The $N$-queens problem asks for the number $Q\left(N\right)$ of ways to place $N$ mutually nonattacking queens on an $N\times N$ chessboard. Simkin [M. Simkin, Adv. Math. 427, 109127 (2023)] proved $Q\left(N\right)\sim {(N/e^{\gamma })}^N$, and subsequent work [P. Nobel, A. Agrawal, and S. Boyd, Optim. Lett. 17, 1229 (2023)] refined the constant to $\gamma =1.94400\left(1\right)$, 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 $T^{*}\approx 0.235\mathrm{J}$, associated with a defect-proliferation crossover. This convergence, combined with the trivially exact high-temperature entropy $S\left(\infty \right)/N=\frac{1}{N}ln\left(\genfrac{}{}{0pt}{}{N^2}{N}\right)$, enables a thermodynamic integration of $C_v/T$ that extracts the ground-state entropy and yields ${\gamma }_{\mathrm{MC}}=1.946\pm 0.003$ 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 $Q\left(N\right)$. 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.

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• Classical statistical mechanics
• Entropy
• Monte Carlo methods