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%I #37 Jan 16 2024 08:33:33
%S 0,1,0,0,2,5,4,7,6,7,8,11,12,13,14,15,16,17
%N The maximum number of sets of n queens which can be placed on an n X n chessboard such that no queen attacks another queen in the same set.
%C a(n) is nonzero for n >= 4 (there is always at least one solution to the n-queens problem). a(n) <= n (because n sets of n queens fill up the board). a(n) = n if n = 1 or 5 (mod 6).
%C a(n) is at least two for all even n >= 4 since a solution and its reflection will fit on the same board. - _Charlie Neder_, Jul 24 2018
%C In addition a(18) >= 16 and a(20) = 20. - _Benjamin Butin_, Dec 11 2023
%H Benjamin Butin, <a href="/A171760/a171760.txt">Solution for a(14) = 14</a>
%H Giovanni Resta, <a href="/A171760/a171760.c.txt">A C program for computing a(1)-a(11)</a>
%e a(4) = 2 because there are only two solutions to the 4-queens problem and they can both fit on the same board:
%e 0 1 2 0
%e 2 0 0 1
%e 1 0 0 2
%e 0 2 1 0
%e a(8) = 6 since at least 6 solutions to the 8-queens problem can fit on the same board but 7 solutions can't:
%e 3 0 5 2 1 6 0 4
%e 0 1 4 0 5 3 2 6
%e 4 6 0 1 2 0 5 3
%e 5 2 3 6 0 4 1 0
%e 6 4 1 5 0 2 3 0
%e 2 5 0 3 4 0 6 1
%e 0 3 2 0 6 1 4 5
%e 1 0 6 4 3 5 0 2
%e a(9) = 7
%e 7 5 6 3 1 . . 2 4
%e 6 3 . 4 2 7 1 . 5
%e . . 2 7 5 6 3 4 1
%e 4 7 5 1 . 2 . 6 3
%e 3 1 4 . 6 . 7 5 2
%e . 6 . 5 3 4 2 1 7
%e 2 4 7 6 . 1 5 3 .
%e 5 . 1 2 7 3 4 . 6
%e 1 2 3 . 4 5 6 7 .
%e a(10) = 8
%e 3 4 2 8 . . 1 7 5 6
%e 6 . 7 1 5 4 8 2 . 3
%e . 1 5 6 7 2 3 4 8 .
%e 2 8 4 . 3 6 . 5 1 7
%e 7 . 6 5 1 8 4 3 . 2
%e 8 3 . 4 2 7 5 . 6 1
%e 5 6 8 7 . . 2 1 3 4
%e 4 7 3 . 8 1 . 6 2 5
%e . 5 1 2 6 3 7 8 4 .
%e 1 2 . 3 4 5 6 . 7 8
%Y Cf. A000170.
%K nonn,hard,more
%O 0,5
%A _Howard A. Landman_, Dec 17 2009
%E a(6) and known a(7) added by _Charlie Neder_, Jul 24 2018
%E a(8)-a(10) and known a(11)-a(13) from _Giovanni Resta_, Jul 26 2018
%E a(14) from _Benjamin Butin_, Nov 07 2023
%E a(15)-a(17) from _Benjamin Butin_, Dec 11 2023
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