The modular \(n\)-queens problem in higher dimensions (Q1903728)

A modular chessboard is a square board with opposite edges identified like a torus. A queen moves on such a chessboard like in an ordinary chessboard but it may continue along the extended diagonals. The previous results on placements of non-attacking queens are summarized in Theorem 1: (a) \(M(n,2) = n\) if \(\text{gcd} (n,6) = 1\), (b) \(M(n,2) = n - 1\) if \(\text{gcd} (n,12) = 2\), (c) \(M(n,2) = n - 2\) otherwise, where \(M(n,d)\) is the maximum number of non-attacking queens on a \(d\)-dimensional modular chessboard. The author studied the higher dimensional cases and proved Theorem 2: (a) \(M(n,d) = n\) if \(\text{gcd} (n, (2^d - 1)!) = 1\), (b) \(M(n,d) < n\) if \(\text{gcd} (n, (2^d - 1)!) > 1\), (c) \(M(n,d) = 1\) if \(n \leq 2^d - 1\), (d) if \(\text{gcd} (n, (2^d - 1)!) > 1\), there are no \(n\) non- attacking queens on a straight line.