The modular n-queens problem in higher dimensions Original Research Article

Let M(n, d) denote the maximum number of queens on a d-dimensional modular chessboard such that no two attack each other. We show that if gcd(n, (2d - 1)!) = 1 then M (n, d) = n. We also prove that if gcd(n, (2d - 1)!) > 1 then there are no complete linear solutions, and if gcd(n, (2d - 1)!) > 1 then M (n, d) < n. Moreover, if n ⩽ 2d - 1 we show M (n, d) = 1.