OFFSET
0,5
COMMENTS
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).
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
In addition a(18) >= 16 and a(20) = 20. - Benjamin Butin, Dec 11 2023
LINKS
Benjamin Butin, Solution for a(14) = 14
Giovanni Resta, A C program for computing a(1)-a(11)
EXAMPLE
a(4) = 2 because there are only two solutions to the 4-queens problem and they can both fit on the same board:
0 1 2 0
2 0 0 1
1 0 0 2
0 2 1 0
a(8) = 6 since at least 6 solutions to the 8-queens problem can fit on the same board but 7 solutions can't:
3 0 5 2 1 6 0 4
0 1 4 0 5 3 2 6
4 6 0 1 2 0 5 3
5 2 3 6 0 4 1 0
6 4 1 5 0 2 3 0
2 5 0 3 4 0 6 1
0 3 2 0 6 1 4 5
1 0 6 4 3 5 0 2
a(9) = 7
7 5 6 3 1 . . 2 4
6 3 . 4 2 7 1 . 5
. . 2 7 5 6 3 4 1
4 7 5 1 . 2 . 6 3
3 1 4 . 6 . 7 5 2
. 6 . 5 3 4 2 1 7
2 4 7 6 . 1 5 3 .
5 . 1 2 7 3 4 . 6
1 2 3 . 4 5 6 7 .
a(10) = 8
3 4 2 8 . . 1 7 5 6
6 . 7 1 5 4 8 2 . 3
. 1 5 6 7 2 3 4 8 .
2 8 4 . 3 6 . 5 1 7
7 . 6 5 1 8 4 3 . 2
8 3 . 4 2 7 5 . 6 1
5 6 8 7 . . 2 1 3 4
4 7 3 . 8 1 . 6 2 5
. 5 1 2 6 3 7 8 4 .
1 2 . 3 4 5 6 . 7 8
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Howard A. Landman, Dec 17 2009
EXTENSIONS
a(6) and known a(7) added by Charlie Neder, Jul 24 2018
a(8)-a(10) and known a(11)-a(13) from Giovanni Resta, Jul 26 2018
a(14) from Benjamin Butin, Nov 07 2023
a(15)-a(17) from Benjamin Butin, Dec 11 2023
STATUS
approved