Extended circular nim
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915765116993536 |
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| author | Suetsugu, Koki |
| author_facet | Suetsugu, Koki |
| contents | Circular nim $CN(m, k)$ is a variant of nim, in which there are $m$ piles of tokens arranged in a circle and each player, in their turn, chooses at most $k$ consecutive piles in the circle and removes an arbitrary number of tokens from each pile. The player must remove at least one token in total. For some cases of $m$ and $k$, closed formulas to determine which player has a winning strategy have been found. Almost all cases are still open problems. In this paper, we consider a variant of circular nim, extended circular nim. In extended circular nim $ECN(m_S, k)$, there are $m$ piles of tokes arranged in a circle. $S$ is a set of positive integers less than or equal to half of $m$. In each turn, a player chooses an integer $s \in S$. Then the player selects at most $k$ piles among those located every $s$-th position on the circle, and removes an arbitrary number of tokens from each selected pile. We show some closed formulas to determine which player has a winning strategy for the cases where the number of piles is no more than eight, and for a few generalized cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12045 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extended circular nim Suetsugu, Koki Combinatorics 91A46 Circular nim $CN(m, k)$ is a variant of nim, in which there are $m$ piles of tokens arranged in a circle and each player, in their turn, chooses at most $k$ consecutive piles in the circle and removes an arbitrary number of tokens from each pile. The player must remove at least one token in total. For some cases of $m$ and $k$, closed formulas to determine which player has a winning strategy have been found. Almost all cases are still open problems. In this paper, we consider a variant of circular nim, extended circular nim. In extended circular nim $ECN(m_S, k)$, there are $m$ piles of tokes arranged in a circle. $S$ is a set of positive integers less than or equal to half of $m$. In each turn, a player chooses an integer $s \in S$. Then the player selects at most $k$ piles among those located every $s$-th position on the circle, and removes an arbitrary number of tokens from each selected pile. We show some closed formulas to determine which player has a winning strategy for the cases where the number of piles is no more than eight, and for a few generalized cases. |
| title | Extended circular nim |
| topic | Combinatorics 91A46 |
| url | https://arxiv.org/abs/2501.12045 |