Define an $(a,b)$-knight as a chess piece that moves $a$ squares horizontally and $b$ squares vertically, or vice versa, in either order. Thus a normal chess knight is a $(1,2)$-knight.

For which $a$ and $b$, does an $(a,b)$-knight admit a (closed) knight's tour?

Assume $a \le b$, and $1 \le a,b \le 7$ for an $8 \times 8$ board. For example, a $(7,7)$-knight can only jump between opposite corners.

Of course the question can be generalized to $n \times n$ boards.

Likely this question has been studied...? Thanks for pointers!

The paper JMac31 found ("Generalized knight's tours on rectangular chessboards") directly addresses my question. For example, for a tour to exist on an $n \times n$ board, two necessary conditions are that $a+b$ must be odd, and $a + b \le n$.


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