Queens on Non-square Tori

We prove that for m < n, the n × m rectangular toroidal chessboard admits gcd(m,n) nonattacking queens except in the case m = 3, n = 6. The classical n-queens problem is to place n queens on the n × n chessboard such that no pair is attacking each other. Solutions for this problem exist for all for n = 2, 3 [1]. The queens problem on a rectangular board is of little interest; on the n ×m board for m < n, one can obviously place at most m nonattacking queens and for 4 ≤ m < n, one can just take a solution on the m ×m board and extend the board. Moreover, the reader will easily find solutions on the 3 × 2 and 4 × 3 boards and so these give solutions on the n× 2 and n× 3 boards for all 3 ≤ n and 4 ≤ n respectively. In chess on a torus, one identifies the left and right edges and the top and bottom edges of the board. The n-queens problem on the n×n toroidal board has solutions when n is not divisible by 2 or 3 [2]. The traditional n-queens problem and the toriodal n-queens problem are closely related, both logically and historically (see [3]). However, unlike the rectangular traditional board, the queens problem on the rectangular toroidal board is interesting and non-trivial and yet it seems that it has not been studied. In order to work on the toroidal board we use the ring Zi = Z/(i), which we identify with {0, . . . , i − 1}, and the natural ring epimorphism : Z → Zi;x → [x]i. We give the squares of the n × m toroidal board coordinate labels (x, y), x ∈ Zm, y ∈ Zn, in the obvious way. The positive (resp. negative) diagonal is the subgroup P = {([x]m, [x]n) ; x ∈ Z} (resp. N = {([x]m, [−x]n) ; x ∈ Z}). Notice that the diagonals are both subgroups of Zm×Zn of index gcd(m,n). An addition, there is the vertical subgroup V = {(0, [x]n) ; x ∈ Z} which has index m, and the horizontal subgroup H = {([x]m, 0) ; x ∈ Z} which has index n. Queens at distinct positions (x1, y1), (x2, y2) are nonattacking if and only if (x1, y1) and (x2, y2) belong to distinct cosets of V,H, P and N . In particular, the m × n toroidal board can support no more than gcd(m,n) nonattacking queens. The aim of this paper is to prove the Theorem. For m < n, the n×m rectangular toroidal chessboard admits gcd(m,n) nonattacking queens except in the case m = 3, n = 6. Proof. First suppose that gcd(m,n) = 3. Notice that in order to place gcd(m,n) nonattacking queens on the n × m toroidal board, it suffices to place gcd(m,n) nonattacking queens on the 2 gcd(m,n)×gcd(m,n) toroidal board. So without loss of generality, we may assume that n = 2m. In this case gcd(m,n) = m.