A q - QUEENS PROBLEM V . THE BISHOPS ’ PERIOD
نویسنده
چکیده
Part I showed that the number of ways to place q nonattacking queens or similar chess pieces on an n× n square chessboard is a quasipolynomial function of n. We prove the previously empirically observed period of the bishops quasipolynomial, which is exactly 2 for three or more bishops. The proof depends on signed graphs and the Ehrhart theory of inside-out polytopes.
منابع مشابه
A q - QUEENS PROBLEM V . THE BISHOPS ’ PERIOD
Part I showed that the number of ways to place q nonattacking queens or similar chess pieces on an n× n square chessboard is a quasipolynomial function of n. We prove the previously empirically observed period of the bishops quasipolynomial, which is exactly 2 for three or more bishops. The proof depends on signed graphs and the Ehrhart theory of inside-out polytopes.
متن کاملA q-QUEENS PROBLEM IV. QUEENS, BISHOPS, NIGHTRIDERS (AND ROOKS)
Parts I–III showed that the number of ways to place q nonattacking queens or similar chess pieces on an n × n chessboard is a quasipolynomial function of n whose coefficients are essentially polynomials in q and, for pieces with some of the queen’s moves, proved formulas for these counting quasipolynomials for small numbers of pieces and highorder coefficients of the general counting quasipolyn...
متن کاملA q - QUEENS PROBLEM I . GENERAL THEORY August
By means of the Ehrhart theory of inside-out polytopes we establish a general counting theory for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen, on a polygonal convex board. The number of ways to place q identical nonattacking pieces on a board of variable size n but fixed shape is (up to a normalization) given by a quasipolynomial function of n, ...
متن کاملA 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 q...
متن کاملA q-QUEENS PROBLEM III. PARTIAL QUEENS
Parts I and II showed that the number of ways to place q nonattacking queens or similar chess pieces on an n× n square chessboard is a quasipolynomial function of n in which the coefficients are essentially polynomials in q. We explore this function for partial queens, which are pieces like the rook and bishop whose moves are a subset of those of the queen. We compute the five highest-order coe...
متن کامل