Discord or GitHub Issues.

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    History

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    1.0

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    On April 24th, 2020, Hactar wrote a small program called Keyboard Speedcube that simulated a 33 using keybinds inspired by Ryan Heise’s Rubik’s Cube Simulator. In October 2021, rudimentary 4D support was added and a screenshot was posted on the Hypercubers Discord server. A month later, Rowan Fortier asked Hactar if he could receive an early version of the program, and showcased it in a YouTube video. Shortly before the video’s release, the project was renamed to Hyperspeedcube.

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    first mention +Early build

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    The first official release was v0.1.0 in January 2022, featuring customizable keybinds and support for the 33 and 34. Over the next year, many community-requested features were added, including mouse controls and more n3 and n4 puzzles. (See the changelog for a complete list of releases and changes.)

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    Hyperspeedcube began to see widespread use after the addition of mouse controls and more advanced piece filters in August 2022, kickstarting the hyperspeedsolving revolution. The web version was released in January 2023.

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    Shortly after the web version was released, Hactar and Luna began reading the MPU source code, intending to make a tool to ease the process of writing MPU puzzle definitions. Instead, they were able to replicate MPU’s puzzle generation algorithms, and Hactar set to work on a new puzzle simulator using this backend. Over the next year, they adapted the algorithm to use Conformal Geometric Algebra in the hopes of supporting curved cuts, but this proved too challenging in higher dimensions. Together with Milo, they built a puzzle definition system using Lua, since it is a well-known programming language with a simple type system that is easy to embed, sandbox, and extend with custom types.

    +

    The first dev build of the new version, Hyperspeedcube v2.0.0-pre.1, was released on July 16th, 2023, and supported Lua shape definitions but not twists. As of mid-2024, HSC 2.0 is still in active development.

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    2.0

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    HSC 2.0 is the next major update in the works, with no set release date as of yet. Below is a list of upcoming features:

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    • Build nearly any puzzle in 3D to 7D space
      • +
    • Complete overhaul of the graphics engine
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    • Built-in timer for speedsolves, including a configurable autosplitter
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    • Timeline of progress during solve
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    • More piece filter customization
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    +

    See Hactar’s website for more details.

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    Development screenshots

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    Early shape generation and twisting prototypes (late 2022)

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    Hypercuboid twisted in an incorrect way (2022-10-13) +3D Jing's pyraminx puzzle, with a popup saying that the program crashed (2022-10-13) +{3}×{5} duoprism (2022-10-13) +Uncolored 120-cell puzzle

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    Graphics debugging (early 2024)

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    + + + +

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    UI development (mid 2024)

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    Prototype of the global color palette (2024-06-27) +Concept sketch of the piece filters UI (2024-08-05) +First puzzle loaded from a Lua file (2024-08-27)

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    Software

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    Since building physical representations of higher-dimensional puzzles is challenging, we use computer software to simulate them.

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    What program should I download?

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    We recommend Hyperspeedcube to get started.

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    If you want macro support (recorded sequences of moves) or other 4D puzzles, Magic Cube 4D is a good option.

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    For hypercubing on mobile devices, see Android apps or iOS apps.

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    General cubing & hypercubing

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    +What do the icons mean? +
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    • Mouse controls
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    • Keyboard controls
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    • Customizable keyboard controls
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    • Find/filter piece by color
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    • Find/filter pieces by type
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    • Timer
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    • Macros
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    • Move input
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    • Custom colors
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    • Custom puzzles
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    • VR support
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    ProgramPlatformsFeaturesPuzzles
    Hyperspeedcube • {1-9}{3-4}
    Magic Cube 4D • 4D via Schläfli symbol
    MagicTile • 2D tilings
    Magic Puzzle Ultimate • 3D+ doctrinaire
    Magic Cube 7D • {3-5}{4-7}
    Magic Cube 5D • {2-7}5
    Magic Cube 4D VR • 34
    Flat Hypercube • {1-19}{1-10}
    Gelatinbrain1 • many cursed things
    pCubes nearly every 3D puzzle
    Ultimate Magic Cube platonic 3D
    Ultimate Magic Cube 2 • platonic + misc 3D
    Twizzle Explorer • many 3D puzzles
    IsoCubeSim • AxBxC, N-layer megaminx
    Geraniums Pot • rotating-circle puzzles
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    Specific puzzles

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    ProgramPlatformsFeaturesPuzzle
    Magic Hyperbolic Tile {6,3,3} (7 quotients)
    Magic Simplex 5D • 5D simplex (+ recuts)
    Magic120Cell • 120-cell (+ recolorings)
    Nan Ma’s 11-cell • 11-cell
    Twisty Prisms • certain 3D prisms
    +

    1D and 2D puzzles

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    Complex & laminated puzzles

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    Other

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    Android apps

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    Hypercubing on mobile devices is not recommended; however, if this is your only option, these apps are available for Android:

    + +

    iOS apps

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    There are no known apps for hypercubing on iOS.

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    If you’re able to connect an external mouse & keyboard, the web version of Hyperspeedcube may work on mobile devices but it’s not officially supported.

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    Tools

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      +
    • Orb, for planning puzzle cut depths
      • +
    • RocKeT, for finding RKT cancels for 3D algorithms
      • +
    • Hypersolve, for generating short 24 solutions and scrambles
      • +
    +

    Games

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    This is not a complete list of 4D games. This is specifically 4D software that hypercubers often recommend for understanding 4D geometry better.

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    Visualizations

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    1. +

      Full name: gelatinbrain’s Virtual Magic Polyhedra (permutationpuzzles) ↩

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    Magic Cube 4D

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    Magic Cube 4D (MC4D) was one of the first ever programs capable of simulating higher dimensional twisty puzzles. It was first released in 1988 and was developed by Melinda Green, Don Hatch, Jay Berkenbilt, and Roice Nelson.

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    +3×3×3×3 puzzle in Magic Cube 4D

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    34 puzzle in Magic Cube 4D

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    Download/installation

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    Magic Cube 4D requires Java to be installed. Once you have Java, go to the Superliminal website and click on the Click here to download MagicCube4D button. Save the executable mc4d-4-3-343.jar file somewhere you will remember, as this file is how you open the program.

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    Alternative versions

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    Raynefork

    +

    Raynefork is Raymond Zhao’s version of MC4D, which has the updated default colour scheme (standard 33 colours plus pink/purple), and some more options in the settings menu. It can be found from its GitHub here.

    +

    Don’s version

    +

    Don’s version contains some 2D puzzles, many 3D puzzles, and some interesting 4D puzzles. It also has a menu for 5D and 6D puzzles, however they don’t work when you select them. Don’s version can be downloaded from its GitHub here.

    +

    Troubleshooting

    +

    Try reading MC4D’s FAQ on the superliminal website. If you’re still having an issue, try asking for help on the Discord server or mailing list.

    +
    +my computer wants to open the .jar file with notepad +

    This means that you haven’t installed Java properly. Make sure to get the newest version from Java’s website. After the installation, right click on the mc4d-4-3-343.jar file, and click open with Java(TM) Platform SE Binary.

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    +
    +facecolors.txt doesn’t change my colors +
      +
    1. Try closing and reopening the program, and then reloading the puzzle using the menu.
      2. +
    2. If you’re on windows and you have show file extensions off in settings, then you may have accidentally named the file facecolors.txt.txt. Try renaming it to just facecolors. (Alternatively, enable show file extensions in settings)
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    Magic Puzzle Ultimate

    +

    Magic Puzzle Ultimate (MPU or MPUlt) is a higher-dimensional puzzle simulator developed by Andrey Astrelin that can simulate nearly any symmetric doctrinaire puzzle and supports, including user-defined ones.

    +

    Download MPU

    +

    32-cell in MPU

    +

    32-cell puzzle in MPU

    +

    Records

    +

    Some records for the shortest and first solutions of a puzzle are kept on the Superliminal Wiki page.

    +

    Virtual Puzzles

    +

    While we’re waiting for Hyperspeedcube 2.0 to release, Magic Puzzle Ultimate is a great program to get your feet wet in the world of other 4D puzzles.

    +

    Open the location where your MPUlt folder is. Next, open the MPUlt_puzzles.txt file. This file is a big list of all the puzzles in the program. Now you can copy & paste the puzzles that you want below under their appropriate Block. Just save the file, reopen MPUlt, and the puzzle you added will be in the menu.

