Search: a260319 -id:a260319
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A260318
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Number of doubly symmetric characteristic solutions to the n-queens problem.
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+10
4
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1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 32, 64, 0, 0, 240, 352, 0, 0, 1664, 1632, 0, 0, 16448, 21888, 0, 0, 203392, 333952, 0, 0, 2922752, 4325376, 0, 0, 38592000, 50746368, 0, 0, 630794240, 897616896, 0, 0, 10758713344, 17514086400, 0, 0, 203437559808, 326022221824, 0, 0, 4306790547456, 6265275064320, 0, 0, 97204813266944, 145913049251840, 0, 0
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OFFSET
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1,12
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COMMENTS
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The problem of placing eight queens on a chessboard so that no one of them can take any other in a single move is a particular case of the more general problem: On a square array of n X n cells place n objects, one on each of n different cells, in such a way that no two of them lie on the same row, column, or diagonal.
There are no (interesting) doubly centrosymmetric solutions for n < 4, and there is just one complete set for n = 4: 2413, 3142 and one for n = 5: 25314, 41352.
On the ordinary chessboard of 8 X 8 cells there are a total of 92 solutions, consisting of 11 sets of equivalent ordinary solutions and one set of equivalent symmetric solutions. There are no doubly symmetric solutions in this case.
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REFERENCES
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Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, pp. 247-255 (The Problem of the Queens).
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LINKS
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M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47. [Incomplete annotated scan of title page and pages 18-51]
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FORMULA
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A260320
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Number of asymmetric characteristic solutions to the n-queens problem.
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+10
4
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0, 0, 0, 0, 1, 0, 4, 11, 42, 89, 329, 1765, 9197, 45647, 284743, 1846189, 11975869, 83259065, 621001708, 4878630533, 39333230881, 336375931369, 3029241762900, 28439270037332, 275986675209470, 2789712437580722
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OFFSET
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1,7
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COMMENTS
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The problem of placing eight queens on a chessboard so that no one of them can take any other in a single move is a particular case of the more general problem: On a square array of n X n cells place n objects, one on each of n different cells, in such a way that no two of them lie on the same row, column, or diagonal.
There are no ordinary solutions for n < 5, and there is just one complete set of ordinary solutions for n = 5: 13524, 52413, 24135, 35241, 53142, 14253, 42531, 31425 (generated by reflection and rotation).
On the ordinary chessboard of 8 X 8 cells there are a total of 92 solutions, consisting of 11 sets of equivalent ordinary solutions and one set of equivalent symmetric solutions. There are no doubly symmetric solutions in this case. These sets may be generated in the ordinary case by 15863724, 16837425, 24683175, 2571384, 25741863, 26174835, 26831475, 27368514, 27581463, 35841726, 36258174 and in the symmetric case by 35281746.
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REFERENCES
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W. Ahrens, Mathematische Unterhaltungen und Spiele, second edition (1910), see page 231.
Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, pp. 247-255 (The Problem of the Queens).
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LINKS
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M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47. [Incomplete annotated scan of title page and pages 18-51]
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FORMULA
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A261596
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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)).
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+10
2
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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
(list;
table;
graph;
refs;
listen;
history;
text;
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OFFSET
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1,16
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COMMENTS
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REFERENCES
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Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, p. 247-255 (The Problem of the Queens).
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LINKS
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EXAMPLE
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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;
...
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CROSSREFS
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KEYWORD
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STATUS
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approved
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