Counting $N$ Queens

Nick Polson; Vadim Sokolov

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arXiv · 2407.08830

Counting $N$ Queens

Nick Polson·Vadim Sokolov

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Abstract

Gauss proposed the problem of how to enumerate the number of solutions for placing $N$ queens on an $N\times N$ chess board, so no two queens attack each other. The N-queen problem is a classic problem in combinatorics. We describe a variety of Monte Carlo (MC) methods for counting the number of solutions. In particular, we propose a quantile re-ordering based on the Lorenz curve of a sum that is related to counting the number of solutions. We show his approach leads to an efficient polynomial-time solution. Other MC methods include vertical likelihood Monte Carlo, importance sampling, slice sampling, simulated annealing, energy-level sampling, and nested-sampling. Sampling binary matrices that identify the locations of the queens on the board can be done with a Swendsen-Wang style algorithm. Our Monte Carlo approach counts the number of solutions in polynomial time.

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Nick Polson, Vadim Sokolov. 2024-07-11. Counting $N$ Queens. https://arxiv.org/abs/2407.08830

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