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    Puzzle Definitions

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    +3D Puzzles +
    +3x3x3 +
    3x3x3
    Puzzle 3x3x3
+Dim 3
+NAxis 1
+Faces 1,0,0
+Group 1,0,0/1,1,0 1,0,0/1,0,1
+Axis 1,0,0
+Twists 0,1,0/0,1,1
+Cuts -0.33 0.33
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    +Skewb +
    Skewb
    Puzzle Skewb
+Dim 3
+NAxis 1
+Faces 1,0,0
+Group 1,0,0/1,1,0 1,0,0/1,0,1
+Axis 1,1,1
+Twists 1,-1,0/1,0,-1
+Cuts 0
+
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    +Compy Rainbow +
    Compy Rainbow
    Puzzle Compy_Rainbow
+Dim 3
+NAxis 1
+Faces 1,0,0 0.6667,0.6667,0.6667
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+Axis 1,1,1
+Twists 1,-1,0/1,0,-1
+Cuts -0.45 0.45
+FixedMask 2
+
    +
    +
    +Cuboctahedron +
    Cuboctahedron
    Puzzle Cuboctahedron
+Dim 3
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+Faces 1,0,0 0.667,0.667,0.667
+Group 1,0,0/1,1,0 1,0,0/1,0,1
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+Cuts 0.5 -0.5
+Axis 1,1,1
+Twists 1,-1,0/1,0,-1
+Cuts 0.5 -0.5
+
    +
    +
    +
    +4D Puzzles +
    +Tesseract Family +
    +2x2x1x1 +
    2x2x1x1
    Puzzle 2x2x1x1
+Dim 0
+NAxis 2
+Faces 1,0,0,0 0,0,0,0.5
+Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,1,1
+
    +
    +
    +2x2x2x1 +
    2x2x2x1
    Puzzle 2x2x2x1
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0,0,0,0.5
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1
+Cuts 0 0
+Axis 0,0,0,1
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0
+Cuts
+
    +
    +
    +2x2x2x3 +
    2x2x2x3
    Puzzle 2x2x2x3
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0,0,0,1.5
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1
+Cuts 0
+Axis 0,0,0,1
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0
+Cuts 0.5 -0.5
+
    +
    +
    +2x2x3x3 +
    2x2x3x3
    Puzzle 2x2x3x3
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0,0,1.5,0
+Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1
+Axis 1,0,0,0
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+Cuts 0
+Axis 0,0,1,0
+Twists 1,0,0,0/1,1,0,0 0,0,0,1/1,0,0,0 0,0,0,1/1,1,0,0
+Cuts 0.5 -0.5
+
    +
    +
    +2x2x3x4 +
    2x2x3x4
    Puzzle 2x2x3x4
+Dim 4
+NAxis 3
+Faces 1,0,0,0 0,0,1.5,0 0,0,0,2
+Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1
+Cuts 0
+Axis 0,0,1,0
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0
+Cuts 0.5 -0.5
+Axis 0,0,0,1
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0
+Cuts 1 0 -1
+
    +
    +
    +2x3x4x5 +
    2x3x4x5
    Puzzle 2x3x4x5
+Dim 4
+NAxis 4
+Faces 1,0,0,0 0,1.5,0,0 0,0,2,0 0,0,0,2.5
+Group 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1
+Cuts 0
+Axis 0,1,0,0
+Twists 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,0,1,0/0,0,0,1
+Cuts 0.5 -0.5
+Axis 0,0,1,0
+Twists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0
+Cuts 1 0 -1
+Axis 0,0,0,1
+Twists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0
+Cuts 1.5 0.5 -0.5 -1.5
+
    +
    +
    +3x3x3x1 +
    3x3x3x1
    Puzzle 3x3x3x1
+Dim 4
+NAxis 2
+Faces 1.5,0,0,0 0,0,0,0.5
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1
+Cuts 0.5 -0.5
+Axis 0,0,0,1
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0
+Cuts
+
    +
    +
    +3x3x3x2 +
    3x3x3x2
    Puzzle 3x3x3x2
+Dim 4
+NAxis 2
+Faces 1.5,0,0,0 0,0,0,1
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1
+Cuts -0.5 0.5
+Axis 0,0,0,1
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0
+Cuts 0
+
    +
    +
    +4x4x4x4 +
    4x4x4x4
    Puzzle 4^4
+Dim 4
+NAxis 1
+Faces 1,0,0,0
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,-1,0/0,0,0,1 0,2,-1,-1/0,1,1,-2
+Cuts 0.5 0 -0.5
+
    +
    +
    +
    +Other +
    +{4}x{4} 3 +
    {4}x{4} 3
    Puzzle {4}x{4} 3
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0,0,1,0
+Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,0,1,0 0,0,1,0/0,0,1,1
+Cuts 0.5 -0.5
+Axis 0,0,1,0
+Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1
+Cuts 0.5 -0.5
+
    +
    +
    +3^4 Skewb +
    3^4 Skewb
    Puzzle 3^4 Skewb
+Dim 4
+NAxis 1
+Faces 1,0,0,0
+Simplified
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1
+Axis 1,1,1,1
+Twists 0,2,-1,-1/0,1,1,-2 1,-1,0,0/0,0,1,-1
+Cuts 0
+
    +
    +
    +5-5_Duotegum +
    5-5_Duotegum
    Puzzle 5-5_Duotegum
+Dim 4
+NAxis 1
+Faces -1,1,0,0
+Group 1,0,0,0/0.809016994,0,0.587785252,0 1,1,0,0/0,0,1,1
+Axis -1,1,0,0
+Twists 0,0,1,0/0,0,0,1 1,1,0,0/0,0,1,1 1,1,0,0/0,0,-1,1
+Cuts 0.65
+
    +
    +
    +16-cell Face Turning +
    16-cell Face Turning
    Puzzle 16-cell_FT
+Dim 4
+NAxis 1
+Faces 1,1,1,1
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1
+Axis 1,1,1,1
+Twists 0,2,-1,-1/0,1,1,-2 1,-1,0,0/0,0,1,-1
+Cuts 0.6 -0.6
+FixedMask 2
+
    +
    +
    +Chamfered Pentagonal Duoprism +
    Chamfered Pentagonal Duoprism
    Puzzle Chamfered_Pentagonal_Duoprism
+Dim 4
+NAxis 2
+Faces -1.41429,0,0,0 1.41429,0,0,0 -1,1,0,0
+Group 1,0,0,0/0.809016994,0,0.587785252,0 1,1,0,0/0,0,1,1
+Axis -1,0,0,0
+Twists 0,1,0,0/0,0.809016994,0,0.587785252 0,0,1,0/0,0,0,1
+Cuts 1.3 -1.23
+Axis -1,1,0,0
+Twists 0,0,1,0/0,0,0,1 1,1,0,0/0,0,1,1 1,1,0,0/0,0,-1,1
+Cuts 0.85
+
    +
    +
    +Chamefered Tesseract +
    Chamfered Tesseract
    Puzzle Chamfered_Tesseract
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0.70710678,0.70710678,0,0
+Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,1,0,1 0,0,1,0/0,0,1,1
+Cuts 0.85 -0.85
+Axis 1,1,0,0
+Twists 0,0,1,0/0,0,1,1 1,-1,0,0/0,0,1,0 1,-1,0,0/0,0,1,1
+Cuts 0.57 -0.57
+
    +
    +
    +Octahedral Prism +
    Octahedral Prism
    Puzzle Octahedral_Prism
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0,1,1,1
+Group 1,0,0,0/0,1,0,0 0,1,0,0/0,1,1,0 0,1,0,0/0,1,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,1,0 0,1,1,0/0,0,1,1 0,1,0,0/0,0,1,1
+Cuts 0.5 -0.5
+Axis 0,1,1,1
+Twists 0,1,-1,0/0,1,0,-1 1,0,0,0/0,1,-1,0
+Cuts 0.5 -0.5
+
    +
    +
    +Snub 24-cell +
    Snub 24-cell
    Puzzle Snub24cell
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0.809017,0.809017,0,0 0.9045085,0.6545085,0.25,0
+#Faces 1,0,0,0 0.809017,0.809017,0,0 0.9045085,0.6545085,0.25,0 0.9045085,0.6545085,-0.25,0
+Group 0,2,-1,-1/0,1,1,-2 0,1,1,2/0,2,-1,1 2,-2,-2,0/1,-1,-1,3
+Axis 1,0,0,0
+Twists 0,2,-1,-1/0,1,1,-2 0,0,1,0/0,0,0,1 0,1,1,2/0,2,-1,1
+Cuts 0.9 -0.9
+FixedMask 2
+Axis 0.809017,0.809017,0,0
+Twists 2,-2,-2,0/1,-1,-1,3 1,-1,-1,-3/2,-2,-2,0 0,0,0,1/0,0,1,0
+Cuts 0.95 -0.95
+FixedMask 2
+
    +
    +
    +Square Antiprism Prism +
    Square Antiprism Prism
    Puzzle Square_Antiprism_Prism
+Dim 4
+NAxis 3
+Faces -0.5,0,0,0 0,0,0.42044820,0 0,0.56903559,0.14014940,0
+Group 0,0,1,0/0,0.38268343,0,0.92387953 0,0,0,1/0,1,0,1 1,0,0,0/0,0,0,1
+Axis 1,0,0,0
+Twists 0,1,0,0/0,1,0,1 0,0,1,0/0,0.38268343,0,0.92387953 0,0,1,0/0,-0.38268343,0,0.92387953
+Cuts 0.1666 -0.1666
+Axis 0,0,1,0
+Twists 0,1,0,0/0,1,0,1 1,0,0,0/0,1,0,0 1,0,0,0/0,1,0,1
+Cuts 0.1235 -0.1235
+Axis 0,0.56903559,0.14014940,0
+Twists 1,0,0,0/0,0,0,1
+Cuts 0.621
+
    +
    +
    +Triangular Antitegmatic Icoschoron +
    Triangular Antitegmatic Icosachoron
    Puzzle Triangular-antitegmatic_Icosachoron
+Dim 4
+NAxis 1
+Faces 1,0,0,0
+Group 1,0,0,0/0.5,0.866025404,0,0 0,0.577350269,0.816496581,0/0,0,0.612372436,0.790569415
+#1,0,0,0/0.5,sqrt(3)/2,0,0 0,1/sqrt(3),sqrt(2/3),0/0,0,1/sqrt(6),sqrt(5/6)
+Axis 1,0,0,0
+Twists 0,0.577350269,0.816496581,0/0,0,0.612372436,0.790569415 0,0.790569,-0.559017,0.25/0,0.57735,1.22474,0.912871
+Cuts 0.75 -0.75
+
    +
    +
    +
    +
    +5D Puzzles +
    +Penteract Family +
    +1x1x1x1x2 +
    1x1x1x1x2
    Puzzle 1x1x1x1x2
+Dim 5
+NAxis 2
+Faces 0.5,0,0,0,0 0,0,0,0,1
+Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/1,0,0,1,0 1,0,0,0,0/0,0,0,0,1
+Axis 1,0,0,0,0
+Twists 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,0,0,0,1
+Cuts
+Axis 0,0,0,0,1
+Twists 1,0,0,0,0/1,1,0,0,0
+Cuts 0
+
    +
    +
    +1x1x1x2x2 +
    1x1x1x2x2
    Puzzle 1x1x1x2x2
+Dim 5
+NAxis 2
+Faces 0.5,0,0,0,0 0,0,0,1,0
+Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1
+Axis 1,0,0,0,0
+Twists 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1
+Cuts
+Axis 0,0,0,1,0
+Twists 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/0,0,0,0,1
+Cuts 0
+
    +
    +
    +1x1x2x2x2 +
    1x1x2x2x2
    Puzzle 1x1x2x2x2
+Dim 5
+NAxis 2
+Faces 0.5,0,0,0,0 0,0,1,0,0
+Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0 0,0,1,0,0/0,0,1,0,1
+Axis 1,0,0,0,0
+Twists 0,1,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0
+Cuts
+Axis 0,0,1,0,0
+Twists 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/0,0,0,1,0 0,0,0,1,0/0,0,0,1,1
+Cuts 0
+
    +
    +
    +1x2x2x2x2 +
    1x2x2x2x2
    Puzzle 1x2x2x2x2
+Dim 5
+NAxis 2
+Faces 0.5,0,0,0,0 0,1,0,0,0
+Group 1,0,0,0,0/0,1,0,0,0 0,1,0,0,0/0,1,1,0,0 0,1,0,0,0/0,1,0,1,0 0,1,0,0,0/0,1,0,0,1
+Axis 1,0,0,0,0
+Twists 0,1,0,0,0/0,1,1,0,0
+Cuts
+Axis 0,1,0,0,0
+Twists 1,0,0,0,0/0,0,1,0,0 0,0,1,0,0/0,0,1,1,0
+Cuts 0
+
    +
    +
    +2x2x2x2x2 +
    2x2x2x2x2
    Puzzle 2^5
+Dim 5
+NAxis 1
+Faces 1,0,0,0,0
+Group 1,0,0,0,0/1,1,0,0,0 1,0,0,0,0/1,0,1,0,0 1,0,0,0,0/1,0,0,1,0 1,0,0,0,0/1,0,0,0,1
+Axis 1,0,0,0,0
+Twists 0,1,0,0,0/0,1,1,0,0
+Cuts 0
+
    +
    +
    +
    +Other +
    +Simplex Prism +
    Simplex Prism
    Puzzle Simplex_Prism
+Dim 5
+NAxis 2
+Faces 0,0,0,0,1 1,0,0,0,0
+Group 1,0,0,0,0 0,0,1,1,0/0,0,1,-1,0 0,2,-1,-1,0/0,1,1,-2,0 0,2,-2,0,0/0,1,1,-1,-2.236068
+Axis 0,0,0,0,1
+Twists 1,0,0,0,0/0,1,-1,0,0 0,0,1,1,0/0,0,1,-1,0 0,2,-1,-1,0/0,1,1,-2,0
+Cuts 0
+Axis 1,0,0,0,0
+Twists 0,2,-1,-1,0/0,1,1,-2,0
+Cuts 0
+
    +
    +
    +{3,3}x{4} +
    {3,3}x{4}
    Puzzle {3,3}x{4}
+Dim 5
+NAxis 2
+Faces 1,1,1,0,0 0,0,0,1.73205081,0
+Group 1,1,0,0,0/1,0,-1,0,0 1,1,0,0,0/0,1,-1,0,0 0,0,0,1,0/0,0,0,1,1
+Axis 1,1,1,0,0
+Twists 0,0,0,1,0/0,0,0,1,1 1,-1,0,0,0/1,0,-1,0,0 1,-1,0,0,0/0,0,0,1,0
+Cuts -0.33333
+Axis 0,0,0,1,0
+Twists 1,1,0,0,0/1,0,-1,0,0 0,0,0,0,1/1,-1,0,0,0
+Cuts 0
+
    +
    +
    +
    +

    Making your own puzzles

    +

    If you figure out the notation, you can try adding your own puzzles. You define the symmetry group, rotational axes, cutting planes, possible twists and puzzle boundary.

    +

    For some definitions of various hypercuboids, see hypercuboids.

    +

    There is also a general formula for 4D duoprism puzzles, made by Luna: +

    {p}x{q}
    Puzzle {p}x{q}
+Dim 4
+NAxis 2
+Faces 1,0,0,0 0,0,1,0
+Group 1,0,0,0/1,tan(pi/p),0,0 0,0,1,0/0,0,1,tan(pi/q)
+Axis 1,0,0,0
+Twists 0,1,0,0/0,0,0,1 0,0,1,0/0,0,1,tan(pi/q)
+Cuts ...
+Axis 0,0,1,0
+Twists 0,0,0,1/0,1,0,0 1,0,0,0/1,tan(pi/p),0,0
+Cuts ...
+

    + + + + + + + + + + + + + + + + + + + + +
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    +
    + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/stylesheets/extra.css b/stylesheets/extra.css new file mode 100644 index 00000000..38b3eb0a --- /dev/null +++ b/stylesheets/extra.css @@ -0,0 +1,12 @@ +.md-nav__title { + box-shadow: none !important; +} + +.md-footer__inner { + display: none !important; +} + +.hidden { + opacity: 50%; + scale: 70%; +} diff --git a/techniques/F2L/index.html b/techniques/F2L/index.html new file mode 100644 index 00000000..bfb64d35 --- /dev/null +++ b/techniques/F2L/index.html @@ -0,0 +1,4043 @@ + + + + + + + + + + + + + + + + + + + + + + + + + F2L - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    + + + + + + + + + + + + +

    F2L

    +

    +
    + +
    +

    +

    First 2 Layers (F2L) is a solving technique for cubes and some other puzzles that involves pairing pieces and inserting those pairs into slots to build up the first 2 layers (of a 3-layered puzzle). The first known description of this technique was by John Conway in 1979, but is most commonly credited to Jessica Fridrich due to the massive popularity her website describing the CFOP method had in the early 1990s.

