Latin squares and superqueens

Let L be a Latin square of order n with entries from {0, 1,…, n ? 1}. In addition, L is said to have the (n, k) property if, in each right or left wrap around diagonal, the number of cells with entries smaller than k is exactly k. It is established that a necessary and sufficient condition for the existence of Latin squares having the (n, k) property is that of (2|n ? 2| k) and (3|n ? 3| k). Also, these Latin squares are related to a problem of placing nonattacking queens on a toroidal chessboard.