A Subtraction Nim with a Pass
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916011968561152 |
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| author | Larsson, Urban Manabe, Hikaru Miyadera, Ryohei |
| author_facet | Larsson, Urban Manabe, Hikaru Miyadera, Ryohei |
| contents | We consider a subtraction Nim with subtraction set {s_1,s_2,s_3={2,4n,4n+2}, where n is a positive integer such that n >= 3. We do not treat the case that n=1 or n=2 in this article. We show that this game satisfies the reverse-mex property of Grundy numbers, i.e., G(x)=mex{G(x+s_1), G(x+s_2), G(x+s_3)}, where the mex is taken over successors rather than predecessors. We modify the rule of this subtraction Nim to allow a one-time pass, that is, a passing move usable at most once during the game, unavailable from terminal positions; once used by either player, it becomes unavailable. In classical Nim, the introduction of a pass move complicates the game, and finding a formula that describes the set of P-positions in traditional three-pile Nim with a pass remains an important open question. In the case of subtraction Nim with a pass, however, the introduction of a pass move does not complicate the game. We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14321 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Subtraction Nim with a Pass Larsson, Urban Manabe, Hikaru Miyadera, Ryohei Combinatorics 91A46 We consider a subtraction Nim with subtraction set {s_1,s_2,s_3={2,4n,4n+2}, where n is a positive integer such that n >= 3. We do not treat the case that n=1 or n=2 in this article. We show that this game satisfies the reverse-mex property of Grundy numbers, i.e., G(x)=mex{G(x+s_1), G(x+s_2), G(x+s_3)}, where the mex is taken over successors rather than predecessors. We modify the rule of this subtraction Nim to allow a one-time pass, that is, a passing move usable at most once during the game, unavailable from terminal positions; once used by either player, it becomes unavailable. In classical Nim, the introduction of a pass move complicates the game, and finding a formula that describes the set of P-positions in traditional three-pile Nim with a pass remains an important open question. In the case of subtraction Nim with a pass, however, the introduction of a pass move does not complicate the game. We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available. |
| title | A Subtraction Nim with a Pass |
| topic | Combinatorics 91A46 |
| url | https://arxiv.org/abs/2605.14321 |