n queen topics

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

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Prepared by Matthias Engelhardt

Mail to Matthias Engelhardt

last change: 2010-10-31

Address of page: http://nqueens.de/sub/ConflictTables.en.html