A q-queens problem VI. The bishops’ period

The number of ways to place q nonattacking queens, bishops, or similar chess pieces on an n × n square chessboard is essentially a quasipolynomial function of n (by Part I of this series). The period of the quasipolynomial is difficult to settle. Here we prove that the empirically observed period 2 for three to ten bishops is the exact period for every number of bishops greater than 2. The proof depends on signed graphs and the Ehrhart theory of inside-out polytopes.