Day 21 docs done

src/AoC_2023/Dazbo's_Advent_of_Code_2023.ipynb

@@ -8192,7 +8192,7 @@
     "\n",
     "$$\n",
     "\\begin{align}\n",
-    "y &= ax^2 + bx + c \\nonumber \\\\\n",
+    "P(x) &= ax^2 + bx + c \\nonumber \\\\\n",
     "\\nonumber \\\\\n",
     "\\text{Therefore:} \\nonumber \\\\\n",
     "P(0) &= a.(65)^2 + b.(65) + c \\nonumber \\\\\n",
@@ -8201,17 +8201,27 @@
     "\\end{align}\n",
     "$$\n",
     "\n",
-    "We can rearrange these formulae to come up with a standard set of equations to determine the coefficients $a$, $b$ and $c$:\n",
+    "We can rearrange these formulae to come up with a standard set of equations to determine the coefficients $a$, $b$ and $c$. (We have three unknowns, so we need three equations to combine.)\n",
+    "\n",
+    "To find $c$, we can subtitute 0 for $x$:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "P(0) &= a.(0)^2 + b.(0) + c \\\\\n",
+    "c &= P(0)\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "We can then substitute and combine the equations to solve for $b$ and $c$:\n",
     "\n",
     "$$\n",
     "\\begin{align}\n",
-    "c &= P(0) \\\\\n",
     "b &= \\frac{(4.P(1) - 3.P(0) - P(2))}{2} \\\\\n",
     "a &= (P(1) - P(0) - b) \\\\\n",
     "\\end{align}\n",
     "$$\n",
     "\n",
-    "So this will give us the actual coefficients.\n",
+    "So this will give us the actual coefficients. Note that these coefficients are only valid for our _real data_ grid, since these coefficients are calculated based on our specific grid sizes and resulting _diamond_ sizes.\n",
     "\n",
     "But what value to use for $x$? Here, we want the number of whole tile lengths."
    ]
@@ -8241,8 +8251,7 @@
     "    b = (4*plot_counts[1] - 3*plot_counts[0] - plot_counts[2]) // 2\n",
     "    a = plot_counts[1] - plot_counts[0] - b\n",
     "    \n",
-    "    x = (steps - grid_size//2) // grid_size\n",
-    "    \n",
+    "    x = (steps - grid_size//2) // grid_size # number of whole tile lengths\n",
     "    return a*x**2 + b*x + c\n",
     "\n",
     "ans = solve_quadratic(input_data, plot_counts=[ct[1] for ct in plot_counts], steps=26501365)\n",