    +
    +

    The goal of this page is to give you a generalized understanding of higher dimensional F2L, not to hand out algorithms for certain cases. There are so many cases for higher dimensional F2L that it would be nearly impossible to list out all the cases. It’s recommended to have a fairly strong intuitive understanding of 3D F2L before reading this page.

    +
    +

    Dimensional Analogy

    +

    In 3D F2L, whenever you twist a side axis, a 1x1x3 block of pieces (containing the F2L slot) goes to the top layer (essentially a 1D line). In 4 dimensions, 2 adjacent sides intersect at a 2D plane. So when you do a twist you’re bringing a 1x1x3x3 block to the top layer, which contains 3 F2L pairs: that being two F2L-b slots and an F2L-a slot. This makes inserting a single pair trickier, because now you need to use more moves to not disturb the other two subslots.

    +

    Continuing this analogy into 5D, you bring a whole 3D cube’s worth of F2L slots to the top (1x1x3x3x3) that you mustn’t distrub while inserting a single pair. This pattern continues for higher and higher dimensions, meaning that it gets trickier to insert a single pair. Pro solvers of higher dimensional puzzles will wait until they have 3 (or multiples of 3) pairs, then pairs all those pairs into a gigapair, and then insert that gigapair like inserting a lower dimensional pair.

    +

    Terminology

    +

    To talk about F2L in a dimensionally neutral way, we need some more terms. Most importantly, the following:

    +
      +
    • Head: the piece in the pair with the most colours (corner in 3D)
      • +
    • Body: the piece in the pair with one less colour (edge in 3D)
      • +
    +

    For more terms, see the glossary.

    +

    Regardless of dimension and type of pair, they are broken down into the following categories:

    +
      +
    1. both in top, head facing side axes
      2. +
    2. both in top, head facing top axis
      3. +
    3. body in slot, head in top facing side axis
      4. +
    4. body in slot, head in top facing top axis
      5. +
    5. head in slot facing side axis, body in top
      6. +
    6. head in slot facing base axis, body in top
      7. +
    7. both stuck in slot
      8. +
    +

    F2L-4a

    +

    F2L-a pairs consist of a 2c and a 3c, exactly the same as in 3D. The solutions feel very similar to 3D cases, but there are a few extra tricks that are possible.

    +
    +Insert +

    F2L-4a hide reveal pair

    +

    Normal F2L insert: IU RU IU' RU'

    +
    +
    +Hide and reveal +

    F2L-4a hide reveal pair

    +

    Notice: same color on top.

    +

    Can either use RKT on the I cell to pair it or use hide and reveal (preferred).

    +
    RU IU RU' //pair
+IU2 RU IU' RU' //insert
+
    +
    +
    +3-mover +

    F2L-4a hide reveal pair +3-move insert. Still works if the body is at IU, IF, or ID except you’ll just need an extra RKT move.

    +

    Notice how the F sticker of the pair matches the F cell colour. (If it doesn’t, then the pair will be flipped after inserting!)

    +
    +
    +Capping +

    F2L-4a hide reveal pair

    +
      +
    • Body stuck in slot, solved
      • +
    • Head has base color on top
      • +
    • Cap head on top of body
      • +
    • Push, rotate pair, pull
      • +
    +

    Solution: IU2 IF RU IR2 RU'

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    Commutators

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    A commutator (or “comm”) is a sequence of moves where you do a sequence A, then a sequence B, then the inverse of A, and finally the inverse of B. Usually, you want the intersection of the pieces affected by sequence A and the pieces affected by sequence B to be small but non-empty. In turn, creating a commutator with those sequences makes it easier to solve a puzzle by only affecting a few specific pieces at a time.

    +

    One of the earliest solutions for the 3x3x3x3 was The Ultimate Solution to a 3x3x3x3 by Roice Nelson, which made heavy use of intuitive setup moves and commutators. Commutators were also much easier to use in MC4D due to its Macro feature, which allowed users to record sequences of moves and then replay them later from any angle.

    +

    Notation

    +

    Commutator notation extends traditional twisty puzzle notation by adding the following definitions:

    +
      +
    • Commutators are notated [A, B], which expands to A B A' B'.
      • +
    • Conjugates are notated [A: B], which expands to A B A'.
      • +
    +

    Here are some example commutators that can be executed on a 3x3x3:

    +
      +
    • [R, U] expands to R U R' U'.
      • +
    • [M', U2] expands to M' U2 M U2, which cycles 3 edges on the M slice.
      • +
    • [D: [M', U2]] expands to D (M' U2 M U2) D', which cycles 3 slightly different edges. Conjugates are often combined with commutators to influence different sets of pieces.
      • +
    • [F: [R, U]] expands to [F: R U R' U'], which expands further to F (R U R' U') F'.
      • +
    • [D, [R, U]] expands to D (R U R' U') D' (U R U' R'), which cycles 3 corners on the F layer.
      • +
    +

    (Parentheses are added for readability but have no special meaning.)

    +

    Strategy

    +

    Once you find commutators for cycling different types of pieces on a puzzle (2c, 3c, etc) then you can almost solve the whole thing! (There may be some situations where you need an algorithm to swap 2 pieces of a type and 2 of a different type.)

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    Solving Techniques

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    Hypercubing methods typically rely on a few fundamental techniques. By mastering these, you can even develop your own solutions to new puzzles.

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    Learn Notation before trying to learn any of these techniques

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    RKT

    +
    +

    You must know 34 notation in order for this page to make sense!

    +
    +

    RKT is a technique that lets you treat a single cell of a (cell-turning) higher dimensional puzzle as if it were a lower dimensional puzzle. This is very useful to do moves that damage fewer pieces. For example: RKT lets you treat a side of a 34 just like a 33 cube, meaning that you can use all the 3D algorithms you already know to solve the full 4D puzzle.

    +

    It has been invented independently several times, but was mainly popularized by Raymond Zhao in his article here.

    +

    Naming

    +
    +

    Other proposed names:

    +
      +
    • SFM (Single Facet Manipulation)
      • +
    • FRM (Facet Redution Method)
      • +
    +
    +

    The name RKT comes from the move set <RK,T*> on n4 puzzles (in the old notation, the Inside and Outside cells were called Top and Kata). In the new notation it should be called ROI, but the old name stuck.

    +
    +

    Grant

    +

    Also it’s the part of the solve where you can get ReKT

    +
    +

    Doing moves with RKT (n4)

    +

    RKT lets us do any n3 sequence of moves on one side of an n4. The beginner’s way to learn this is by “translating” 3D algorithms into RKT (although this is not the best way to think about how RKT works beacuse you can really perform any sequence of moves). Below is a simple 3 step guide to translate your algorithms into RKT.

    +
    +

    Example: Sune algorithm with RKT

    +

    Say you want to do the Sune algorithm R U R' U R U2 R' with RKT. The first step is to try and think of how you can execute the algorithm using only R moves and cube rotations. R U R' U R U2 R' becomes R z R z' R' z R z' R z R2 z' R'. Now we replace R with RO, and cube rotations with I cell rotations: RO IF RO IF' RO' IF RO IF' RO IF RO2 IF' RO'

    +

    Sune with RKT

    +
    +
      +
    1. Rewrite the algorithm to only use cube rotations and R moves
      2. +
    2. Rewrite cube rotations in this new algorithm (x, y, or z) to their respective I cell rotation (IR, IU, IF)
      3. +
    3. Replace all R moves with their RO counterpart
      4. +
    +

    This can definitely be very confusing and disorienting to beginners, as you have to think about 3x3x3 moves with only R moves and rotations. It takes a lot of practice to get comfortable with RKT, but it is an extremely powerful technique.

    +

    The best way to learn RKT before doing your first 34 solve is by manually scrambling the last cell using any RO moves and random I cell rotations until it looks scrambled. Then practice doing PLC by solving it with RKT.

    +
    +

    Tip

    +

    Even though this technique is called RKT, it is not limited to those types of turns. Thus, you can use LO and I- moves instead, or rotate your view and use RU and D- instead of RK and I-.

    +
    +

    Another popular way to do RKT is to use wide O cell turns (instead of normal I cell turns). This has the benefit of keeping the cell you’re doing RKT to in the same orientation the entire time. Instead of rotating a side of the I cell to the RI plane, you rotate all the outside layers over the side that you’re twisting.

    +

    Parity

    +
    +

    RKT Parity

    +

    RKT parity on the last layer of the last cell of a 3×3×3×3

    +
    +

    RKT parity is a state you can get to that appears to rotate a single layer of a cell by 180°. +If you try and fix this with RKT, then the outer layers will be off by 180°. This means that you must use an algorithm (or intuition) to solve it.

    +

    You can use a 3D supercube algorithm that rotates the U center 180° ((R U R' U)5, or (L R U2 R' L' U)2).

    +

    Harder to memorize (but much lower in movecount) is this 9-mover RKT parity alg: +IU UR IU' IF' UO' IF RF UR RF' UIR, found by Tetrian22.

    + +

    On 24, the algorithm is shorter because it doesn’t have to worry about messing up other pieces besides corners. A commonly used one is R2 B2 R2 U R2 B2 R2 U'RKT.

    +

    On bigger n4 puzzles (where \(n>3\)), it can look like a single slice layer of a cell is off by 180°. An intuitive way to solve this is to do the 24 RKT parity algorithm with wide moves, and then the normal 34 RKT parity algorithm. It can also be avoided by just lining up your slice the same way you line up centers in 3D before finishing last 4 edges (when using freeslice).

    +

    Debt

    +

    RKT Debt is when the R cell isn’t aligned with the rest of the puzzle aftering performing an algorithm or sequence of moves with RKT. For example: after executing a T perm algorithm with RKT the R cell will be misaligned by 90°. RKT Debt always has to be “paid back” at some point during the solve. During complicated setup moves for fancy inserts, RKT debt can be used as “ammo”. That is, undoing the debt in a useful way to help solve the puzzle. An easy way for beginners to “avoid” debt is to just do a move on a layer that you don’t care about messing up at that point in the solve. For example: in a CFOP solve during the final F2L-b pair insertion, you could do any U* move that fixes the debt.

    +

    Cancels

    +
    +

    Sune with RKT: left = normal RKT, right = RKT cancel

    +

    Sune with RKT

    +
    +

    RKT cancelling is a technique that reduces the move count of certain RKT algorithms and triggers by abusing symmetry of rotations. HactarCE made a program called RocKeT to find cancels for 34 algorithms. Often, it just involves inserting some flipping moves at certain points throughout the algorithm.

    +

    Consider R U R' U'. Conventional RKT rotates after every move, but we don’t have to do that. We can build up multiple moves of RKT debt and then cancel them later:

    +
      +
    • RO UO — do R U using RKT, building up two moves of RKT debt (R U)
      • +
    • IF RO' — do R' using RKT, undoing the debt from U
      • +
    • IF2 UO' — do U' using RKT, undoing the debt from the R
      • +
    +

    But we can do even better!

    +
      +
    • RO UO — do R U using RKT
      • +
    • IUR — swap R with U
      • +
    • UO' RO' — do R' U' using RKT (because we swapped R and U), undoing the debt from R U
      • +
    +

    If you squint, you might notice that this is a conjugate [RO UO: IUR]. This corresponds to the fact that R U R' U' can be written as a conjugate where one part is a pure rotation: [R U: z x2]. When executing this algorithm, we apply it to I with the rotation (so it’s equivalent to R U R' U') and to the outer layers without the rotation (so [RU: _] expands to R U U' R', which completely cancels out). This is the fundamental theory behind how to do RKT cancels: by rewriting algorithms as conjugates and commutators with pure rotations.

