Knights Problem


The problem of determining how many nonattacking knights K(n) can be placed on an n×n chessboard. For n=8, the solution is 32 (illustrated above). In general, the solutions are

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

giving the sequence 1, 4, 5, 8, 13, 18, 25, ... (OEIS A030978, Dudeney 1970, p. 96; Madachy 1979).


The minimal number of knights needed to occupy or attack every square on an n×n chessboard (i.e., domination numbers for the n×n knight graphs) are given for n=1, 2, ... by 1, 4, 4, 4, 5, 8, 10, 12, 14, ... (OEIS A006075), with corresponding numbers of such solutions given by 1, 1, 2, 3, 8, 22, 3, ... (OEIS A006076).

See also

Bishops Problem, Chess, Domination Number, Kings Problem, Knight Graph, Queens Problem, Rooks Problem

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Dudeney, H. E. "The Knight-Guards." §319 in Amusements in Mathematics. New York: Dover, p. 95, 1970.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 38-39, 1979.Moser, L. "King Paths on a Chessboard." Math. Gaz. 39, 54, 1955.Sloane, N. J. A. Sequences A006075/M3224, A006076/M0884, and A030978 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. Figure M3224 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 196-197, 1991.Watkins, J. Across the Board: The Mathematics of Chessboard Problems. Princeton, NJ: Princeton University Press, 2004.Wilf, H. S. "The Problem of Kings." Electronic J. Combinatorics 2, No. 1, R3, 1-7, 1995.

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Knights Problem

Cite this as:

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

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