Nim on Integer Partitions and Hyperrectangles
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913877994766336 |
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| author | Gottlieb, Eric Krnc, Matjaž Muršič, Peter |
| author_facet | Gottlieb, Eric Krnc, Matjaž Muršič, Peter |
| contents | We describe PNim and RNim, two variants of Nim in which piles of tokens are replaced with integer partitions or hyperrectangles. In PNim, the players choose one of the integer partitions and remove a positive number of rows or a positive number of columns from the Young diagram of that partition. In RNim, players choose one of the hyperrectangles and reduce one of its side lengths.
For PNim, we find a tight upper bound for the Sprague-Grundy values of partitions and characterize partitions with Sprague-Grundy value one. For RNim, we provide a formula for the Sprague-Grundy value of any position. We classify both games in the Conway-Gurvich-Ho hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nim on Integer Partitions and Hyperrectangles Gottlieb, Eric Krnc, Matjaž Muršič, Peter Combinatorics 91A46, 05A17 We describe PNim and RNim, two variants of Nim in which piles of tokens are replaced with integer partitions or hyperrectangles. In PNim, the players choose one of the integer partitions and remove a positive number of rows or a positive number of columns from the Young diagram of that partition. In RNim, players choose one of the hyperrectangles and reduce one of its side lengths. For PNim, we find a tight upper bound for the Sprague-Grundy values of partitions and characterize partitions with Sprague-Grundy value one. For RNim, we provide a formula for the Sprague-Grundy value of any position. We classify both games in the Conway-Gurvich-Ho hierarchy. |
| title | Nim on Integer Partitions and Hyperrectangles |
| topic | Combinatorics 91A46, 05A17 |
| url | https://arxiv.org/abs/2506.04991 |