Metric spaces in chess and international chess pieces graph diameters
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911736910577664 |
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| author | Ripà, Marco |
| author_facet | Ripà, Marco |
| contents | This paper aims to study the graph radii and diameters induced by the $k$-dimensional versions of the well-known six international chess pieces on every finite $\{n \times n \times \dots \times n\} \subseteq \mathbb{Z}^k$ lattice since they originate as many interesting metric spaces for any proper pair $(n,k)$. For this purpose, we finally discuss a mathematically consistent generalization of all the planar FIDE chess pieces to an appropriate $k$-dimensional environment, finding (for any $k \in \mathbb{Z}^+$) the exact values of the graph radii and diameters of the $k$-rook, $k$-king, $k$-bishop, and the corresponding values for the $3$-queen, $3$-knight, and $3$-pawn. We also provide tight bounds for the graph radii and diameters of the $k$-queen, $k$-knight, and $k$-pawn, holding for any $k \geq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00016 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Metric spaces in chess and international chess pieces graph diameters Ripà, Marco History and Overview 05C12 (Primary) 00A08, 05C57 (Secondary) This paper aims to study the graph radii and diameters induced by the $k$-dimensional versions of the well-known six international chess pieces on every finite $\{n \times n \times \dots \times n\} \subseteq \mathbb{Z}^k$ lattice since they originate as many interesting metric spaces for any proper pair $(n,k)$. For this purpose, we finally discuss a mathematically consistent generalization of all the planar FIDE chess pieces to an appropriate $k$-dimensional environment, finding (for any $k \in \mathbb{Z}^+$) the exact values of the graph radii and diameters of the $k$-rook, $k$-king, $k$-bishop, and the corresponding values for the $3$-queen, $3$-knight, and $3$-pawn. We also provide tight bounds for the graph radii and diameters of the $k$-queen, $k$-knight, and $k$-pawn, holding for any $k \geq 4$. |
| title | Metric spaces in chess and international chess pieces graph diameters |
| topic | History and Overview 05C12 (Primary) 00A08, 05C57 (Secondary) |
| url | https://arxiv.org/abs/2311.00016 |