Search: id:A006075
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%I A006075 M3224
%S A006075 1,4,4,4,5,8,10,12,14,16,21,24,28,32,36,40,46,52,57,62
%N A006075 Minimal number of knights needed to cover an n X n board.
%C A006075 How many knights are needed to occupy or attack every square of an n 
               X n board?
%C A006075 Upper bounds for the terms after a(20) = 62 are: 68, 75, 82, 88, 96, 
               102, ... (see Frank Rubin's web site).
%D A006075 David C. Fisher, On the N X N Knight Cover Problem, Ars Combinatoria 
               69 (2003), 255-274.
%D A006075 M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 194.
%D A006075 Anderson H. Jackson and Roy P. Pargas, Solutions to the N x N Knights 
               Cover Problem, J. Rec. Math., Vol. 23 #4, pp. 255-267, 1991
%D A006075 Bernard Lemaire, Knights Covers on N X N Chessboards, J.Recreational 
               Mathematics, Vol. 31-2, pp. 87-99, 2003.
%D A006075 Frank Rubin, Improved knight coverings, Ars Combinatoria 69 (2003), 185-196.
%D A006075 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006075 J. Danaher, <a href="http://www.mit.edu/~jsd/coverage.txt">Results for 
               15 X 15 board</a>
%H A006075 Lee Morgenstern, <a href="http://home.earthlink.net/~morgenstern">Knight 
               Domination</a> [Much material, including optimality proofs for the 
               values given in this entry]
%H A006075 Frank Rubin, Contest Center Web Site, <a href="http://www.contestcen.com/
               knight.htm">Knight Coverings for Large Chessboards</a> [Much material, 
               including many illustrations]
%H A006075 Frank Rubin, <a href="a006075a.html">Illustration of three 52-knight 
               coverings of an 18 X 18 board</a> (See Frank Rubin's web site, from 
               which this is taken, for many further examples)
%H A006075 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               KnightsProblem.html">Link to a section of The World of Mathematics 
               (1).</a>
%H A006075 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               KnightsTour.html">Link to a section of The World of Mathematics (2).</
               a>
%e A006075 Illustrations for a(3) = 4, a(4) = 4, a(5) = 5 (o = empty square, X = 
               knight):
%e A006075 ooo .. oooo .. ooooo
%e A006075 oXo .. oXXo .. ooKoo
%e A006075 XXX .. oXXo .. oKKKo
%e A006075 ...... oooo .. ooKoo
%e A006075 .............. ooooo
%Y A006075 A006076 gives number of inequivalent ways to cover the board using a(n) 
               knights, A103315 gives total number.
%Y A006075 Sequence in context: A036858 A131957 A127932 this_sequence A074904 A010304 
               A164821
%Y A006075 Adjacent sequences: A006072 A006073 A006074 this_sequence A006076 A006077 
               A006078
%K A006075 nonn,hard,nice
%O A006075 1,2
%A A006075 N. J. A. Sloane (njas(AT)research.att.com).
%E A006075 Comment from John Danaher (jsd(AT)mit.edu), Oct 24 2000: 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.
%E A006075 Terms (or bounds) through a(26) updated by Frank Rubin (contestcen(AT)aol.com), 
               May 22 2002
%E A006075 a(20) = 62 added from the Context Center web site by N. J. A. Sloane 
               (njas(AT)research.att.com), Mar 02 2006

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