A171760 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.

0, 1, 0, 0, 2, 5, 4, 7, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17

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

Table of n, a(n) for n=0..17.

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

Cf. A000170.

Sequence in context: A100116 A366682 A107921 * A085801 A023843 A153990

Adjacent sequences: A171757 A171758 A171759 * A171761 A171762 A171763

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