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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2404.18237 |
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| _version_ | 1866911954576080896 |
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| author | Williams, Kada |
| author_facet | Williams, Kada |
| contents | Define a queen on $\mathbb{Z}_n^d$ with admissible moves parallel to $\mathbf{x}\in\{-1,0,1\}^d$ at arbitrary length. How many queens can be placed on $\mathbb{Z}_n^d$ without any two in conflict? In two dimensions, this problem was initiated by Pólya in 1918 and resolved by Monsky in 1989. We give the first known results in $d$ dimensions, showing that the trivial upper bound $n^{d-1}$ cannot be attained if $n$ is a multiple of $5$, not $25$. We demonstrate, for every $d$, how $n^{d-1}-O(n^{d-2})$ queens can be placed independently. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torus Queen Independence Williams, Kada Combinatorics Define a queen on $\mathbb{Z}_n^d$ with admissible moves parallel to $\mathbf{x}\in\{-1,0,1\}^d$ at arbitrary length. How many queens can be placed on $\mathbb{Z}_n^d$ without any two in conflict? In two dimensions, this problem was initiated by Pólya in 1918 and resolved by Monsky in 1989. We give the first known results in $d$ dimensions, showing that the trivial upper bound $n^{d-1}$ cannot be attained if $n$ is a multiple of $5$, not $25$. We demonstrate, for every $d$, how $n^{d-1}-O(n^{d-2})$ queens can be placed independently. |
| title | Torus Queen Independence |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.18237 |