Recursive Patterns in the Chocolate Game

Abstract

We study the recursive structure of P-positions in the chocolate game $C_{m,m}$, an impartial game played on an $m \times m$ chocolate bar. We show that the set of P-positions exhibits self-similar patterns that can be described and enumerated recursively. We further establish a correspondence between these patterns and the cross-sections of a three-dimensional Sierpiński octahedron. Finally, we show that the P-positions can be generated by a second-order cellular automaton, analogous to the onedimensional Rule-60 automaton. Our results reveal deep connections between combinatorial games, fractal geometry, and discrete dynamical systems.

Figures (8)

• Figure 1: The scaled pattern $\frac{1}{m}R_{m,m}$
• Figure 2: $R_{3,3}$, $R_{5,5}$, and $R_{11,11}$
• Figure 3: Operation $T$ on a regular octahedron
• Figure 4: A horizontal section of the Sierpiński octahedron of order 4
• Figure 5: $H_{n,5}$, $H_{n,5.5}$, and $H_{n,6}$

Theorems & Definitions (25)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• proof