A $q$-Queens Problem. VI. The Bishops' Period

TL;DR

This paper proves that the observed period 2 in the quasipolynomial counting function for placing multiple bishops on a chessboard is actually the exact period for all numbers of bishops greater than 2, using advanced mathematical tools.

Contribution

It establishes the exact period of the quasipolynomial counting function for bishops, confirming empirical observations with a rigorous proof.

Findings

The period for bishops is exactly 2 for all numbers greater than 2.

The proof uses signed graphs and Ehrhart theory of inside-out polytopes.

The result clarifies the structure of the counting function for bishops.

Abstract

The number of ways to place $q$ nonattacking queens, bishops, or similar chess pieces on an $n\times n$ 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.