A051756 Consider the problem of placing N queens on an n X n board so that each queen attacks precisely 3 others. Sequence gives maximal number of queens.

4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 34, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 88, 90, 92, 94, 96, 100, 102, 104, 106, 108, 112, 114, 116, 118, 120, 124, 126, 128, 130, 132, 136, 138, 140, 142, 144

Offset: 2

Author

Robert Trent (trentrd(AT)hotmail.com), Aug 23 2000

Keywords

• nonn
• nice

Comments

a(n) <= 2[(6n-2)/5]. - Jud McCranie, Aug 12 2001

Conjecture: a(n) = 2[(6n-2)/5] for n >= 2; verified up to n = 100. - Alexander D. Healy, Feb 11 2024

Examples

			Examples from _R. J. Mathar_, May 01 2006: (Start)
==== n = 3
6 queens:
Q Q Q
Q - -
Q - Q
6 queens:
Q Q Q
- - -
Q Q Q
==== n = 4
8 queens:
Q Q Q Q
Q - - -
Q - - -
Q - - Q
8 queens:
Q Q Q Q
Q - - -
- - Q -
Q - - Q
8 queens:
Q Q Q Q
- - - -
- - - -
Q Q Q Q
8 queens:
Q Q - Q
- Q - -
- - Q -
Q - Q Q
==== n = 7
16 queens:
Q Q Q - Q - Q
- - - - - - Q
- - - Q - - -
Q - - - - - Q
- - - Q - - -
Q - - - - - -
Q - Q - Q Q Q
16 queens:
Q Q Q - - Q Q
- - - Q - - -
- - - - - - Q
Q - - - - - Q
Q - - - - - -
- - - Q - - -
Q Q - - Q Q Q
(End)
		

References

• Martin Gardner, The Last Recreations, Copernicus, NY, 1997, 274-283.
• Peter Hayes, A Problem of Chess Queens, Journal of Recreational Mathematics, Baywood, 24(4), 1992, 264-271.

Links

• Alexander D. Healy, Examples of optimal placements for n <= 61

Crossrefs

Cf. A051754, A051755, A051757, A051758, A051759, A051567, A051568, A051569, A051570, A051571, A019654.

Extensions

More terms from Jud McCranie, Aug 12 2001

a(10)-a(61) from Alexander D. Healy, Feb 11 2024