A q-queens problem. VI. The bishops' period

Seth Chaiken, Christopher R. H. Hanusa, Thomas Zaslavsky

Abstract


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.


Keywords


Nonattacking chess pieces, Ehrhart theory, inside-out polytope, arrangement of hyperplanes, signed graph

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DOI: https://doi.org/10.26493/1855-3974.1657.d75

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications