Official knight move rule definition

Question

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?

Answer 1

Im not sure if this is an answer, but it is too long for a comment.

I have seen it noted that a Knight, on the center of a 5x5 set of squares, covers all of the squares that would not be covered by the pieces (Bishop and Rook) that move in straight lines. This would certainly generalize, but might not be best.

I think the problem starts with the Bishops. Even with three dimensions, there are two sorts of diagonal, so we have two sorts of Bishop. Once the Rooks and Bishops are sorted out, the Knights could take whatever is left.

In $d$ dimensions I think there are $d-1$ kinds of diagonal. Together with the Rooks, they give a choice of $4d$ destinations to the linear pieces. This leaves the Knights with $5^d-4d-1$ destinations, which eventually becomes most of the $5^d$ hypercube. In very high dimensions, the Knight would have access to a fraction $(5/8)^d$ of the board.

Answer 2

What do you think, is this the best way to describe a multidimensional knight?

I'd use a more precise fairy chess notation to describe multi-dimensional chess movement. There are a few different notations, but they are all more precise than the FIDE description. For example, the extended Partlett notation for a knight is ~1/2 which means "this piece jumps over any obstacles to land and capture on its destination location, which is 1 unit away in a dimension and 2 units away in another." Another term for this is a (1,2) leaper.

In 2D, this yields the 8 regular knight moves. In more than one dimension, the same analogy would apply (so a 3D knight could move 2 units "upward" and 1 unit "inward").

If instead you wanted to translate the confusing FIDE definition into multiple dimensions, you are certainly welcome to try. But the terms "rank," "file," and "diagonal" will need more precise definition in higher dimensions. And depending on how you define those terms, you may just end up with a regular old (1,2) leaper after all the work.