Recursive Patterns in the Chocolate Game
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910030379352064 |
|---|---|
| author | Okubo, Tomoro Kashiwagi, Yuzuri Niida, Nobumitsu |
| author_facet | Okubo, Tomoro Kashiwagi, Yuzuri Niida, Nobumitsu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20182 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Recursive Patterns in the Chocolate Game Okubo, Tomoro Kashiwagi, Yuzuri Niida, Nobumitsu Combinatorics 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. |
| title | Recursive Patterns in the Chocolate Game |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.20182 |