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

TL;DR

This paper proves that the total number of solutions to the n-queens problem, Q(n), is divisible by 4 for all n ≥ 6, revealing a new divisibility property of these solutions.

Contribution

The paper establishes a novel divisibility property of Q(n), showing it is divisible by 4 for all n ≥ 6, which was previously unknown.

Findings

Q(n) is divisible by 4 for all n ≥ 6

The result applies to all sufficiently large n

Provides insight into the structure of n-queens solutions

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 $n \ge 6$.