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We apply Monte Carlo simulations to count the numbers of solutions of twowell-known combinatorial problems: the N-queens problem and Latin squareproblem. The original system is first converted to a general thermodynamicsystem, from which the number of solutions of the original system is obtainedby using the method of computing the partition function. Collective moves areused to further accelerate sampling: swap moves are used in the N-queensproblem and a cluster algorithm is developed for the Latin squares. The methodcan handle systems of $10^4$ degrees of freedom with more than $10^10000$solutions. We also observe a distinct finite size effect of the Latin squaresystem: its heat capacity gradually develops a second maximum as the sizeincreases.

Counting Solutions for the N-queens and Latin Square Problems by Efficient Monte Carlo Simulations