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arXiv · 2602.20182

Recursive Patterns in the Chocolate Game

Tomoro Okubo·Yuzuri Kashiwagi·Nobumitsu Niida
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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.

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Tomoro Okubo, Yuzuri Kashiwagi, Nobumitsu Niida. 2026-02-20. Recursive Patterns in the Chocolate Game. https://arxiv.org/abs/2602.20182

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