Kings Problem


The problem of determining how many nonattacking kings can be placed on an n×n chessboard. For n=8, the solution is 16, as illustrated above (Madachy 1979). In general, the solutions are

 K(n)={1/4n^2   n even; 1/4(n+1)^2   n odd

(Madachy 1979), giving the sequence of doubled squares 1, 1, 4, 4, 9, 9, 16, 16, ... (OEIS A008794). This sequence has generating function


The minimal number of kings needed to occupy or attack every square on an n×n chessboard (i.e., domination numbers for the n×n king graphs) are given for n=1, 2, ... by 1, 1, 1, 4, 4, 4, 9, 9, 9, 16, ... (OEIS A075561), with the gamma(K_(8,8))=9 case illustrated above and noted by (Madachy 1979, p. 39). In general, for an m×n chessboard,


See also

Bishops Problem, Chess, Hard Hexagon Entropy Constant, Knights Problem, Queens Problem, Rooks Problem

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Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 39, 1979.Sloane, N. J. A. Sequences A008794 and A075561 in "The On-Line Encyclopedia of Integer Sequences."Watkins, J. Across the Board: The Mathematics of Chessboard Problems. Princeton, NJ: Princeton University Press, 2004.

Referenced on Wolfram|Alpha

Kings Problem

Cite this as:

Weisstein, Eric W. "Kings Problem." From MathWorld--A Wolfram Web Resource.

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