A q - QUEENS PROBLEM I . GENERAL THEORY MARCH 3 , 2013

We establish a general counting theory for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen, and we apply the theory to square boards. We show that the number of ways to place q nonattacking queens on a chessboard of variable size n but fixed shape is a quasipolynomial function of n. The period of the quasipolynomial is bounded by a function of the queen's move directions. Similar conclusions hold for any piece whose moves have unlimited length. We apply our theory to the square board, to show that the highest-order coefficients of the counting quasipolynomial do not depend on the size of the board. On the other hand, we present simple pieces for which the fourth quasipolynomial coefficient is periodic.