On the modular n-queen problem

Abstract Let M ( n ) denote the maximum number of queens on a modular chessboard such that no two attack each other. We prove that if 4 or 6 divides n then M ( n ) ⩽ n − 2 and if gcd( n , 24) = 8 then M ( n ) ⩾ n − 2. We also show that M (24) = 22.