    +

    Recall how in 3D you can replace a U y' with Dw. We can do the same thing here: replace IUR with {1-2}OUR, which is a rotation of the outer layers instead of the inner layers. Now it just looks like a normal algorithm with a flip thrown in the middle: RO UO {1-2}OUR RO' UO'. So we can notate it a little more simply: R U {1-2}OUR R' U'

    +

    Here’s a more complex example, the Sune algorithm: RO UO RO' {1-2}OUR UO RO {1-2}OUR UO2 RO'. This works because you can rewrite Sune using conjugates with rotations: [R U: [R': z x2] [U: z x2]]. Notice how if you remove the rotations, the whole thing cancels out. (You don’t need to expand the conjugates to see this.)

    +

    RKT Cancel Algorithms

    +
    +

    These won’t help you solve a puzzle for the first time

    +

    You do not need to learn RKT cancel algs to solve any puzzle. RKT can and should be done completely intuitively unless you are already speedsolving at a high level and wish to improve this step.

    +
    +
      +
    • See Commutator notation first.
      • +
    • +

      Single letter moves are the “big 3d” moves. (R = RO, U = UO, etc.)

      +
      • +
    • +

      TODO: explain xyz notation and {1-2}O notation in notation page. put a big caveat that its purpose is for execution, and it’s totally arbitrary as far as what’s actually going on with the puzzle.

      +
      • +
    +
    +44 edge swap parity + + + + + + + + + + + + + + + +
    ImageDescriptionRKT Cancel Algorithm
    4^4 edge swap parityUF and UR edges swapped[f' l': [[r' U' l': D2], Iy2]]
    +
    +
    +OCLL + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    ImageNameRKT Cancel Algorithm
    Sune OCLLSuneR U R' {1-2}Ozx2 U R {1-2}Ozx2 U2 R'
    Antisune OCLLAntisuneR U2 {1-2}Ozx2 R' U' {1-2}Ozx2 R U' R'
    H OCLLHR U2 R' U' {1-2}Ozx2 R U R' U' {1-2}Ozx2 R U' R'
    U OCLLU(R U R' U R U2 R') Iy (R U2 R' U' R U' R') Iy'
    T OCLLT(R U R' U R U2 R') Iy' (R U2 R' U' R U' R') Iy
    L OCLLL(R U R' U R U2 R') Iy2 (R U2 R' U' R U' R') Iy2
    Pi OCLLPiR U2 {1-2}Ozx2 R2 U' R2 U' R2 {1-2}Ozx2 U2 R
    +
    +
    +CPLL + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    ImageNameRKT Cancel Algorithm
    Aa CPLLAa(Ix) R' U {1-2}Oz R' D2 R {1-2}Oz' U' R' D2 {1-2}Oz' R2
    Ab CPLLAb(Ix) R2 {1-2}Oz D2 R U {1-2}Oz R' D2 R {1-2}Oz' U' R
    Na CPLLNaR U' L U2 R' U L' {1-2}Oy2 R U' L U2 R' U L'
    TODONb
    +
    +
    +EPLL + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    ImageNameRKT Cancel Algorithm
    Ub EPLLUbR2 U R U {1-2}Ozx2 R' U' R' U' R' {1-2}Ozx2 U R'
    Ua EPLLUaR U' {1-2}Ozx2 R U R U R {1-2}Ozx2 U' R' U' R2
    Z EPLLZ(R U' R U R U R U' R' U' R2) Iy2 (R2 U R U R' U' R' U' R' U R') Iy2
    alternate: M2 U' M U2 M' Iy2 M U2 M' U M2
    H EPLLHM2 U' M2 U2 M Iy2 M' U2 M Iy2 M U M2
    +
    +

    Simultaneous RKT

    +

    RKT can be done by using 2 opposite sides (e.g. RO and LO turns) as well as normal I cell turns. This would allow you to execute <R,U,L> gen algorithms easier, at the cost of having to fix RKT debt on both layers. Another interesting way to do this is in a method like Belt Method. After solving the belt, orienting opposite sides, and separating the colours, the user is left with solving 2 opposite cells that can be solved simultaneously using RKT. If you turn one of the sides with the belt going through it, it essentially does a twist to both cells. One case that can be annoying is if you want to do a 180° turn on one side, and a 90° turn on the other (you would have to wait until both cells’ debt lined up).

    +

    Higher Dimensional RKT

    +

    The term Double/Triple/Quadruple/etc… RKT is used to refer to using RKT to do RKT in 5D+ puzzles. Because RKT lets us treat a single side of an nd like an nd-1, using RKT on the 35 lets us treat a single 4D cell as a 34. And if you know how to use RKT to treat a 34 like a 33, then you can do Double RKT. This generalizes to any number of dimensions, but the movecount doubles each time, making it impractical. This is why bigger nd puzzles are mostly solved using commutators.

    +

    RKT in 3D

    +

    If RKT treats a single layer of an nd puzzle like an nd-1 puzzle, then technically the last step of the 33 Roux method counts as RKT. The last step of Roux is to permute the M-slice like a 32 (with mirroring moves allowed).

    +

    Another thing that “feels like RKT” is solving 3x3xn cuboids where \(n>3\). A typical strategy for these is to solve from the innermost layers to the outermost layers, treating it as several nested 3x3x2 puzzles. Doing an R2 on the whole puzzle does an R2 to each of the subpuzzles.

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    God’s Number

    +

    God’s number is the minimum number of moves that is sufficient to solve a twisty puzzle from any starting position. For 3x3x3, this has been proven to be 20 HTM (or 26 QTM).

    +

    God’s number for 33 took lots of creative mathematical work and 35 years of CPU time to prove \(\sim 4.3 \times 10^{19}\) states reachable in 20 moves or less. For comparison, the 24 has \(\sim 3.4 \times 10^{27}\) states and 43 has \(\sim 7.4 \times 10^{45}\) states. There isn’t a single nontrivial 4D puzzle for which God’s number is known, let alone remotely possible to compute.

    +

    There are three strategies we can use to estimate it:

    +
      +
    1. Setting a lower bound using the branching factor of move sequences
      2. +
    2. Setting an upper bound by analyzing the worst-case solution of every stage in a given method
      3. +
    3. Estimation by measuring move counts produced by a near-optimal solver
      4. +
    +

    Summary

    + + + + + + + + + + + + + + + + + + + + + + + +
    PuzzleLower BoundUpper BoundEstimate
    2415 STM37 STM~20 ±5
    3451 STM570 STM~125 ±50
    +

    2x2x2x2

    +

    Lower bound

    +

    We can compute a lower bound of 15 in the STM by computing a bound on the number of positions reachable by algorithms of a certain length.

    +
    +Derivation +

    Moves here will be measured using STM. Our focus here is on positions that can be reached by algorithms of a certain length. Note that on the 24, we consider one piece fixed when counting permutations. There are \(\sim 3.4 \times 10^{27}\) positions on the 24.

    +

    One position, the solved position, is reachable without making any moves. There are 23 moves on each of the 4 cells that do not affect the fixed piece, so at most 92 one-move positions are reachable by 1 move. For subsequent moves, there are only 69 non-canceling moves since any move on the same axis as the previous move will cancel. Therefore there are at most \(92 \times 69^{k-1}\) reachable by exactly \(k\) moves (where \(k\ge 1\)). Let \(P_n\) denote the number of positions reachable in \(n\) moves or less. Based on the previous analysis,

    +
    \[P_n \le 1+\sum_{k=1}^n 92 \times 69^{k-1} = \frac{23}{17} (69^n-1)+1.\]
    +

    Let \(n_G\) be God’s number. By definition, \(P_{n_G}\) must be equal to the total number of positions. Combining this with the bound on \(P_n\), we find

    +
    \[P_{n_G} = 3.4\times 10^{27} \le \frac{23}{17} (69^{n_G}-1)+1,\]
    +

    which implies \(n_G \ge 15\).

    +
    +

    Upper bound

    +

    We can compute an upper bound of 37 in the STM by computing the worst-case move count of a known method.

    +
    +Derivation +

    Hypersolve uses a method that has a worst-case of 39 STM. Furthermore, it has been shown that all cases with move counts more than 37 STM are avoidable. Therefore God’s number is at most 37 STM.

    +
    +

    Estimate

    +

    Hypersolve typically produces solutions in the range of 20-30 STM. Note that this solver does not produce optimal solutions1, but based on this, God’s number for 24 is probably not higher than 20-30.

    +

    Conclusion

    +

    God’s number for 24 is definitely between 15 and 37 inclusive, and probably around \(\sim 20 \pm 5\).

    +
    +

    Could this be improved?

    +
      +
    • A better method or lots of compute time might improve slightly on the upper bound. More worst-case scenarios can be checked for avoidability.
      • +
    • Unless there is some fundamental breakthrough in our understanding of computation, there’s likely no way to improve on the lower bound. If you’re an expert in quantum computing then perhaps you can devise some clever quantum algorithm to help, but as of 2023 quantum computers haven’t solved a single real-world problem faster than a classical computer so we remain skeptical.
      • +
    +
    +

    3x3x3x3

    +

    Lower bound

    +

    We can compute a lower bound of 51 in the STM. This works by showing that algorithms of limited length can generate, at most, only a subset of the possible positions on the 34. We will also demonstrate some known optimizations (often useful for other puzzles), although they do not immediately improve the lower bound in the case of 34.

    +
    +Lower Bound, Winning Ways Method +

    Turns here are measured in the Slice Turn Metric (STM).

    +

    We can describe turns on the \(3^4\) as happening on one of four axes, and with one of three layers. Each layer can be turned in 23 ways, so we have \(4 \times 3 \times 23 = 276\) turns measured as one move in the STM. After turning a layer, we want subsequent turns to be noncancelling (turning the same layer twice can be written as a single move), so we have \(276-23=253\) choices for subsequent turns.

    +

    From here, we can proceed without the Winning Ways improvement, where we find the number of positions reachable by 50 turns or fewer is at most

    +
    \[1 + 276 \times \sum_{k=0}^{49}253^k \approx 1.57 \times 10^{120} < \frac{\left(24!\times 2^{24}\right) \times \left(32! \times 6^{32}\right) \times (16! \times 12^{16})}{48} \approx 1.78 \times 10^{120}.\]
    +

    This shows there are positions on the \(3^4\) which require 51 turns or more to solve. +In the book Winning Ways for Your Mathematical Plays Vol. 4, the authors optimize this argument (as it applies to the Rubik’s Cube in the half turn metric) by accounting for relations like \(LR = RL\). We can apply a similar optimization for the \(3^4\).

    +

    We will refer to the set of possible new positions reachable after exactly \(n\) turns as \(T_n\). “New” here means that \(T_n\) does not include positions that were in \(T_{n-1}\). “Exactly” means that positions reachable in more than \(n\) turns are not in \(T_n\). The number of elements in \(T_n\) is \(|T_n|\).

    +

    By the computation from before, there are 276 distinct puzzle states after a single turn. So \(|T_1|=276\). Finding \(|T_2|\) is more complicated. First, we need to setup a more detailed way of describing turning axes.

    +

    A primary axis is a line through the core of a puzzle in any of the four cardinal directions. Primary axes have three layers, which we might label \(-1,\ 0,\) and \(1\). The white-yellow primary axis’ three primary layers would correspond to the yellow cell (\(-y\)), white cell (\(y\)), and the slice between the white and yellow cells. A \textbf{secondary} axis is a line through the center of a facet and any of the pieces in that facet. Importantly, the secondary axis can be described using a single cardinal direction when it is through a ridge piece (the \(90^\circ\) degree cell turns and their doubles). A primary axis, secondary axis, and an angle which we turn by is enough to describe any turn in the STM.

    +

    We have just 253 noncancelling chocies for a second turn. Some of these second turns commute with a first turn, which means we will overcount postions reachable by two algorithms differing by the order of commuting turns. Noncancelling commuting pairs of turns can happen in two ways:

    +
      +
    1. +

      The first and second turns happen in the same primary axis and on different layers (for example, \(UO\ \ \{2\}UO' = \{2\}UO'\ \ UO\).\

      +
      2. +
    2. +

      The first turn has primary and secondary axes \(a\) and \(b\), while the second turn has primary and secondary axes \(b\) and \(a\) (for example, \(OL\ LO= LO\ OL\)).

