Not a trivial question: How would you define the knight move rule on a d-dimensional chessboard (say an 8 X 8 X ... X 8 board) by extending the official knight move rule that states how a knight moves on a regular 8 X 8 chessboard?
The most strict and consistent proposal I have found is to take the Euclidean distance between the centers of adjacent cells (as an example, just consider the distance between (0,0,...,0) and (1,0,...,0) or between (0,0,...,0) and (0,1,0,...,0), generalizing the distance between the centers of a1 and b1 or between the centers of a1 and a2 on a planar chessboard) and state that our knight can only move from a cell of our chessboard to another one of the same chessboard, under the constraint that this distance have to be equal to sqrt(5).
I used the Euclidean distance in the definition above, since Article 3.6, Section E, of the FIDE Handbook defines the 2D knight move rule as follows: "The knight may move to one of the squares nearest to that on which it stands but not on the same rank, file or diagonal". Now, if we do not accept my generalization, we should assume that a 3-dimensional knight can move from the (1,1,1) to (0,0,1) (or (2,0,1) or (2,2,1) and so forth...) and I do not think that this would be reasonable.
What do you think, is this the best way to describe a multidimensional knight?