Problem of the Week
Hilton Memorial Lecture
In how many ways can q queens be placed on an n × n chessboard so no two queens attack each other? What about other chess pieces, like bishops or knightriders (a fairy chess piece)? This generalization of the famous n-queens problem can be treated by a hyperplane-arrangement generalization of Ehrhart's theory of counting lattice points in a convex polytope. An ingredient in the Ehrhart-type formula is the least common denominator of the “vertices” of the polytope + the arrangement; this number depends on the Kronecker product of two matrices.