      +
      3. +
    +

    In describing commuting cases of the first kind, the second turn is one that happens on the same primary axis as the first and on a different layer. There are two other layers, and we can make one of \(23\) turns on that layer. This means that after a single turn is made, \(2\times 23 = 46\) of the following turns may lead to commuting cases of the first kind. However, two of these actually lead to single turn puzzle states (like \(UO\ \{2\}UO = DO\)) which were counted in \(T_1\) and should not be part of \(T_2\). So, only 44 of these turns lead to distinct new puzzle states. This means there are \(\frac{44\times 276}{2}\) distinct new puzzle states reachable by two commuting moves of this first kind.

    +

    Commuting cases of the second kind can only happen if the first turn’s secondary axis goes through a ridge piece. There are \(108\) turns like this. The second turn’s primary/secondary axes are determined by transposing the first turn’s primary/secondary axes, and then can happen in one of three ways on one of three layers. So, we have \(\frac{9\times 108}{2}\) distinct new puzzle states reachable by two commuting moves of this second kind. NOTE: If the first turn is one of the \(276-108= 168\) other turns that cannot be followed by a commuting turn of the second kind, those nine second turns happen on a different primary axis and will neither cancel nor commute.

    +

    Now we can compute an upper bound on \(|T_2|\). There are \(\frac{44\times 276}{2}\) distinct new puzzle states reachable by two commuting moves of the first kind, and \(\frac{9\times 108}{2}\) distinct new puzzle states reachable by two commuting moves of the second kind. There are \(276-23-44-9=200\) noncancelling, noncommuting second turns. In total, we have \(|T_2| \leq \frac{44\times 276}{2} + \frac{9\times 108}{2} + 9\times 168 + 200\times 276 = 63,270\). As mentioned in the “NOTE”, following one of the other 168 first moves (with secondary axis \textit{not} through a ridge peice) with one of those nine turns suggested by commuting case of the second kind will never cancel or commute, so we add \(9\times 168\) to complement the “leftovers” that were ignored in the \(\frac{9\times 108}{2}\) term.

    +

    To complete the argument, we can derive a recurrence relation (more accurately a recurrence estimate) that gives an upper bound on the size of \(|T_{n+2}|\) in terms of \(|T_n|\) and \(|T_{n+1}|\).

    +

    In the case where the \(n+2\)th turn will not commute or cancel with the \(n+1\)th turn, we have at most \(9\times 168|T_{n}| + 200\times |T_{n+1}|\) possible puzzle states. In the case where the \(n+2\)th turn \textit{will} commute (but still not cancel) with the \(n+1\)th turn, we have at most \(\left(\frac{44\times 253}{2} + \frac{9\times 108}{2}\right)|T_n|\) possible puzzles states. This gives us the estimate \(|T_{n+2}| \leq 200|T_{n+1}| + 7,582|T_n|.\)

    +

    This is an order 2 linear recurrence estimate with constant coefficients. We can show that

    +
    \[|T_n| \leq A\lambda_1^n + B\lambda_2^n,\]
    +

    where

    +
    \[ +\begin{array}{c c} +\lambda_{1}=100+\sqrt{17,582}, & \lambda_{2}=100-\sqrt{17,582}, \\ +A=\displaystyle \frac{4,035}{7,582}+\frac{160,704}{3,791}\sqrt{\frac{2}{8,791}}, & +B=\displaystyle \frac{4,035}{7,582}-\frac{160,704}{3,791}\sqrt{\frac{2}{8,791}}. \\ +\end{array} +\]
    +

    Finally, the number of turns reachable by 50 turns or fewer is at most

    +
    \[1+\sum_{n=1}^{50}|T_n| \leq 2.52\times 10^{118} < \frac{\left(24!\times 2^{24}\right) \times \left(32! \times 6^{32}\right) \times (16! \times 12^{16})}{48} \approx 1.78 \times 10^{120}.\]
    +

    This shows there are positions on the \(3^4\) which require 51 turns or more to solve.

    +

    Unfortunately, we did not improve the lower bound here. It is worth noting that in the estimate from the beginning (with few optimizations), the actual value of the sum on the LHS is quite close to the number of scrambles possible on the \(3^4\) (within the same order of magnitude). Compare this to our new estimate, where the sum is bounded above by a number two orders of magnitude smaller than the number of scrambles possible on the \(3^4\). So as we would have expected, our improvements do reduce the size of the estimate, just not quite enough to bring down the number of turns derived.

    +
    +

    Below is an example of a similar argument used to derive a lower bound of 56 turns in the OBTM. Note that this argument is missing some optimizations that were applied in the STM lower bound derivation.

    +
    +OBTM Lower Bound, Winning Ways Method +

    Moves here will be measured using something equivalent (as far as this discussion is concerned) to OBTM. Our focus here is on positions that can be reached by algorithms of a certain length. Wide moves contribute to algorithm length in the same way that single cell turns do (wide move = single cell move + cube rotation, where cube rotations count as 0 moves), so we will make a simplification by restricting to single cell turns. There are 23 moves on each of the 8 cells, so 184 one-move algorithms are possible at any given time.

    +

    We know the 34 can be scrambled in over 1.7 novemtrigintillion ways. When solving, we have 184 choices for the first turn, and 161 (non cancelling) choices for the following turns. The number of positions reachable by 54 turns or fewer is at most

    +
    \[1 + 184 \times \sum_{n=0}^{53} 161^n \approx 1.695 \times 10^{119} < 1.757 \times 10^{120}.\]
    +

    This shows that there exist positions on the 34 that require 55 or more turns to solve.

    +

    In the book Winning Ways for Your Mathematical Plays Vol. 4, the authors optimize this argument (as it applies to the Rubik’s Cube in the half turn metric) by accounting for relations like \(LR = RL\). We can apply a similar optimization for the 34.

    +

    We will refer to the number of possible 34 positions after \(n\) moves as \(u_n\). It is not too hard to convince ourselves that \(u_1 = 184\). After turning one cell, moving the same cell again is what we might call a “cancelling move”, since the two moves could have been measured as a single move. So, we have \(161\) non cancelling choices for a second move. We should be able to reduce that \(161\) further by avoiding double counting positions reachable by two algorithms which differ only by the order of commuting elements.

    +

    There are different ways that a pair of moves can commute on 34. One is by the cell turns affecting completely distinct subsets of pieces, which happens when turning opposite cells, like \(OL\ IL = IL\ OL\). This is analogous to the commuting cases on the Rubik’s Cube as addressed in Winning Ways. There is at least one more way, which can involve certain adjacent cell moves. \(OL\ LO = LO\ OL\) is one example. We have not yet reached a description of all commuting cases of this kind. Accounting for these may allow us to further improve the lower bound. For now, we will factor out the commutativities that we can describe, which we will call “opposite-commuting” moves.

    +

    First, there are \(161-23 = 138\) possible non-cancelling, non-opposite-commuting choices for the second move. We’ll add the opposite-commuting cases separately.

    +

    There are four “flavors” of opposite-commuting cases: I cell moves with O cell moves, R moves with L moves, U moves with D moves, and B moves with F moves. In any of these, choose any one of the 23 moves possible on each cell (order doesn’t matter), so there are \(23 \times 23 = 529\) opposite-commuting cases of each flavor. Then across the four flavors, we have \(529\times 4 = 2,116\) distinct cases after the second move, when it opposite-commuted with the first. So we have that the number of \(3^4\) positions after \(n\) moves is at most \(138u_1 + 2,116 = 27,508\). That is, we can say \(u_2 \leq 27,508\).

    +

    As a sanity check, we could note at this point the \(27,508\) positions after two turns is better than the \(184\times 161 = 29,624\) positions we would have estimated without accounting for opposite-commutativity of two successive turns here.

    +

    To complete the argument, we can derive a recurrence relation (more accurately a recurrence estimate) that gives us an upper bound on the size of \(u_{n+2}\) in terms of \(u_n\) and \(u_{n+1}\)

    +

    In the case where the \(n+2\)th move will not opposite-commute (or cancel) with the \(n+1\)th move, we have \(138u_{n+1}\) possible puzzle states. If the \(n+2\)th move is going to opposite-commute with the \(n+1\)th move, we might have \(529\times 4 = 2,116\) cases to follow up the \(n\)th move. However, one of those four flavors of opposite commuting cases would have cancelled with the \(n\)th move, so this time we only have \(529 \times 3 = 1,587\) cases to follow the \(n\)th move. This gives us the recurrence estimate \(u_{n+2} \leq 138u_{n+1} + 1,587u_{n}\).

    +

    Note that as we have defined it, \(u_n\) is the number of positions accessible by exactly \(n\) moves. If we sum the \(u_n\)s from 1 to \(N\), that will gives us (an upper bound on) the number of positions accessible by \(n\) moves or fewer.

    +

    In this case, our recurrence estimate is an order 2 linear recurrence with constant coefficients. There exists a method to derive a general formula for \(u_n\). In the end, we find

    +
    \[u_n = C\lambda_1^n + D\lambda_2^n,\]
    +

    where

    +
    \[ +\begin{array}{c c} +\lambda_{1}=69+46\sqrt{3}, & \lambda_{2}=69-46\sqrt{3}, \\ +C=\displaystyle \frac{1}{4}\left(-22+13\sqrt{3}\right), & D=\displaystyle \frac{1}{4}\left(-22-13\sqrt{3}\right). \\ +\end{array} +\]
    +

    Finally, the number of positions reachable by 55 turns or fewer is at most

    +
    \[1\ +\ \sum_{n=1}^{55}u_n \approx 3.865 \times 10^{118} < 1.757 \times 10^{120}.\]
    +

    This shows that there exist positions on the \(3^4\) that require 56 or more turns to solve.

    +

    We should address the other side of the inequality,

    +
    \[1.757 \times 10^{120} <1\ +\ \sum_{n=1}^{56}u_n \approx 5.746 \times 10^{120}.\]
    +

    So, the argument as applied here can’t necessarily show that there are positions requiring 57 or more turns to solve.

    +
    +

    Upper bound

    +

    Hactar computed a generous upper bound of 570 STM using CFOP.

    +
    +Step-by-step breakdown +
      +
    • cross: 6 pieces x 4 STM per piece = 24 STM
      • +
    • F2L-a: 12 pairs x 13 STM per pair = 156 STM
        +
      • 3 STM to put corner on I (with corner first-cell color not on I)
        • +
      • 3 STM to put edge on I
        • +
      • 3 STM to pair + 2 STM to fix RKT debt
        • +
      • 4 STM to insert
        • +
      +
      • +
    • F2L-b: 8 pairs x 26 STM per pair = 208 STM
        +
      • 6 STM to put corner on I (with corner first-cell color not on I)
        • +
      • 6 STM to put edge on I
        • +
      • 4 RKT = 7 STM to pair them and fix RKT debt
        • +
      • 7 STM to insert
        • +
      +
      • +
    • ridge OLC: worst case (all ridges bad) is 17 STM
      • +
    • edge OLC:
        +
      • solve at least 3/4 pieces in each slice using OCLL: 3 slices x 11 STM = 33 STM
        • +
      • worst case is 3 unsolved edges remaining: 5 RKT + 9 STM = 18 STM
          +
        • 2 RKT to arrange any two edges so that they are adjacent in the same slice but do not form a valid 2-corner OLL case
          • +
        • 2 RKT to place the last edge in a way that forms a valid 3-corner OLL case
          • +
        • 1 RKT to fix RKT debt
          • +
        • 9 STM to solve the OLL case (consider OCLLs with at least one corner solved)
          • +
        +
        • +
      +
      • +
    • corner OLC: 58 STM
        +
      • 11 RKT = 23 STM to solve a 2^3 into a state where the D and U layer each contains a valid 3-corner OLC case
          +
        • if there is exactly one remaining corner unsolved, then misorient two corners when solving one of those 3-corner OLC cases.
          • +
        • if there are exactly two remaining corners unsolved, then misorient one corner when solving one of the 3-corner OLC cases.
          • +
        • arrange the cases so that the 3 unsolved corners left will form a sune
          • +
        +
        • +
      • 13 STM to solve each OLL case on U/D using RKT cancels (worst case is double antisune)
        • +
      • 9 STM to solve the remaining sune case
        • +
      +
      • +
    • ridge PLC: 14 STM (2x U perm in the worst case)
      • +
    • PLC: 42 STM
        +
      • solve a 3^3 using God’s algorithm in 20 moves
        • +
      • adjust I cell to solve
        • +
      +
      • +
    +
    +

    Estimate

    +

    We do not have a near-optimal 34 solver. Good FMC speedsolves average 200 STM, so God’s number is probably lower than that.

