Recursive Patterns in the Chocolate Game

TL;DR

This paper explores the recursive and fractal structure of P-positions in the chocolate game, revealing self-similarity, connections to Sierpiński octahedra, and cellular automaton generation.

Contribution

It introduces a recursive description of P-positions, links them to fractal geometry, and demonstrates their generation via cellular automata, advancing understanding of combinatorial game structures.

Findings

P-positions exhibit self-similar patterns

P-positions correspond to cross-sections of a Sierpiński octahedron

P-positions can be generated by a second-order cellular automaton

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 Sierpi\'nski 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.