The N-queens problem demands a way to place $N$ queens on an $N$ × $N$ chessboard so that no two queens attack each other (lie on the same row, column, or diagonal). In this project, we implement a Metropolis–Hastings algorithm for sampling a solution to the problem, for any board size. Moreover, we use this algorithm to estimate the number of solutions, by sequentially approximating the partition function. Our method finds a solution for $N \leq 1000$ in less than $20 s$ and approximates correctly the number of solutions for all the exact values up to $N = 26$. For large $N$, instead, our estimates scale as $N!/e^N$, coherently with previous literature.

Read more about the project in our report.

This work is part of the course "Markov Chains and Algorithmic Applications" at EPFL, and is the joint effort of Francesco Salvi, Neta Singer and Perrine Vantalon.