Ben Green's Open Problem 97

In how many ways (asymptotically) Q(n) may n non-attacking queens be placed on an n × n chessboard? This is the n-queens problem, see [Gr24, Problem 97]. We read "asymptotically" as: find an explicit function green_97.answer with Q(n) ∼ green_97.answer n.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_97 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_97`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 open, 3 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-green-97,
  title        = {Ben Green's Open Problem 97},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-97}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

In how many ways (asymptotically) $Q (n)$ may $n$ non-attacking queens be placed on an $n \times n$ chessboard?

This is the $n$ -queens problem, see [Gr24, Problem 97]. We read "asymptotically" as: find an explicit function green_97.answer with $Q (n) \sim$ green_97.answer n. The function $n \mapsto Q (n)$ itself is not an explicit answer. [Gr24] notes that an asymptotic "rather than an asymptotic for the log" is open. What is known is $Q (n) = ((1 + o (1)) n e^{- α})^{n}$ for a constant $1.94 < α < 1.9449$ [Si23], see green_97.variants.simkin.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«97».

theorem green_97 :
    (fun n : ℕ ↦ (numQueens n : ℝ)) ~[atTop] green_97.answer

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

green_97.variants.alpha — Find an explicit value for the constant α of green_97.variants.simkin, that is, the number with Q(n)^1/n/n → e^-α.
green_97.variants.toroidal — Find an asymptotic for the number T(n) of toroidal n-queens configurations, for n ≡ 1, 5 pmod 6.

References

[Gr24] Green, Ben. "100 open problems." (2024).
[Si23] Simkin, M., The number of $n$ -queens configurations. Adv. Math. 427 (2023), Paper No. 109127. arXiv:2107.13460
[BoKe21] Bowtell, C. and Keevash, P., The $n$ -queens problem. arXiv:2109.08083 (2021).
[NAB23] Nobel, P., Agrawal, A. and Boyd, S., *Computing tighter bounds on the $n$ -queens constant via Newton's method*. Optim. Lett. 17 (2023), 1229--1240. arXiv:2112.03336
[EMM19] Eberhard, S., Manners, F. and Mrazović, R., *Additive triples of bijections, or the toroidal semiqueens problem*. J. Eur. Math. Soc. 21 (2019), 441--463. arXiv:1510.05987

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.