A006075 Minimal number of knights needed to cover an n X n board.

1, 4, 4, 4, 5, 8, 10, 12, 14, 16, 21, 24, 28, 32, 36, 40, 46, 52, 57, 62, 68

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

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 as follows: 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 annealing-based 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

Examples

			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
		

References

• David C. Fisher, On the N X N Knight Cover Problem, Ars Combinatoria 69 (2003), 255-274.
• 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. Recreat. Math., Vol. 23(4), 1991, 255-267.
• Bernard Lemaire, Knights Covers on N X N Chessboards, J. Recreat. Math., Vol. 31-2, 2003, 87-99.
• Frank Rubin, Improved knight coverings, Ars Combinatoria 69 (2003), 185-196.
• 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

• J. Danaher, Results for 15 X 15 board.
• Andy Huchala, Python program.
• 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 52-knight 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.

Crossrefs

A006076 gives number of inequivalent ways to cover the board using a(n) knights, A103315 gives total number.

Cf. A075458, A075561, A189889, A342576.

Extensions

Terms (or bounds) through a(26) updated by Frank Rubin (contestcen(AT)aol.com), May 22 2002

a(20) added from the Contest Center web site by N. J. A. Sloane, Mar 02 2006

a(21) added by Andy Huchala, Jun 06 2021