Statistical mechanics of the N-queens problem

The N-queens problem asks for the number Q(N) of ways to place N mutually nonattacking queens on an N×N chessboard. Simkin [M. Simkin, Adv. Math. 427, 109127 (2023)0001-870810.1016/j.aim.2023.109127] proved Q(N)∼(N/e^{γ})^{N}, and subsequent work [P. Nobel, A. Agrawal, and S. Boyd, Optim. Lett. 17, 1229 (2023)1862-447210.1007/s11590-022-01933-2] refined the constant to γ=1.94400(1), settling the asymptotic behavior of this classical combinatorial problem. In this work, we pursue a thermodynamic-integration route in an unrestricted lattice-gas ensemble. We map the N-queens problem to a lattice gas with pairwise repulsive interactions along shared rows, columns, and diagonals, and we perform extensive Monte Carlo simulations for N=8-1024 with 10^{8} sweeps per temperature point. The specific heat per queen C_{v}/N converges to a size-independent function with a broad, nondivergent maximum at T^{*}≈0.235J, associated with a defect-proliferation crossover. This convergence, combined with the trivially exact high-temperature entropy S(∞)/N=1/Nln(N^{2}/N), enables a thermodynamic integration of C_{v}/T that extracts the ground-state entropy and yields γ_{MC}=1.946±0.003 at N=1024, within 0.1% of the precise combinatorial value. The Simkin-Nobel value enters only as a benchmark; the thermodynamic route requires no prior knowledge of Q(N). We further present a transfer-matrix-based tensor-network formulation that encodes the nonattacking constraints into a rank-9 site tensor with 17 nonzero elements, providing a complementary exact-enumeration route.