

A006075


Minimal number of knights needed to cover an n X n board.
(Formerly M3224)


11



1, 4, 4, 4, 5, 8, 10, 12, 14, 16, 21, 24, 28, 32, 36, 40, 46, 52, 57, 62
(list;
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refs;
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OFFSET

1,2


COMMENTS

How many knights are needed to occupy or attack every square of an n X n board?
Also known as the domination number of the n X n knight graph.  Eric W. Weisstein, May 27 2016
Upper bounds for the terms after a(20) = 62 are: 68, 75, 82, 88, 96, 102, ... (see Frank Rubin's web site).
The value a(15) = 37 given by Jackson and Pargas is wrong. A simulated annealingbased program I wrote found several complete coverages of a 15 X 15 board with 36 knights.  John Danaher (jsd(AT)mit.edu), Oct 24 2000


REFERENCES

David C. Fisher, On the N X N Knight Cover Problem, Ars Combinatoria 69 (2003), 255274.
M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 194.
Anderson H. Jackson and Roy P. Pargas, Solutions to the N x N Knights Cover Problem, J. Rec. Math., Vol. 23 #4, pp. 255267, 1991
Bernard Lemaire, Knights Covers on N X N Chessboards, J.Recreational Mathematics, Vol. 312, pp. 8799, 2003.
Frank Rubin, Improved knight coverings, Ars Combinatoria 69 (2003), 185196.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
John Watkins, Across the Board: The Mathematics of Chessboard Problems (2004), p. 97


LINKS

Table of n, a(n) for n=1..20.
J. Danaher, Results for 15 X 15 board
Lee Morgenstern, Knight Domination [Much material, including optimality proofs for the values given in this entry]
Frank Rubin, Contest Center Web Site, Knight Coverings for Large Chessboards [Much material, including many illustrations]
Frank Rubin, Illustration of three 52knight coverings of an 18 X 18 board (see Frank Rubin's web site, from which this is taken, for many further examples)
Eric Weisstein's World of Mathematics, Domination Number
Eric Weisstein's World of Mathematics, Knight Graph
Eric Weisstein's World of Mathematics, Knights Problem


EXAMPLE

Illustrations for a(3) = 4, a(4) = 4, a(5) = 5 (o = empty square, X = knight):
ooo .. oooo .. ooooo
oXo .. oXXo .. ooXoo
XXX .. oXXo .. oXXXo
...... oooo .. ooXoo
.............. ooooo


CROSSREFS

A006076 gives number of inequivalent ways to cover the board using a(n) knights, A103315 gives total number.
Cf. A075458, A075561, A189889.
Sequence in context: A036858 A131957 A127932 * A241295 A074904 A010304
Adjacent sequences: A006072 A006073 A006074 * A006076 A006077 A006078


KEYWORD

nonn,hard,more,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Terms (or bounds) through a(26) updated by Frank Rubin (contestcen(AT)aol.com), May 22 2002
a(20) = 62 added from the Context Center web site by N. J. A. Sloane, Mar 02 2006


STATUS

approved



