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# Number of solutions, ordered by number of conflicts

This is A181499 in the OEIS.

 0 .. 2 3 4 5 6 7 8 9 .. Sum 1 1 .. .. 1 .. .. .. .. .. .. .. .. .. .. .. 4 0 .. 2 0 0 .. 2 5 10 .. 0 0 0 0 .. 10 6 0 .. 0 4 0 0 0 .. 4 7 28 .. 0 8 4 0 0 0 .. 40 8 0 .. 64 0 28 0 0 0 0 .. 92 9 0 .. 232 0 96 24 0 0 0 0 .. 352 10 0 .. 240 0 372 112 0 0 0 0 .. 724 11 88 .. 0 328 1,252 872 140 0 0 0 .. 2,680 12 0 .. 0 3,016 5,140 4,696 1,316 32 0 0 .. 14,200 13 4,524 .. 0 5,296 22,816 24,656 14,804 1,616 0 0 .. 73,712 14 0 .. 15,008 0 103,432 130,864 94,728 20,884 680 0 .. 365,596 15 0 .. 41,424 33,616 293,188 734,632 800,324 338,296 37,648 56 .. 2,279,184 16 0 .. 174,048 196,880 1,949,600 4,213,848 4,973,764 2,710,988 535,788 17,596 .. 14,772,512 .. .. .. .. .. .. .. .. .. .. .. 4,651 .. 231,018 239,148 2,375,928 5,109,704 5,885,076 3,071,816 574,116 17,652 .. 17,509,109

Interpretation: there is a number 4,696 in a central place in the table. It is in the line beginning with "12", and under "5" as column head. That means that there are 4,696 solutions of the n-queens problem having exactly 5 conflicts, on a 12×12 board.

As a consequence, we get the number of all solutions for a given size in the sum column on the right. The number of torus solutions is found in the leftmost column, under row header "0", as torus solutions do note have conflicts.

# Number of solutions, ordered by number of queens engaged in conflicts

This is A181500 in the OEIS.

 0 .. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .. Sum 1 1 .. .. 1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4 0 .. 0 2 .. 2 5 10 .. 0 0 0 .. 10 6 0 .. 0 4 0 0 .. 4 7 28 .. 0 0 0 12 0 .. 40 8 0 .. 0 64 0 28 0 0 .. 92 9 0 .. 0 232 8 32 48 32 0 .. 352 10 0 .. 96 144 0 152 240 76 16 0 .. 724 11 88 .. 0 0 0 656 616 708 456 156 0 .. 2,680 12 0 .. 0 0 1,464 3,068 2,928 3,436 2,120 1,000 136 48 .. 14,200 13 4,524 .. 0 0 424 9,292 10,952 19,156 14,144 10,244 3,976 1,000 0 .. 73,712 14 0 .. 3,648 11,360 1,008 15,788 63,360 81,592 80,608 63,560 32,960 10,432 1,136 144 .. 365,596 15 0 .. 0 41,424 856 63,808 147,896 377,248 474,896 514,244 389,992 198,008 60,120 10,692 0 .. 2,279,184 16 0 .. 0 193,296 100,240 270,916 1,002,376 2,103,052 2,772,032 3,215,736 2,638,096 1,629,880 664,112 164,664 17,016 1,096 .. 14,772,512 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4,651 .. 3,744 246,526 104,000 363,752 1,228,416 2,585,300 3,344,272 3,804,940 3,065,160 1,839,368 725,368 175,500 17,016 1,096 .. 17,509,109

Example: as last number in row "12", you see 48. There is a 12 also in the column head. That means: there are 48 solutions on the 12×12 board in which all 12 queens are engaged in conflicts. Four of these solutions are shown as image on page "Conflicts", and you can get further 28 by rotation and reflection (i.e. action of the dihedral group). The other 16 solutions come from shift operations. The images contain small red circles, for separation points. If you shift so that the separation point lies on a corner, then all conflicts are separated again. All four solution have an additional separation point, and that should yield 32 further solutions - but shifting does not always yield new solutions.

# Number of solutions, ordered by number of connection components in the conflict graph

This is A181501 in the OEIS.

 0 1 2 3 4 5 6 7 8 .. Sum 1 1 0 .. 1 .. .. .. .. .. .. .. .. .. .. .. 4 0 0 2 0 0 .. 2 5 10 0 0 0 0 0 .. 10 6 0 4 0 0 0 0 0 .. 4 7 28 0 4 8 0 0 0 0 .. 40 8 0 0 92 0 0 0 0 0 0 .. 92 9 0 8 272 56 16 0 0 0 0 .. 352 10 0 96 344 240 44 0 0 0 0 .. 724 11 88 0 424 1,216 872 80 0 0 0 .. 2,680 12 0 36 3,696 6,120 3,920 380 48 0 0 .. 14,200 13 4,524 480 6,076 28,176 27,272 6,664 520 0 0 .. 73,712 14 0 4,796 37,148 129,468 142,264 48,268 3,508 144 0 .. 365,596 15 0 1,312 113,540 531,576 975,516 554,000 99,336 3,904 0 .. 2,279,184 16 0 25,756 681,992 2,982,828 6,198,844 3,919,556 912,052 50,388 1,096 .. 14,772,512 .. .. .. .. .. .. .. .. .. .. .. 4,651 32,488 843,590 3,679,688 7,348,748 4,528,948 1,015,464 54,436 1,096 .. 17,509,109

# Number of solutions, ordered by maximal size of a connection component in the conflict graph

This is A181502 in the OEIS.

 .. 1 2 3 4 5 6 7 .. Sum 1 .. 1 .. 1 .. .. .. .. .. .. .. .. .. 4 .. 0 2 0 0 .. 2 5 .. 10 0 0 0 0 .. 10 6 .. 0 0 0 4 0 0 .. 4 7 .. 28 8 4 0 0 0 0 .. 40 8 .. 0 64 24 4 0 0 0 .. 92 9 .. 0 248 80 16 8 0 0 .. 352 10 .. 0 172 484 36 32 0 0 .. 724 11 .. 88 812 1,308 400 72 0 0 .. 2,680 12 .. 0 3,288 8,480 2,204 192 36 0 .. 14,200 13 .. 4,524 17,908 37,000 11,432 2,656 192 0 .. 73,712 14 .. 0 62,132 219,948 69,032 13,712 748 24 .. 365,596 15 .. 0 406,496 1,308,060 478,020 77,832 8,560 216 .. 2,279,184 16 .. 0 2,423,304 9,004,508 2,824,568 481,624 36,916 1,592 .. 14,772,512 .. .. .. .. .. .. .. .. .. .. 4,651 2,914,434 10,579,896 3,385,716 576,128 46,452 1,832 .. 17,509,109
Prepared by Matthias Engelhardt

last change: 2010-10-31