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

TL;DR

This paper studies the divisibility of the total number of $n$-queen configurations, $Q(n)$. Using the dihedral symmetry group $D_4$ acting on the $n\times n$ board, the solution set is partitioned into symmetry-fixed subsets $F_r$, $F_{r^2\setminus r}$, and $F_e$, enabling an orbit-stabilizer analysis. It proves that for $n \ge 6$, $|F_r|$ is divisible by $4$ by constructing border configurations $U_\ell$ and showing the corresponding complements $D^{(\ell)}$ pair up under the reflection $s$, so $Q(n)$ is divisible by $4$. This provides a sharper parity statement and reveals a structured combinatorial decomposition that may inform further arithmetic properties of $Q(n)$.

Abstract

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 $n \ge 6$.

Figures (3)

• Figure 1: An illustrative diagram of the $D_4$ symmetry group.
• Figure 2: Examples of 5-queens configurations with (left) and without (right) rotational symmetry under $r$, the $90^\circ$ rotation of the $5\times 5$ board.
• Figure 3: The diagonals and anti-diagonals induced by the border queens of $U_\ell$ (shown here for $n=12$ and $\ell = 4$), where each diagonal is paired with its corresponding anti-diagonal under the reflection $s$.

Theorems & Definitions (10)

• Theorem 1
• Lemma 2
• proof
• Proposition 3
• proof
• Theorem 4
• proof
• Lemma 5
• Lemma 6
• proof