    +

    Conclusion

    +

    God’s number for 34 is definitely between 56 and 570 inclusive, and probably \(\sim 125 \pm 50\)[citation needed].

    +
    +

    Could this be improved?

    +
      +
    • A better method could easily improve the upper bound, probably to around 300[citation needed].
      • +
    • A better manual computation could give a slightly better lower bound.
      • +
    • An automated 34 solver or search program could give a much better estimate.
      • +
    +
    + + + + + + + +
    +
    +
      +
    1. +

      It does eventually converge on optimal solutions when run for a sufficient amount of time, but this amount of time is impractical for all but the simplest scrambles. ↩

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    Grip theory

    +

    Grip theory is a framework for analyzing many different kinds of twisty puzzles in a geometry-independent way. It was first developed on the TwistyPuzzles forum in 2009 in this thread. Grip theory was originally used to describe doctrinaire puzzles in 3D, but it can be extended to describe bandaged puzzles, jumbling puzzles, higher-dimensional puzzles, and more.

    +

    Basic grip theory

    +

    The 3x3x3

    +

    Consider a cube in 3-dimensional euclidean space. Rotations of this 3-dimensional space form a group: \(SO(3)\). This group is said to be acting on the 3D space. Consider a subset of these rotations that sends the cube to itself. These rotations form a subgroup of order 24, which is isomorphic to the symmetric group \(S_4\), although it goes by many other names. To see how this subgroup acts on the cube, we can define the 6 sides of the cube (as subsets of 3D space), and label them e.g. \(R\), \(U\), \(F\), \(L\), \(D\), and \(B\). Because we restricted our attention to rotations that preserve the cube, the image of any side under any rotation will always also be one of the sides. Thus we can define an action of the group on the 6-element set of sides \(\{R, U, F, L, D, B\}\).

    +

    A piece of the 3x3x3 can be thought of as a subset of the cube, and it is incident with one or more of its faces. Corners are incident with 3 faces, edges with 2, centers with 1, and the core with 0. At any time each piece has 3 important properties:

    +
      +
    • Set of faces with which it was incident when solved. This defines which piece it is, and is reflected in e.g. the stickers that are on the piece.
      • +
    • An element of the aforementioned rotation group which sends the piece from its solved state to its current state.
      • +
    • Set of faces with which the piece is incident now. This determines whether the piece moves or not when a particular twist is performed.
      • +
    +

    These properties are redundant: suppose we call these \(X, g, Y\) respectively. Then \(Y = \{ x \cdot g \mid x \in X\}\), or simply \(Y = X \cdot g\); and likewise and \(X = Y \cdot g^{-1}\).

    +

    When we perform a twist, we choose a face \(f\), and a rotation \(r\) of the cube that sends the chosen face to itself: \(f \cdot r = f\). Then for each piece one of two things happen:

    +
      +
    • If the piece is currently incident with \(f\), it gets acted on by \(r\), sending \((X, g, Y) \mapsto (X, gr, Y \cdot r)\).
      • +
    • If the piece is currently not incident with \(f\), it remains unchanged.
      • +
    +

    For example when we do an R twist, it’s as if the pieces currently incident with the \(R\) face underwent an x cube rotation, and other pieces remained static.

    +

    Note that when twisting around \(f\), we didn’t change whether any piece is incident with \(f\), meaning the twist can be inverted by using the same face \(f\) and the rotation \(r^{-1}\).

    +

    The grips of the 3x3x3

    +

    Instead of working with geometric shapes embedded in 3D space, we can work with abstract sets. Consider a set \(\Gamma\) of 6 abstract objects, called grips:

    +
    \[\Gamma = \{R, U, F, L, D, B\}.\]
    +

    Next consider a group acting on this set, generated by two 4-cycles that cycle \(F, U, B, D\), and \(R, F, L, B\) respectively. More precisely, we mean the subgroup \(G\) of \(S_\Gamma\) generated by \((F,U,B,D)\) and \((R,F,L,B)\). We will call this the grip group. This group and its action are isomorphic to the rotations of the cube and their action on the 6 faces we defined above.

    +

    A piece is identified by a set \(X \subseteq \Gamma\) – the set of solved grips. At any time a piece has some attitude \(g\), which is an element of the grip group \(G\). From that we can also compute the set of active grips, which is defined to be \(Y = X \cdot g\). The grip group acts on the piece like so:

    +
    \[(X, g, Y) \cdot r = (X, gr, Y \cdot r),\]
    +

    so the active grips follow the piece.

    +

    The attitude of a piece with active grip set \(Y\) can be decomposed into two components: the orientation of the piece, which is an element of the stabilizer subgroup \(\operatorname{Stab}(Y)\); and the position, which is the set of active grips \(Y\) itself. By the orbit-stabilizer theorem we can identify the orbit \(X \cdot G\) (the possible positions of this piece) with the cosets in \(G / \operatorname{Stab}(Y)\).

    +

    For example, \((\{R, U\}, (U,R)(L,D)(F,B), \{R, U\})\) is an abstract description of the up-right edge in its solved position, but wrong orientation. We can verify its set of active grips:

    +
    \[\{R, U\} \cdot (U,R)(L,D)(F,B) = \{R, U\}.\]
    +

    Note that the groups \(\operatorname{Stab}(X)\) and \(\operatorname{Stab}(X \cdot g) = \operatorname{Stab}(Y)\) are conjugate, and thus isomorphic, and in general we have a fiber bundle:

    +
    \[\operatorname{Stab}(X) \hookrightarrow G \rightarrow G / \operatorname{Stab}(X).\]
    +

    Next, define a twist to be a pair of: a grip \(f\) we twist around, or axis, and element of \(\operatorname{Stab}(f)\), the transformation. Equivalently, we can choose an arbitrary transformation \(r \in G\) first, and then an axis \(f\) from the set \(\Gamma^r\) of fixed points of \(r\). As before, the action of a twist \((f, r)\) on a piece is:

    +
    \[(X, g, Y) \mapsto \begin{cases} +(X, gr, Y \cdot r), & \text{if } f \in X \cdot g, \\ +(X, g, Y), & \text{if } f \notin X \cdot g. \\ +\end{cases}\]
    +

    For instance, applying the twist with axis \(U\) and transformation \((R,F,L,B)\), which fixes \(U\), to the piece from above, we get a new attitude

    +
    \[(U,R)(L,D)(F,B) * (R,F,L,B) = (R,U,F)(L,D,B),\]
    +

    and a new active grip set

    +
    \[\{U, R\} \cdot (R,F,L,B) = \{U, F\}.\]
    +

    The set of all positions our piece can move to are all sets of two grips except \(\{R, L\}\), \(\{U, D\}\), and \(\{F, B\}\). These exactly correspond to the edge positions of the 3x3x3. An edge piece is only affected by a twist if one of its colored faces is on the face corresponding to the axis of the twist, and after the twist, the positions of its colored faces moves. This is exactly the behavior of the active grips. Therefore, the piece above is an edge piece.

    +

    We can similarly construct three more sets of pieces: the corners, with active grips like \(\{R, U, F\}\), the centers, with active grips like \(\{R\}\), and the core, with empty active grip set. Note that even though the centers have only one grip, they still have attitude, so the centers on this puzzle are actually super centers.

    +

    States

    +

    Above we have chosen some set of pieces \(P\), a subset of \(2^\Gamma\). Notably it is an invariant subset under the action of the grip group: \(P \cdot G = P\). The state of the puzzle comprises of the states of all pieces, i.e. some set of triples \((X \in P, g \in G, Y \in P)\) such that \(X \cdot g = Y\). This defines a relation between \(P\), \(P\), and \(G\). Due to the redundancy, there are three ways to look at it:

    +
      +
    • A function from \(X\) to \(g\), called \(a\), indicating that the piece \(X\) has attitude \(a(X)\), and has active grips \(Y(X) = X \cdot a(X)\).
      • +
    • A function from \(Y\) to \(g\), also called \(g\), indicating that the piece whose active grips are \(Y\) has attitude \(g(Y)\) and its solved grips were \(X(Y) = Y \cdot g(Y)^{-1}\).
      • +
    • A permutation of \(X\)’s into \(Y\)’s, \(\pi : P \to P\), together with a choice of \(g\) at each fiber such that \(\pi(X) = X \cdot g\).
      • +
    +

    Denoting the set of such relations \(G \wr P\), the first description allows us to straightforwardly define a group structure on it. If we treat elements of \(G \wr P\) as functions from \(P\) to \(G\) in the first sense, then given \(a, b \in G \wr P\), their product \(ab\) is defined by:

    +
    \[ab(X) = a(X) b(X \cdot a(X)).\]
    +

    The second and third descriptions allow us to identify \(G \wr P\) as a subgroup of the wreath product \(G \wr_P S_P\) (hence the notation). Indeed, suppose \(g, h : P \to G\) are maps sending active grips to attitudes, and \(\pi, \sigma \in S_P\) are permutations of pieces. Defining multiplication as:

    +
    \[(\pi, g) * (\sigma, h) = (\pi\sigma, Y \mapsto g(\sigma^{-1}(Y)) h(Y)),\]
    +

    we end up with the group structure of the semidirect product \(S_P \rtimes G^P\), with the action of reindexing, i.e. exactly the wreath product \(G \wr_P S_P\). The group \(G \wr P\) is then the subgroup consisting of pairs \((\pi, g)\) that agree:

    +
    \[\forall X \in P, \pi(X) = X \cdot g(\pi(X)).\]
    +

    Note that the \(\pi\)’s are limited to those that preseve the orbits \(P / G\), else there doesn’t exist an agreeing \(g\). The wreath product thus splits into a product of independent groups, one for each orbit. In turn, for a piece with active grips \(Y\), the set of agreeing \(g(Y)\)’s is some coset of the stabilizer \(\operatorname{Stab}(Y)\) in \(G / \operatorname{Stab}(Y)\). Within an orbit, all these stabilizers are isomorphic, thus we can write \(G \wr P\) as:

    +
    \[\prod_{O \in P / G} \operatorname{Stab}(Y \in O) \wr_O S_O,\]
    +

    provided we choose a designated element \(Y\) in each orbit \(O\), and an identification of \(\operatorname{Stab}(Y)\) with its cosets.

    +

    For our 3x3x3, there are four orbits, corresponding to the four types of piece we defined above:

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    • 12 edges with stabilizers \(\mathbb{Z}_2\),
      • +
    • 8 corners with stabilizers \(\mathbb{Z}_3\),
      • +
    • 6 faces with stabilizers \(\mathbb{Z}_4\),
      • +
    • and 1 core with stabilizer \(G\).
      • +
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    The group of puzzle states is thus:

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    \[(\mathbb{Z}_2 \wr S_{12}) \times (\mathbb{Z}_3 \wr S_8) \times (\mathbb{Z}_4 \wr S_6) \times (G \wr S_1).\]
    +

    Twists

    +

    A twist acts on a piece, and thus can act on the puzzle state – a set of pieces. Because the set of states \(G \wr P\) admits a group structure, the action of a twist \((f, r)\) is actually equivalent to multiplication on the right by a certain element of \(G \wr P\) – the state you get if you apply the twist to the solved puzzle:

    +
    \[\pi(X) = \begin{cases} +X \cdot r, & \text{if } f \in X, \\ +X, & \text{if } f \notin X; \\ +\end{cases}\]
    +
    \[g(Y) = \begin{cases} +r, & \text{if } f \in Y, \\ +1, & \text{if } f \notin Y. \\ +\end{cases}\]
    +

    This defines a group homomorphism \(\operatorname{twist}_f : \operatorname{Stab}(f) \to G \wr P\). To be able to talk about a sequence of twists about different axes, we can define the twist group as the free product (coproduct) of the groups of twists about each axis:

    +
    \[\coprod_{f \in \Gamma} \operatorname{Stab}(f),\]
    +

    which comes with a free homomorphism \(\operatorname{twist} : \coprod\limits_{f \in \Gamma} \operatorname{Stab}(f) \to G \wr P\), which simply applies multiple twists in sequence.

