Editing Eight queens puzzle (section)

=== Asymptotic enumeration ===
In 2021, Michael Simkin proved that for large numbers ''n'', the number of solutions of the ''n'' queens problem is approximately <math>(0.143n)^n</math>.<ref>{{Cite web|last=Sloman|first=Leila|date=2021-09-21|title=Mathematician Answers Chess Problem About Attacking Queens|url=https://www.quantamagazine.org/mathematician-answers-chess-problem-about-attacking-queens-20210921/|access-date=2021-09-22|website=Quanta Magazine|language=en}}</ref>  More precisely, the number <math>\mathcal{Q}(n)</math> of solutions has [[asymptotic growth]]
<math display="block">
\mathcal{Q}(n) = ((1 \pm o(1))ne^{-\alpha})^n
</math>
where <math>\alpha</math> is a constant that lies between 1.939 and 1.945.<ref>{{Cite arXiv|last=Simkin|first=Michael|author-link=Michael Simkin|date=2021-07-28|title=The number of $n$-queens configurations|class=math.CO|eprint=2107.13460v2|language=en}}</ref> (Here ''o''(1) represents [[little o notation]].)

If one instead considers a [[flat torus|toroidal]] chessboard (where diagonals "wrap around" from the top edge to the bottom and from the left edge to the right), it is only possible to place ''n'' queens on an <math>n \times n</math> board if <math>n \equiv 1,5 \mod 6.</math>  In this case, the asymptotic number of solutions is<ref>{{Cite arXiv|last=Luria|first=Zur|date=2017-05-15|title=New bounds on the number of n-queens configurations|class=math.CO|eprint=1705.05225v2|language=en}}</ref><ref>{{Cite arXiv|last1=Bowtell|first1=Candida|last2=Keevash|first2=Peter|authorlink2 = Peter Keevash|date=2021-09-16|title=The $n$-queens problem|class=math.CO|eprint=2109.08083v1|language=en}}</ref>
<math display="block">T(n) = ((1+o(1))ne^{-3})^n.</math>