Table of contents

• Introduction
• Solver
  • Algorithm
  • Benchmarks
  • Potential Improvements

Introduction

The project contains a basic framework to create different 3D twisty puzzles. It includes definitions for general sized permutation puzzles based on the platonic solids: Tetrahedral, Cubic & Dodecahedral puzzles as well as Face turning Octahedra & Axis turning Icosahedra.

The contained defnitions can be extended to model other known shapes or to create completely new permutation puzzles with crazy structures!

Live demo: https://aditya-r-m.github.io/twisty-polyhedra

Techniques used to build the interface: https://medium.com/@adityam.rtm/creating-3d-twisty-puzzles-using-programming-5800fb7eaab8

The interface relies on cursor/touch gestures and works well with pc/mobile setups running chrome/firefox/etc.

Solver

The project contains a polynomial time solver for the 5 platonic solids. The solver stays agnostic to the shape it's given & the solution length is quadratic in terms of edge length (i.e. bounded by the number of clusters with solution of each cluster bounded by some constant).

Algorithm

The solver proceeds cluster by cluster. It breaks down the stickers into connected components & then does the following,

Pre-processing the puzzle

1. Generate Conjugates from simple twists.

   X . Y . X' where X & Y are simple twists.

2. Generate Commutators by merging Conjugates & simple twists.

   C . X . C' . X' where C is a conjugate & X is a simple twist.

The solver preserves the commutators which perform very small swaps - for example, 3 sticker cycles which leave all other stickers unaffected. Once a puzzle is pre-processed, the information stays cached & need not be computed again until the tab is killed.

Processing the given state

1. Identify & align the face centers.
2. Identify & fix odd parity clusters.
3. Use pre-built commutators to solve clusters piece by piece.
   • (L0) Attempt application of simple commutators & look for improvements.
   • (L1) Attempt application of commutators conjugated by simple twists & look for improvements.

     X . Cm . X' where X is simple twist & Cm is a commutator.

   • (L2) Find pairs of commutators whose chained application results in improvement.

     Cm . Cn where Cm & Cn are different commutators.

   • (L3) Find pairs of commutators conjugated by simple twists & look for improvements.

     X . Cm . Cn . X' where X is simple twist & Cm, Cn are different commutators.

   • (L4) Find pairs of commutators conjugated by simple twist pairs & look for improvements.

     X . Y . Cm . Cn . Y' . X' where X & Y are simple twists & Cm, Cn are different commutators.

For (L2+) algorithms, we can also consider the conjugates built by the two commutators without any significant overhead.

Note: We have to be careful with special clusters of the following kind,

1. No commutators result in small cycles for the cluster (ex. - corner & near-corner clusters of the Tetrahedra).
2. Commutators result in small cycles within the cluster, but all of them dislocate pieces of other clusters (ex. - star clusters on Dodecahedra).

We identify which cluster fall in these categories just by looking at the results of the preprocessing & solve these before considering the normal clusters. Since there are no useful commutators for type (1) special clusters, we substitute their simple atomic twists into the patterns which generally apply commutators to improve puzzle state.

At the code level, any given puzzle state can be represented by a permutation (ComposableCycle) & any allowed twist is a permutation itself. The solved state is naturally represented by the identity permutation. The key task is composing the starting state with different permutations while constantly moving towards smaller & smaller permutations.

Benchmarks (Chrome 83 | Intel i7-9750H @2.60GHz)

Puzzle	Edge size	Preprocessing time (ms)	Processing time (ms)	Solution length
Tetrahedron	3	468	11	38
Tetrahedron	4	1837	15	106
Tetrahedron	5	5654	19	250
Cube	2	149	23	64
Cube	3	1066	32	148
Cube	4	7052	62	340
Octahedron	2	216	3	33
Octahedron	3	2301	13	172
Octahedron	4	9839	19	366
Dodecahedron	2	3729	499	388
Dodecahedron	4	88374	804	2202
Dodecahedron	6	441504	2837	4050
Icosahedron	2	4317	130	568
Icosahedron	3	26648	272	1312
Icosahedron	4	104973	563	2252

Potential Improvements

The symmetries across different clusters & stickers within the same cluster can be leveraged to avoid a significant fraction of the computation for preprocessing the larger puzzles.