    +

    Often not all puzzle states are reachable by twists. The image of this homomorphism is exactly the set of puzzle states that are. This is a subgroup of \(G \wr P\) and is called the puzzle group.

    +

    Various reasons why the puzzle group is smaller than \(G \wr P\) are called piece invariants. For example, no twist on a cube ever relocates a center, thus in the puzzle group’s \(\mathbb{Z}_4 \wr S_6\) component, the \(S_6\) permutation is always the identity. For more complicated reasons, in the \(\mathbb{Z}_2 \wr S_{12}\) component, the sum across all copies of \(\mathbb{Z}_2\) must be \(0\) – this is the edge orientation parity.

    +

    Complex puzzles

    +

    We do not need to be limited to the pieces that exist on the physical 3x3x3. We can take pieces corresponding to every possible subset of the grip set, thus \(P = 2^\Gamma\), which has \(2^6 = 64\) pieces. The puzzle you get when you include all the pieces of another puzzle in this sense is called the complex version of that puzzle.

    +

    The complex 3x3x3 has 10 types of pieces, which include the core, centers, edges, and corners from the 3x3x3. It also includes the double-axles, with active grips like {R, L}, and the bridges, with active grips like {R, U, L}. The other types of pieces are generated by the anti- automorphism of this puzzle. The anti- automorphism takes every piece with active grip set \(A\) to the piece whose active grip set is \(A^c\), its complement. Given a twist with axis \(a\) and transformation \(g\), the composition of two actions of the puzzle: the first action is a rotation of every piece by \(g^{-1}\), and the second is the original twist. When viewed on an actual 3x3x3, this automorphism maps moves to wide moves of the opposite faces. The anti- version of a piece is the image of a piece under the anti- automorphism. Thus, we can call the other pieces of the complex 3x3x3 anti-edges, anti-double-axles, anti-centers, and the anti-core. Anti-corners and anti-bridges are isomorphic to the corners and bridges, so they do not need to be recounted.

    +

    More puzzles

    +

    This definition is immediately generalizable by replacing the set of grips \(\Gamma\) and the grip group \(G\). For instance, using 12 grips and the grip group of a dodecahedron, we can create a megaminx, and by including all pieces, we can construct the complex megaminx, which has \(2^{12} = 4096\) pieces of 96 types1.

    +

    We can also take 8 grips, R, U, F, O, L, D, B, and I, with the grip group being the 192 symmetries of the hypercube. This allows us to construct the 34. One notable feature of this group that is not present in the previous cases is that when we take two adjacent grips (e.g. R and U), there are non-identity elements of the grip group that fix both of these, even though there are also non-identity elements that only fix one of them. This means that RKT is possible on this puzzle.

    +

    More layers

    +

    More layers can be added to a puzzle by adding more grips. For instance, to build a 5x5x5, we need two grips for every grip of the 3x3x3, which we can call 1R and 2R, and similarly for the others. The grip group will act identically on the layer-1 grips and the layer-2 grips, but will not swap them. In this way, either layer of a face of the 5x5x5 can be selected by choosing which grip on that face to use. The complex 5x5x5 can be constructed out of this grip system.

    +

    Odd-layered puzzles can be constructed like this, but even-layered puzzles are less clear. They can be constructed like the odd-layered puzzles with one more layer, but this results in identical grip systems and complex variants. Alternatively, we can use stored grips, but this results in less symmetrical puzzles.

    +

    Group automorphisms

    +

    There are more automorphisms that these puzzles can have. An inner automorphism of the grip group, which is a conjugate by an element of the group, can be extended to an automorphism of the puzzle by interpreting the inner automorphism as a whole-puzzle rotation. More interesting is what happens when we use a non-inner automorphism. The one we choose should be realizable as conjugation by a permutation of grips, though not necessarily one in the group. We consider the 12-grip complex megaminx. The grip group of this puzzle has an outer automorphism of reflection. By applying this outer automorphism, we obtain a reflection of the whole puzzle, showing that the puzzle is not chiral, even though some pieces are. More interesting is the other outer automorphism of this group, which manifests as an exotic automorphism that takes adjacent grips on the dodecahedron to non-adjacent but non-opposite grips and vice versa. This automorphism is called the co- automorphism, and it shows that complex puzzles can have a higher degree of symmetry than the geometric puzzle they were based on.

    +

    In general, if we have the grip set \(\Gamma\) with group \(G\) acting on it, the normalizer \(N_{S_\Gamma}(G)\) is the subgroup of the automorphism group of \(G\) of automorphisms that can be realized as acting on \(\Gamma\). Thus, the cos of the grip system are elements of the group \(N_{S_\Gamma}(G)/G\). This group is a subgroup of \(\operatorname{Out}(G)\), and manifests as additional symmetries of the grip system and the complex puzzle created with it. For example, the cube has two cos: the identity and the reflection. The dodecahedron has four: the identity, the reflection, and the co mentioned above and its inverse. That co applied twice is a reflection, so in a way, it can be considered a square root of the reflection.

    +

    Advanced example

    +

    Suppose we have 7 grips, labeled 001, 010, 011, 100, 101, 110, and 111, which are acted on by \(GL_3(\mathbb{F}_2)\) in the standard way. This group has order 168. The action of the grip group on the grips can be thought of as the Fano plane: a line is a set of 3 grips where one is the XOR-sum of the other two, and the group is all permutations of the grips that map lines to lines. Using this, we can build the complex puzzle and identify its piece types. First, there is clearly the core, with empty active grips, and 7 centers, with only one active grip each (e.g. {001}). Then, among the 15 2-grip pieces, there is only one orbit since the action is 2-transitive, so there is only one piece type, for instance that with active grip set {001, 010}. Among 3-grip pieces, there are only two types: those that contain three grips on a line like {001, 010, 011}, and those that don’t, like {001, 010, 100}. There are 7 of the former piece type and 28 of the latter type. Finally, for 4-, 5-, 6-, and 7- grip pieces, we can use the anti- automorphism to observe that these pieces are exactly the anti- variants of the 0- through 3-grip pieces. Thus, we have classified all the pieces of the complex Fano plane puzzle.

    +

    Extensions to grip theory

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    Infinite grips

    +

    Consider a set of grips \(\mathbb{Z}^2\), whose grip group are the rotations and translations that preserve the lattice. A puzzle constructed like this would have infinitely many twists, and would be unwieldy. A way to simplify this puzzle is to restrict the axes that twists are allowed on. Now, a grip that is not under an allowed twist is called stored. For instance, if twists are only allowed with axes \((0,0)\) and \((1,0)\), you can make a selection of pieces to make a two-circle puzzle like the Rashkey. However, the complex Rashkey would still have uncountably many pieces, many of which can reach infinitely many attitudes. A solution to this is to only allow pieces with finite active grip sets. If the active grip set of a piece is finite and there are only a finite number of allowed axes, each with finitely many transformations corresponding to a turn, the piece can only reach finitely many attitudes, despite the infinitude of the grip group.

    +

    Bandaging

    +

    The 3x3x3 is a doctrinaire puzzle. If a corner {R, U, F} and edge {R, U} are bandaged together, the puzzle ceases to be doctrinaire. This can be implemented into grip theory by adding a new kind of grip to pieces: bandaged grips. Bandaged grips follow the attitude just like active grips. We remove the corner and edge above and replace them with a new piece with active grips {R, U} and bandaged grips {F}. When we apply a twist, if its axis would turn a piece along a bandaged grip, the entire twist is illegal in this state. This exactly mimics the bandaging of the 3x3x3, because by removing F from the set of active grips of the corner and the set of inactive grips of the edge, it prevent us from performing any twist that would only move one of the two pieces.

    +

    Jumbling

    +

    We can combine these two notions. Consider the curvy copter (ignore overhang bandaging for now). Clearly, it has at least 12 grips, corresponding to the edges of a cube, with a grip group that contains the rotations of a cube acting on them. However, it should also have jumbling moves, rotations of \(\arccos\frac{1}{3}\) around one of the 12 allowed axes. If we take the image of all the grips under repeated application of this type of rotation, we end up with a countably infinite dense set of grips with a countably infinite grip group. Each piece of the curvy copter can be compared with every grip to identify its active and bandaged grips. This produces a description of the curvy copter in grip theory. This description can be simplified so there are only finitely many active and bandaged grips per piece.

    +

    Non-faithful actions

    +

    In all previous examples, the action on the grip group has been faithful. However, this does not have to be the case. Consider the rotational symmetry of the cube, the octahedral group. This group is a quotient of the binary octahedral group of order 48, so the binary octahedral group can act on 6 grips via its quotient. The resulting puzzle has the same pieces as the complex 3x3x3, but each piece can have twice as many additudes, and thus twice as many orientations in a given position. Because the binary octahedral group can be interpreted as a group of spinors, the puzzle with the 3x3x3’s pieces but the binary octahedral grip group is the quaternion cube.

    +

    Multiplex puzzles

    +

    Rather than simply assigning pieces a set of active grips, it is possible to create a set of grips combined with endomorphisms of the transforms associated with each grip. This allows for a puzzle like the 3x3x3 where some pieces perform an R while others perform R2 or R'. A full multiplex puzzle can be built by including a piece for every combination of these grip-endomorphism pairs for a given set of endomorphisms. A simple example is the Triplex Tetrahedron on this page.

    +

    Of particular note are puzzles using automorphisms of the transforms, as these preserve symmetry between different versions of the same grip. These include combination puzzles like Entangled 3x3x3s.

    +

    Lamination

    +

    Lamination is a restriction and generalization of grip theory. It allows fewer types of pieces than are described by grip theory, but the pieces it describes are closer to those of actual puzzles. It has multiple related interpretations.

    +

    Group action interpretation

    +

    Suppose, as above, there is a grip group that acts on the set of grips. We now select a block system of this action, and each block is called an axis. Now, instead of a piece having a selection of active grips, a piece will have a layer on each axis, where a layer is an assignment of a symbol (WLOG element of \(\mathbb{Z}\)) to each grip in that axis. Then, a twist still has an axis, but now it has a selected layer within that axis, as well as a transformation, an element of the grip group, that pointwise stabilizes the axis. A laminated puzzle is one that, given a selection of axes an selection of layers on each axis, contains a piece for every combination of layers. In this way, it is similar to complex puzzles.

    +

    For instance, to construct a 3x3x3, first start with the 6 grips R, U, F, L, D, and B. Then, let the axes be x = {R, L}, y = {U, D}, and z = {F, B}. We can choose a set of layers for each axis: the layers of x will be the assignments {R↦0, L↦0}, {R↦1, L↦0}, and {R↦0, L↦1}, and similary for y and z. Since there are 3 layers per axis and 3 axes, the laminated 3x3x3 has \(3^3 = 27\) pieces, exactly as many as a 3x3x3. In fact, the laminated 3x3x3 is exactly the super 3x3x3.

    +

    Representation interpretation

    +

    Now, we still have a grip group, but instead of acting on a set of grips, it acts linearly on a vector space \(V\). The axes are a set of linearly independent subspaces that span \(V\) (which implies the \(V\) is the direct sum of the axes), and this set should be setwise stabilized by the grip group. A layer is a point in a subspace. A piece as before has one layer per axis, but now, since layers are points in orthogonal subspaces, we can also identify a piece with a point in \(V\) whose projections onto the subspaces are its layers. Thus, a twist, being a selection of axis \(a\), layer \(\lambda\), and transformation \(g\), has the effect of selecting all pieces in an affine space passing through \(\lambda\) parallel to the direct sum of all axes that are not \(a\), and applying \(g\) to them.

    +

    Constructing the 3x3x3 is even more geometrically intuitive than before. Now, we have three axes x, y, z, which correspond to the three coordinate axes of \(\mathbb{R}^3\). The grip group acts on \(\mathbb{R}^3\) as the rotational symmetries of the cube, and the layers of x will be the surfaces \(x=0\), \(x=-1\), and \(x=1\), and similarly for y and z. Like this, the 27 pieces of the laminated 3x3x3 form an arrangement just like that of the actual 3x3x3. However, this approach is not limited to 3 dimensions, and can be used to construct all the puzzles of doctrinaire grip theory.

