The Invariance Reduction Process -- a New Tool to Solve Circular Nim and Related Games
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| author | Kadam, Balaji R. Dufour, Matthieu Heubach, Silvia |
| author_facet | Kadam, Balaji R. Dufour, Matthieu Heubach, Silvia |
| contents | We introduce the notion of invariant vectors of a game and develop the Invariance Reduction Process, which first uses reduction of positions via invariance and then zero and merge reductions of games to arrive at smaller, solved sub-games for closed subspaces of the positions. This process makes it much easier to prove that there are moves from N-positions to P-positions, and can also be used in some cases to show that there are no moves between P-positions. This process is suitable for all variations of the game Nim whose rule sets form a simplicial complex. We rephrase Simplicial Nim as Set Nim SN($n,A$) and derive results on the structure of the P-positions in terms of invariant vectors, without needing the background and notation of simplicial complexes. We also show that invariant vectors differ from the circuits used to describe the P-positions in Simplicial Nim and that invariant vectors have wider applicability compared to circuits. We apply the Invariance Reduction Process to derive results on the P-positions of the family of Path Nim games where play is allowed on at least half the stacks, as well as for the Circular Nim games CN($n,k$) with $n=7, k=3$ and $n=8,k=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02587 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Invariance Reduction Process -- a New Tool to Solve Circular Nim and Related Games Kadam, Balaji R. Dufour, Matthieu Heubach, Silvia Combinatorics 91A46 We introduce the notion of invariant vectors of a game and develop the Invariance Reduction Process, which first uses reduction of positions via invariance and then zero and merge reductions of games to arrive at smaller, solved sub-games for closed subspaces of the positions. This process makes it much easier to prove that there are moves from N-positions to P-positions, and can also be used in some cases to show that there are no moves between P-positions. This process is suitable for all variations of the game Nim whose rule sets form a simplicial complex. We rephrase Simplicial Nim as Set Nim SN($n,A$) and derive results on the structure of the P-positions in terms of invariant vectors, without needing the background and notation of simplicial complexes. We also show that invariant vectors differ from the circuits used to describe the P-positions in Simplicial Nim and that invariant vectors have wider applicability compared to circuits. We apply the Invariance Reduction Process to derive results on the P-positions of the family of Path Nim games where play is allowed on at least half the stacks, as well as for the Circular Nim games CN($n,k$) with $n=7, k=3$ and $n=8,k=3$. |
| title | The Invariance Reduction Process -- a New Tool to Solve Circular Nim and Related Games |
| topic | Combinatorics 91A46 |
| url | https://arxiv.org/abs/2604.02587 |