Metric spaces in chess and international chess pieces graph diameters

Abstract

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 $\{n\times n\times \cdots \times n\}\subseteq {\mathbb{Z}}^k$ 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 $k\ge 4$.

35 pages, 34 figures. References and a figure added; some typos corrected; improvements on grammar and style

Cited by in corpus (1)

• Proving the existence of Euclidean knight's tours on $n\times n\times \cdots \times n$ chessboards for $n<4$