    +

    More layers

    +

    Laminated puzzles, as the name suggests, are made of layers, so adding more layers is trivial. Moreover, even-layered puzzles can be constructed as naturally as odd-layered ones. This makes them a more natural description of these puzzles.

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      https://www.gcsu.edu/sites/files/page-assets/node-808/attachments/powers.pdf ↩

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    Piece invariants

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    Many puzzles exhibit a phenomenon where the position or orientation of some pieces are determined by the position or orientation of the other pieces. This can often be captured with an invariant, a function of position that does not change when moves are applied, and calculating the invariant can identify whether a position is valid or not. Invariants are useful when computing the total numer of positions of a puzzle.

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    The most common kinds of invariants are permutation parity and total orientation.

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    Permutation parity

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    The permutation parity of a permutation is the parity of the number of swaps needed to produce it. It is exhibited well in k-Card Game.

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    Monoflip

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    Monoflip is a phenomenon where a single corner piece on the 3x3x3x3 can be flipped in place, with two 2-swaps of stickers, while all other corners are solved. Conversely, if 15 corners on the puzzle are solved, the last corner must either be solved or in one of 3 orientations of the monoflip. This effect is unique to 4D hypercubes. In 3D, having 7 solved corners implies the last corner must also be solved, and in 5D and above, the orientation of the last corner is independent of the orientations of the others.

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    The monoflip exists because the group of orientations of a single corner, ignoring others, is \(A_4\), and its commutator subgroup is the Klein four-group \(\mathbb{Z}_2 \times \mathbb{Z}_2\). Each element of this group corresponds to either the identity or a monoflip. Its quotient in \(A_4\) is \(\mathbb{Z}_3\), which represents an invariant analogous to total corner twist in 3D. This will allow us to prove that the orientation of the last corner, given that the other 15 are solved, must be identity or monoflip.

    +

    First, we construct a particular set of 8 ridges by choosing two sets of 4 parallel ridges which are mutually orthogonal (for instance, choose two disjoint rings of 4 cells each, and select the ridges which lie between two cells of the same ring). We say these ridges are red. We can take two more sets of 8 ridges the same way both disjoint from the first one, and we call those ridges green and blue. Notice that if we perform a monoflip on a corner colored like this, the arrangement of colors does not change.

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    If we pick a particular corner position, we can call its position even. Then, we specify that positions adjacent to even positions are odd, and positions adjacent to odd positions are even. This is well-defined and unique given our original choice. Coordinate-wise, if the coordinates of the corners are in \(\{-1,1\}^4\), corners whose coordinates have product \(1\) can be even and those with product \(-1\) can be odd. Then, in any position of the 3x3x3x3, we define the variant of a particular corner based on two factors: the color in the solved position of where the red ridge of the corner is, and whether the position of the corner is even or odd.

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    EvenOdd
    Red00
    Green12
    Blue21
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    In effect, in an odd position, the roles of green and blue are swapped. By adding the variant of each corner, we get the invariant of the position.

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    We now analyze how the invariant changes under turns. Since quarter turns form a generating set, we only need to consider those. Each quarter turn fixes one color of ridge and swaps the other two. It also takes corners in even positions to odd positions and vice versa, which swaps blue and green (for the purposes of computing the variant). Thus, it effectively either keeps all colors fixed or 3-cycles them. This has the effect of adding \(k\) to the variants of even-positioned corners and subtracting \(k\) from the odd-positioned corners for some \(k \in \mathbb{Z}_3\). Thus, the invariant stays fixed. Since the invariant of the solved position is \(0\), it is \(0\) in any position and so if 15 corners are solved, the last one can only be in a variant-\(0\) position, that is, identity or monoflip.

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    Physical puzzles

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    Higher-dimensional (“virtual”) puzzles naturally exist in more than 3 dimensions. Physical puzzles are a way to make a 3-dimensional object that functions similarly to this higher-dimensional puzzle. They must be able to perform the moves of the puzzle, so everything that can be done on the virtual puzzle can be done on the physical puzzle. However, most physical puzzles are able to perform other moves that are not allowed on the virtual puzzle, and these must be disallowed by the rules of operating the puzzle. The more rules a puzzle requires, the more complicated operating it becomes, and it becomes less interesting. The most interesting and natural physical puzzles implement as many of the restrictions as possible as part of the mechanism.

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    Warning

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    The relationship between virtual and physical puzzles is reversed from what it usually is in 3 dimensions! In the normal case, the physical puzzle has a mechanism that restricts the possible moves, and the unrestricted moves become legal, and a virtual simulation of the puzzle should copy the moves from the physical puzzle. However, when constructing a virtual higher-dimensional puzzle, the legal moves are defined abstractly, and the physical puzzle needs to have restrictions applied to it by rules.

    +
    +

    The simplest way to construct a physical puzzle is to create the most unrestricted object possible, and apply all restrictions via the rules. We can make one physical piece for each sticker of the virtual puzzle, and observe on the virtual puzzle the permutation of the stickers when applying one move. When we allow this set of moves on the physical puzzle, we get sticker soup. This is the simplest way to construct a physical puzzle, and since it has no mechanism at all, the rules of operation are just the permutation. This is generally considered uninteresting.

    +

    Since in a virtual puzzle, stickers are part of pieces and cannot be separated, it makes sense to make this a restriction of a physical puzzle as well. In this case, a physical piece is a virtual piece. This applies some design constraints to the puzzle. On the virtual puzzle, there are generally moves that reorient a piece, which has the effect of permuting its stickers. This produces the orientation group of the piece. A physical puzzle built like this must have pieces that can represent that symmetry group.

    +

    Melinda’s 24

    +

    Melinda's 2×2×2×2

    +

    The 24 has 16 pieces, each with 4 stickers. By performing moves on a virtual 24, a piece can be brought back to its position with any even permutation of stickers. (It is not just the monoflip because that assumes the rest of the puzzle is solved, which is not important here.) In three dimensions, a tetrahedron with its four vertices representing the stickers has the same symmetry. (This is not a coincidence; a 24 built on the surface of a hypersphere will have its pieces be spherical tetrahedra with the stickers at the vertices.) Thus, the pieces of Melinda’s 24 have tetrahedral symmetry.

    +

    This is not the only design decision resulting in a usable physical puzzle. Since the pieces of the 24 can be arranged in two cubic layers, the pieces are cubes to allow them to stick together in this fashion. This makes many of the moves simply rotating one whole cube of 8 pieces, and the others rotate 4 pieces from each layer in a simple way. These moves are unable to swap a white or yellow sticker with a sticker of a different color, which means the puzzle simulates a cubic prism puzzle: a 23x21 cuboid, where the first three axes can turn into each other but the last one cannot in the same way that a 3x3x3x2 is restricted.

    +

    The virtual 24, on the other hand, does allow moves that take a white or yellow sticker to a sticker of a different color, and the pieces were designed so that this is possible. It would be possible to observe each move that does that, record which piece goes where, and apply that to the physical 24, but this is unwieldy. Instead, Melinda introduced a gyro, a sequence of actions that does the equivalent of a whole-puzzle rotation on the virtual puzzle, but takes a white sticker to a sticker on a different axis. This means other moves are accessible by performing the gyro and then a simple move. On this puzzle, a gyro is usually about six physical twisting actions, but on other puzzles, they can be more complicated, so finding a puzzle that supports a short gyro is valuable.

    +

    Restricted 25

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    Restricted 2×2×2×2×2

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    The pieces of a virtual 25 have 60 orientations, the even permutations of its 5 stickers. It is difficult to design a piece with sufficient symmetry to represent this, so the restricted 25 sets one axis to be unswappable with the others, i.e. it is a 24x21 in the notation of the previous section. This means the pieces only need to be able to evenly permute 4 of their stickers, which is the same symmetry as the pieces of the physical 24. The pieces from that can thus be reused to build the restricted 25, with one sticker added to each piece. The extra sticker needs to be invariant under all rotations of the piece, so it needs to be applied to every face of the piece.

    +

    The design of the puzzle is simple: two copies of the physical 24, where the legal moves are doing a whole puzzle rotation on one, doing the same twist on both, or swapping 8 pieces from each half by twisting a side of the puzzle. The design requires a buffer layer (pictured above in purple), because if it were not present, doing a 90° rotation of the whole puzzle corresponds to three moves on the virtual restricted 25, which is considered illegal in solves.

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    Unrestricted 25

    +

    Grant's 2×2×2×2×2

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    It is not impossible to design a piece with 5 stickers that supports all even permutations of its stickers. The symmetry group of this piece is called \(A_5\), the alternating group on 5 points, and it is isomorphic to the symmetry of the icosahedron. It is not obvious whether there are actually 5 objects permuted by this, but there are: the 5 tetrahedra of the compound of 5 tetrahedra, or the 5 cubes of the compound of 5 cubes. Any even permutation of the 5 components of either shape can be realized as a rotation of the whole shape.

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    The compounds of 5 tetrahedra and 5 cubes1

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    +Compound of 5 tetrahedra +Compound of 5 cubes +

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    Observe that in the compound of 5 cubes, if you take one cube as fixed, the other four cubes each share one pair of opposite vertices with it. In particular, the arrangement of these 4 other cubes has the same symmetry as the 4 stickers of the piece of the physical 24, or the 4 exchangeable stickers of the restricted 25. Thus, by designing a piece based on this object, the true physical 25 can be implemented that shares all the moves of the restricted 25, along with one new gyro move. Because of the complexity of the puzzle, the gyro is very difficult to execute.

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    25 gyro, by Akkei

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    2^5^ gyro, by Akkei

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    26

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    The virtual 26 has pieces, which have 6 stickers each. Each piece must be able to perform all even permutations of all 6 stickers, which results in the group \(A_6\) of order 360. A rigid piece in 3 dimensions must have as its symmetry a chiral point group. However, the only discrete chiral point groups in 3 dimensions of order 360 all contain a rotation by \(\frac{\pi}{90}\), which has order 180. \(A_6\) has no elements of order 180, so none of these point groups are isomorphic to \(A_6\), and a physical 26 with rigid pieces is impossible. (The lowest dimension to have an object with the requisite symmetry is 5, with the 5-simplex.)

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    This does not preclude other designs for a physical 2^6 that have not been developed yet. For instance, a skewb has 6 center pieces, and all even permutations of them are accessible with skewb twists. This means that skewbs can be used as pieces for a physical 26. Whether there is a design for it that does not require manipulating the individual pieces every turn is unknown.

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    Restricted 26

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    Akkeis's restricted 2×2×2×2×2×2

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    There are several ways to restrict a 26 to make it able to be made physical. One way is to make it a 25x21, which can be made physical out of two copies of the 25 in the same way as the 24x21 was made out of the 24. Another way is the 24x22, pictured above. Each piece of the 24x22 has four 4-stickers and two 2-stickers. The four 4-stickers can have an even permutation applied to them, or the two 2-stickers can be swapped and an odd permutation can be applied to the four 4-stickers. This enlarges the symmetry group of the piece from \(A_4\) to \(S_4\), the symmetric group on 4 points. Fortunately, octahedral symmetry acts on the four space diagonals of a cube by \(S_4\), and one endpoint from each diagonal corresponds to the vertices of a tetrahedron, used for the stickers on the 24. Therefore, the 4-stickers can be placed on opposite corners of a cube. The stickers on the periphery of the pieces correspond to the 2-stickers, which swap whenever the piece is rotated in a way that does not preserve the original tetrahedron, i.e. one that does an odd permutation. It may require one or more buffer layers.

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      Images created with Stella software. ↩

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    + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/videos/index.html b/videos/index.html new file mode 100644 index 00000000..f0667e45 --- /dev/null +++ b/videos/index.html @@ -0,0 +1,4018 @@ + + + + + + + + + + + + + + + + + + + + + + + + + Videos - Hypercubing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    YouTube Videos & Playlists

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    List of popular videos and playlists related to hypercubing.

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    YouTube Videos

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    YouTube Playlists

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Once you feel comfortable solving 33 from a full scramble and 34 from a 1-move scramble, you know everything you need to do a full solve. If you get stuck, you can ask for help on the Hypercubers Discord server.