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×n×· · ·×n} ⊆ ℤ k lattice since they originate as many interesting metric spaces for every 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 all k ∈ 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 every k ⩾ 4.