Cyclic impartial games with carry-on moves
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912768555220992 |
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| author | Abuku, Tomoaki Carvalho, Alda Larsson, Urban Nowakowski, Richard J. Santos, Carlos P. Suetsugu, Koki |
| author_facet | Abuku, Tomoaki Carvalho, Alda Larsson, Urban Nowakowski, Richard J. Santos, Carlos P. Suetsugu, Koki |
| contents | In an impartial combinatorial game, both players have the same options in the game and all its subpositions. The classical Sprague-Grundy Theory was developed for short impartial games, where players have a finite number of options, there are no special moves, and an infinite run is not possible. Subsequently, many generalizations have been proposed, particularly the Smith-Frankel-Perl Theory devised for games where the infinite run is possible, and the Larsson-Nowakowski-Santos Theory able to deal with entailing moves that disrupt the logic of the disjunctive sum. This work presents a generalization that combines these two theories, suitable for analyzing cyclic impartial games with carry-on moves, which are particular cases of entailing moves where the entailed player has no freedom of choice in their response. This generalization is illustrated with sc green-lime hackenbush, a game inspired by the classic green hackenbush. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cyclic impartial games with carry-on moves Abuku, Tomoaki Carvalho, Alda Larsson, Urban Nowakowski, Richard J. Santos, Carlos P. Suetsugu, Koki Combinatorics 91A46 In an impartial combinatorial game, both players have the same options in the game and all its subpositions. The classical Sprague-Grundy Theory was developed for short impartial games, where players have a finite number of options, there are no special moves, and an infinite run is not possible. Subsequently, many generalizations have been proposed, particularly the Smith-Frankel-Perl Theory devised for games where the infinite run is possible, and the Larsson-Nowakowski-Santos Theory able to deal with entailing moves that disrupt the logic of the disjunctive sum. This work presents a generalization that combines these two theories, suitable for analyzing cyclic impartial games with carry-on moves, which are particular cases of entailing moves where the entailed player has no freedom of choice in their response. This generalization is illustrated with sc green-lime hackenbush, a game inspired by the classic green hackenbush. |
| title | Cyclic impartial games with carry-on moves |
| topic | Combinatorics 91A46 |
| url | https://arxiv.org/abs/2512.14466 |