OP-00992 Open

An asymptotic formula for the number of n-queens configurations

Let $Q(n)$ be the number of ways to place $n$ non-attacking queens on an $n \times n$ chessboard. Simkin showed that $Q(n) = ((1 + o(1)) n e^{-\alpha})^n$, with $\alpha \approx 1.944$ given by an entropy optimisation problem; Nobel, Agrawal and Boyd showed $\alpha \in [1.944000752, 1.944001082]$, and no explicit form of $\alpha$ is known. For the toroidal board, Bowtell and Keevash showed that the number of configurations is $((1 + o(1)) n e^{-3})^n$ when $n \equiv 1$ or $5 \pmod 6$ (there are none otherwise). An asymptotic for $Q(n)$ itself, rather than for $\log Q(n)$, is open.

Problem. What is the asymptotic behaviour of the number $Q(n)$ of placements of $n$ mutually non-attacking queens on an $n \times n$ board?

Source: The n-queens problem (Bezzel 1848, Nauck 1850 for general n); the asymptotic question goes back to I. Rivin, I. Vardi and P. Zimmermann (1994). Listed as Problem 97 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: CO — Combinatorics
Secondary: PR
Keywords: asymptotic enumeration, entropy methods, n-queens problem, permutations
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{rivin1994queens,
  author = {Rivin, Igor and Vardi, Ilan and Zimmermann, Paul},
  title = {The $n$-queens problem},
  journal = {American Mathematical Monthly},
  volume = {101},
  number = {7},
  pages = {629--639},
  year = {1994},
  doi = {10.2307/2974691}
}

@article{simkin2023,
  author = {Simkin, Michael},
  title = {The number of $n$-queens configurations},
  journal = {Advances in Mathematics},
  volume = {427},
  pages = {109127},
  year = {2023},
  doi = {10.1016/j.aim.2023.109127},
  eprint = {2107.13460},
  archivePrefix = {arXiv}
}

@article{nobel2023queens,
  author = {Nobel, Parth and Agrawal, Akshay and Boyd, Stephen},
  title = {Computing tighter bounds on the $n$-queens constant via {N}ewton's method},
  journal = {Optimization Letters},
  volume = {17},
  number = {5},
  pages = {1229--1240},
  year = {2023},
  doi = {10.1007/s11590-022-01933-2}
}

@misc{bowtell2021keevash,
  author = {Bowtell, Candida and Keevash, Peter},
  title = {The $n$-queens problem},
  year = {2021},
  eprint = {2109.08083},
  archivePrefix = {arXiv}
}

@article{eberhard2019semiqueens,
  author = {Eberhard, Sean and Manners, Freddie and Mrazovi{\'c}, Rudi},
  title = {Additive triples of bijections, or the toroidal semiqueens problem},
  journal = {Journal of the European Mathematical Society},
  volume = {21},
  number = {2},
  pages = {441--463},
  year = {2019},
  doi = {10.4171/JEMS/841}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 97; version of January 2026, most recent update December 2025},
  year = {2026}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00992, "An asymptotic formula for the number of n-queens configurations", clopen.solutions.