A lower bound for the $n$-queens problem

Abstract

The $n$-queens puzzle is to place $n$ mutually non-attacking queens on an $n \times n$ chessboard. We present a simple two stage randomized algorithm to construct such configurations. In the first stage, a random greedy algorithm constructs an approximate \textit{toroidal} $n$-queens configuration. In this well-known variant the diagonals wrap around the board from left to right and from top to bottom. We show that with high probability this algorithm succeeds in placing $(1-o(1))n$ queens on the board. In the second stage, the method of absorbers is used to obtain a complete solution to the non-toroidal problem. By counting the number of choices available at each step of the random greedy algorithm we conclude that there are more than $\left( \left( 1 - o(1) \right) n e^{-3} \right)^n$ solutions to the $n$-queens problem. This proves a conjecture of Rivin, Vardi, and Zimmerman in a strong form.