The Problem of

Let f(n) denote the number of conngurations of n 2 mutually non-attacking kings on a 2n 2n chessboard. We show that log f(n) grows like 2n log n 2n log 2, with an error term of O(n 4=5 log n). The result depends on an estimate for the sum of the entries of a high power of a matrix with positive entries. In chess, two kings can attack one another if their squares are horizontally, vertically, or diagonally adjacent. Consider the problem of placing mutually non-attacking kings on a chessboard with 2m rows and 2n columns. Partitioning the chessboard into 2 2 cells, we see that no cell can contain more than one king, so there can be no more than mn kings: Figure 1 In this note, we estimate the number K(m; n) of conngurations of mn kings. H. Wilf 5] has obtained good estimates in the case that m is xed and n m. We consider the order of growth when both m and n tend to innnity, and especially the case m = n. Our main result is stated at the end of the paper. Before proceeding with the problem of kings, it may be worth a brief look at the analogous problem for other chess pieces. No more than n mutually non-attacking rooks can t on an n n board; the legal conngurations have one rook in each row and column and are therefore given by the n! permutations. It is known that there exist conngurations of n mutually non-attacking queens as long as n 4, and the number of such conngurations is conjecturally super-exponential in n 4]. For bishops, it is not too diicult to see that