The n-queens solution count Q(n) is divisible by 4

We consider the classical $n$-queens problem, which asks how many ways one can place $n$ mutually non-attacking queens on an $n$ x $n$ chessboard. We prove that the total number of solutions to the $n$-queens problem $Q(n)$ is divisible by 4 whenever $n \ge 6$.

💡 Research Summary

The paper “The n‑queens solution count Q(n) is divisible by 4” addresses a long‑standing arithmetic property of the classic n‑queens problem: for board sizes n ≥ 6 the total number of distinct solutions Q(n) is a multiple of four. The author begins with a brief historical overview, noting that while the evenness of Q(n) (i.e., Q(n) ≡ 0 (mod 2) for n ≥ 2) has been known for decades, the stronger divisibility by four had not been formally established.

In Section 2 the author sets up notation. The board is modeled as the Cartesian product