A141843 -id:A141843 - cp's OEIS frontend

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

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

nonn

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.

			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.

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

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

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.

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

A261595 Triangular array T(n, k) read by rows (n >= 1, 1 <= k <= n): row n gives the lexicographically earliest doubly centro-symmetric characteristic solution to the n queens problem, or n zeros if no doubly centro-symmetric characteristic solution exists. The k-th queen is placed in square (k, T(n, k)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 4, 1, 3, 2, 5, 3, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Martin Renner, Aug 25 2015

See the comments under A260318.

			n = 1: 1 is the trivial solution.
2 <= n < 4: no doubly centro-symmetric solutions exist.
n = 4: 2413 is the first and only solution.
       .*..
       ...*
       *...
       ..*.
n = 5: 25314 is the first and only solution.
6 <= n < 12: no doubly centro-symmetric solutions exist.
Triangle starts:
1;
0, 0;
0, 0, 0;
2, 4, 1, 3;
2, 5, 3, 1, 4;
0, 0, 0, 0, 0, 0;
...

Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, pp. 247-255 (The Problem of the Queens).

Cf. A141843, A260318, A261596, A261597.

A261596 Triangular array T(n, k) read by rows (n >= 1, 1 <= k <= n): row n gives the lexicographically earliest symmetric characteristic solution to the n queens problem, or n zeros if no symmetric characteristic solution exists. The k-th queen is placed in square (k, T(n, k)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 1, 3, 5, 2, 5, 1, 4, 7, 3, 6, 3, 5, 2, 8, 1, 7, 4, 6, 2, 4, 9, 7, 5, 3, 1, 6, 8, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9

Offset: 1

Martin Renner, Aug 25 2015

See the comments under A260319.

			1 <= n < 6: no symmetric solutions exist.
n = 6: 246135 is the first and only symmetric solution.
       .*....
       ...*..
       .....*
       *.....
       ..*...
       ....*.
n = 7: 2514736 is the first of two existing symmetric solutions.
n = 8: 35281746 is the first and only symmetric solution.
Triangle starts:
0;
0, 0;
0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0, 0;
2, 4, 6, 1, 3, 5;
2, 5, 1, 4, 7, 3, 6;
3, 5, 2, 8, 1, 7, 4, 6;
...

Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, p. 247-255 (The Problem of the Queens).

Cf. A141843, A260319, A261595, A261597.

A261597 Triangular array T(n, k) read by rows (n >= 1, 1 <= k <= n): row n gives the lexicographically earliest asymmetric characteristic solution to the n queens problem, or n zeros if no asymmetric characteristic solution exists. The k-th queen is placed in square (k, T(n, k)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 2, 4, 0, 0, 0, 0, 0, 0, 1, 3, 5, 7, 2, 4, 6, 1, 5, 8, 6, 3, 7, 2, 4, 1, 3, 6, 8, 2, 4, 9, 7, 5, 1, 3, 6, 8, 10, 5, 9, 2, 4, 7, 1, 3, 5, 7, 9, 11, 2, 4, 6, 8, 10, 1, 3, 5, 8, 10, 12, 6, 11, 2, 7, 9, 4

Offset: 1

Martin Renner, Aug 25 2015

See the comments under A260320.

			1 <= n < 5: no ordinary solutions exist.
n = 5: 13524 is the first and only solution.
       *....
       ..*..
       ....*
       .*...
       ...*.
n = 6: no ordinary solution exists.
n = 7: 1357246 is the first of four existing solutions.
n = 8: 15863724 is the first of eleven existing solutions.
Triangle starts:
0;
0, 0;
0, 0, 0;
0, 0, 0, 0;
1, 3, 5, 2, 4;
0, 0, 0, 0, 0, 0;
1, 3, 5, 7, 2, 4, 6;
1, 5, 8, 6, 3, 7, 2, 4;
...

Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, p. 247-255 (The Problem of the Queens).

Cf. A141843, A260320, A261595, A261596.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A261596 Triangular array T(n, k) read by rows (n >= 1, 1 <= k <= n): row n gives the lexicographically earliest symmetric characteristic solution to the n queens problem, or n zeros if no symmetric characteristic solution exists. The k-th queen is placed in square (k, T(n, k)).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A261597 Triangular array T(n, k) read by rows (n >= 1, 1 <= k <= n): row n gives the lexicographically earliest asymmetric characteristic solution to the n queens problem, or n zeros if no asymmetric characteristic solution exists. The k-th queen is placed in square (k, T(n, k)).

Views

Author

Keywords

Comments

Examples

References

Crossrefs