Impartial Chess on Integer Partitions
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912202992123904 |
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| author | Gottlieb, Eric Krnc, Matjaž Muršič, Peter |
| author_facet | Gottlieb, Eric Krnc, Matjaž Muršič, Peter |
| contents | Berlekamp proposed a class of impartial combinatorial games based on the moves of chess pieces on rectangular boards. We generalize impartial chess games by playing them on Young diagrams and obtain results about winning and losing positions and Sprague-Grundy values for all chess pieces. We classify these games, and their restrictions to sets of partitions known as rectangles, staircases, and general staircases, according to the approach of Conway, later extended by Gurvich and Ho. The games $\rm {R\small OOK}$ and $\rm{Q\small UEEN}$ restricted to rectangles are known to have the same game tree as $2$-pile $\rm N{\small IM}$ and $\rm W{\small YTHOFF}$, respectively, so our work generalizes these well-known games. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14640 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Impartial Chess on Integer Partitions Gottlieb, Eric Krnc, Matjaž Muršič, Peter Combinatorics 91A46, 05A17 G.2.1 Berlekamp proposed a class of impartial combinatorial games based on the moves of chess pieces on rectangular boards. We generalize impartial chess games by playing them on Young diagrams and obtain results about winning and losing positions and Sprague-Grundy values for all chess pieces. We classify these games, and their restrictions to sets of partitions known as rectangles, staircases, and general staircases, according to the approach of Conway, later extended by Gurvich and Ho. The games $\rm {R\small OOK}$ and $\rm{Q\small UEEN}$ restricted to rectangles are known to have the same game tree as $2$-pile $\rm N{\small IM}$ and $\rm W{\small YTHOFF}$, respectively, so our work generalizes these well-known games. |
| title | Impartial Chess on Integer Partitions |
| topic | Combinatorics 91A46, 05A17 G.2.1 |
| url | https://arxiv.org/abs/2501.14640 |