Writing a graph theory preprint about metric spaces in chess, I am struggling with the formalization of how every chess piece would move on a $k$-dimensional ($n \times n \times \dots \times n \subset \mathbb{R}^k$) checkerboard.
Now, for the sake of simplicity, let us consider the following $3$D scenario and then we will move to the questions. We have a $4 \times 4 \times 4$ board, where each of the $64$ cubes/squares is identified by providing the $3$ Cartesian coordinates, each of them is a number between $0$ and $3$.
Let us assume that the 3D board is initially blank and we alternatively put only one piece in position $(1, 0, 0)$, asking which other potitions it could reach in just one move.
Question 1: If we take into account a bishop (B) in position $(1, 0, 0)$ (and we consider a $4 \times 4 \times 4$ board given by the Cartesian product $\{0, 1, 2, 3\} \times \{0, 1, 2, 3\} \times \{0, 1, 2, 3\} \subset \mathbb{R}^3$), would it be more reasonable to assume that it could reach the points $(2, 1, 0)$, $(0, 1, 0)$, $(2, 1, 1)$, $(3, 2, 0)$, $(0, 1, 1)$, $(3, 2, 2)$ or just the points $(0, 1, 1)$, $(2, 1, 1)$, $(3, 2, 2)$?
Question 2: If we take into account a pawn (P) in position $(1, 0, 0)$ (and we consider our $4 \times 4 \times 4$ board, as usual), which positions can be covered in just one move? In this regard, I have found this $3$D chess variant (see https://en.wikipedia.org/wiki/Millennium_3D_chess) that allows some counterintuitive (IMHO) pawn moves (i.e., from $(0, 1, 0)$ to $(0, 2, 1)$). What do you think from a pure mathematical point of view, bearing in mind my goal to study the graph induced by a $k$-dimensional version of the standard pawn?).
Question 3: Is it reasonable to assume that the most natural way to think to a $k$-dimensional rook (R) in position $(1, 0, 0)$ could only modify one of its $k$ coordinates at a time?
Question 4: Is it the most logic way of thinking that the points reachable in only $1$ move by a queen (Q) in position $(1, 0, 0)$ are those that can be covered by a bishop together with a rook from the same starting position (assuming that Question $1$ and Question $3$ have positively been answered), with no exception?
EDIT - I have finally managed to answer most of the questions above in the following preprint, that has just been announced on the arXiv (any comment is welcome): Metric spaces in chess and international chess pieces graph diameters