Closed-Form Expressions for the n-Queens Problem and Related Problems

In this paper, we derive simple closed-form expressions for the n-queens problem and three related problems in terms of permanents of (0, 1) matrices. These formulas are the first of their kind. Moreover, they provide the first method for solving these problems with polynomial space that has a nontrivial time complexity bound. We then show how a closed form for the number of Latin squares of order n follows from our method. Finally, we prove lower bounds. In particular, we show that the permanent of Schur’s complex-valued matrix is a lower bound for the toroidal semi-queens problem, or equivalently, the number of transversals in a cyclic Latin square.