  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 $k \geq 4$.
