Regular solutions of the n-queens problem on the torus

Abstract

The n-queens problem on the torus is the problem of placing n queens on an n × n chessboard drawn on the torus so that no two queens attack each other. This is known to be possible if and only if n ≡ ±1 (mod 6). A solution to this problem is said to be regular if it places queens on all squares with co-ordinates (x + a, kx + b) for some fixed integers k ≠ 0, a and b. We determine the number of non-isometric regular solutions for each n ≡ ±1 (mod 6).