Problem

Source: Polish Math Olympiad 2nd stage 2023 P6

Tags: number theory, prime numbers, combinatorics, Chessboard



Given a chessboard $n \times n$, where $n\geq 4$ and $p=n+1$ is a prime number. A set of $n$ unit squares is called tactical if after putting down queens on these squares, no two queens are attacking each other. Prove that there exists a partition of the chessboard into $n-2$ tactical sets, not containing squares on the main diagonals. Queens are allowed to move horizontally, vertically and diagonally.