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

2,1

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

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

Table of n, a(n) for n=2..61.

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

Example

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)

Crossrefs

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

Sequence in context: A020645 A053226 A227112 * A322839 A074827 A068354

Adjacent sequences: A051753 A051754 A051755 * A051757 A051758 A051759

Keyword

nonn,nice

Author

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

Extensions

More terms from Jud McCranie, Aug 12 2001

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

Status

approved