Counting Kings: As Easy As λ 1 , λ 2 , λ 3 ... Neil J. Calkin, Kevin James, Shannon Purvis, Shaina Race, Clemson; Keith Schneider, UNCA; Matthew Yancey, Virginia Tech July 18, 2006 Abstract Let F (m, n) be the number of distinct configurations of non- attacking kings on an m×n chessboard. Let η2 = limm,n→∞F (m, n) 1 mn . We give rigorous and heuristic bounds for η2. We also give bounds for similar constants in higher dimensions. 1 Introduction We consider the following question: How many different ways can kings be placed on a chessboard so that no two kings can attack each other? How about on an m × n chessboard? The statement of the problem generalizes naturally to a d-dimensional board. In chess a king can attack any of the 8 squares surrounding the square in which the king is placed. Kings in the center of the board can attack any of the 8 surrounding squares while kings on the boundary of the board attack fewer. In this paper, we will first examine the Kings Problem in one dimension, then discuss the problem in two dimensions, and will eventually approach the problem in higher dimensions. Our primary approach will utilize ad- jacency matrices and the dominant eigenvalues of these matrices. We will attempt to bound the entropy constant of each system,η d , where d is the number of dimensions of the board. 2 The One-Dimensional Problem A one dimensional chess board is simply a row or column of squares. Squares at either end are adjacent to exactly one other square and all other This research was funded, in part, by a grant from the NSF. 1