A140450 The count of how many queens must be placed tentatively onto a board while seeking a first solution to the "N-Queens on an N x N chessboard" puzzle.

1, 6, 18, 26, 15, 171, 42, 876, 333, 975, 517, 3066, 1365, 26495, 20280, 160712, 91222, 743229, 48184, 3992510, 179592, 38217905, 584591, 9878316, 1216775, 10339849, 12263400, 84175966, 44434525, 1692888135, 408773285, 2799725104, 4618568460

Offset

1,2

Comments

The term a(4) with the value 26 is the count for a board size of 4 squares by 4 squares. The highest term so far a(45) is the count for a board of 45 squares by 45 squares.

This whole sequence refers only to the number of queen pieces placed tentatively on a board in the hunt for the FIRST POSSIBLE solution for each board size. This sequence makes no reference to queen placements needed to hunt for subsequent solutions that are possible for board sizes above 3x3.

References

CSP Queens - Counting Queen-placements http://queens.cspea.co.uk/

Links

Colin S. Pearson, Table of n, a(n) for n = 1..45 [Corrected Jul 31 2008]

Colin S. Pearson, CSP Queens - Counting Queen-placements

Martin S. Pearson, Queens On A Chessboard

Wikipedia, Eight Queens puzzle

Example

Using a simple, mechanical and naive "one queen at a time" algorithm (in other words, a computer-friendly algorithm), in order to place 4 non-clashing queens on a simple board of 4 x 4 squares, we will need to place a tentative new queen 26 times before we discover the first combination that allows all queens to sit unchallenged. For a board size of 5 x 5 we will need to place tentative new queens just 15 times before we discover the first combination of 5 unchallenged queens. In this extended and corrected sequence, those figures "26" and "15" are the values of terms a(4) and a(5) above.

Prog

A step-by-step example of this algorithm applied to 5 Queens on a 5 X 5 board is displayed on the web site at http://queens.cspea.co.uk/. In addition, a free Microsoft Windows WIN32s-based program can be downloaded from the same website. The program generates and displays these terms for any board size up to 32 squares by 32 squares.

Crossrefs

Cf. A000170 = Number of ways of placing n nonattacking queens on n X n board; A002562 = Number of ways of placing n nonattacking queens on n X n board (symmetric solutions count only once); A141843 = Triangular array of lexicographically earliest solutions to the n queens problem.

Sequence in context: A228104 A028558 A337324 * A157800 A355228 A352061

Adjacent sequences: A140447 A140448 A140449 * A140451 A140452 A140453

Keyword

nonn

Author

Colin S. Pearson and Martin S. Pearson, Jun 26 2008, Jun 30 2008, Jul 03 2008, Jul 31 2008, Aug 16 2008

Extensions

Edited by Colin S Pearson to update the URL for Martin S Pearson's website Colin S. Pearson, Mar 25 2009

Status

approved