Combinatorial games and the golden ratio on digraphs
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arXiv
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| Auteurs principaux: | , , , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911139399467008 |
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| author | Abuku, Tomoaki Inazu, Hiroki Inoue, Hiyu Kimura, Shun-ichi Suetsugu, Koki Watanabe, Kosaku Yamashita, Takahiro |
| author_facet | Abuku, Tomoaki Inazu, Hiroki Inoue, Hiyu Kimura, Shun-ichi Suetsugu, Koki Watanabe, Kosaku Yamashita, Takahiro |
| contents | We introduce a new combinatorial game, named Triangle Game. In this game, a directed $3$-cycle graph is given, and tokens are placed on each vertex. The player chooses an edge and takes at least one token from the initial vertex. At the same time, the player is allowed to return some tokens to the terminal vertex of the edge, as far as the total number of the tokens decreases. We describe the set of \Pps~under both normal play and misère play. The golden ratio $ϕ=\dfrac{1+\sqrt{5}}{2}$ plays an essential role in our description. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04943 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial games and the golden ratio on digraphs Abuku, Tomoaki Inazu, Hiroki Inoue, Hiyu Kimura, Shun-ichi Suetsugu, Koki Watanabe, Kosaku Yamashita, Takahiro Combinatorics We introduce a new combinatorial game, named Triangle Game. In this game, a directed $3$-cycle graph is given, and tokens are placed on each vertex. The player chooses an edge and takes at least one token from the initial vertex. At the same time, the player is allowed to return some tokens to the terminal vertex of the edge, as far as the total number of the tokens decreases. We describe the set of \Pps~under both normal play and misère play. The golden ratio $ϕ=\dfrac{1+\sqrt{5}}{2}$ plays an essential role in our description. |
| title | Combinatorial games and the golden ratio on digraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.04943 |