N Queens Problem Asymptotics

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $[n]=\{1,\ldots,n\}$, and let $S_n$ be the permutations of $[n]$. A configuration of $n$ queens on the ordinary $n\times n$ chessboard has exactly one queen in each row and column, hence is represented uniquely by a permutation $\pi\in S_n$, with a queen at $(i,\pi(i))$. Nonattack along the two diagonal directions is the requirement \[ |\pi(i)-\pi(j)|\ne |i-j|\qquad(1\leq i<j\leq n). \] Define \[ Q(n)=\left|\{\pi\in S_n:|\pi(i)-\pi(j)|\ne|i-j|\text{ for every }i<j\}\right|. \] The board has no toroidal identifications, and configurations related by a rotation or reflection are counted separately.

Statement recorded in the supplied source.

Determine the asymptotic growth of $Q(n)$ as $n\to\infty$. In particular, determine the limit of $Q(n)^{1/n}/n$, or equivalently the constant $\alpha$ in the exponential-scale form \[ Q(n)=n^n\exp\left(-\alpha n+o(n)\right). \] [S2]

Short English statement

Determine the asymptotic number of placements of n mutually nonattacking queens on an ordinary n-by-n board.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 149 (GREEN-061). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michael Simkin, The number of $n$-queens configurations, arXiv:2107.13460v3 (2022), §1, Theorem 1.1. This theorem answers the recorded exponential-growth question: $Q(n)^{1/n}/n\to e^{-\alpha}$ for a variationally characterized constant $1.94<\alpha<1.9449$. https://arxiv.org/abs/2107.13460.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 97 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.