Wolfram Schubert's N-Queens page

Here are a few results of the n-queens puzzle (esp variations of the "original" n-queens puzzle) split into subtasks.

The subtask results are published here to enable a probabilistic aproach of verifying the total solution numbers, i.e. without (re)performing the whole computations.

Number of (180deg) symmetric solutions to nonattacking queens problem on NxN board.
See OEIS sequence A032522. OEIS b-file.

N	total solutions	subtask results
1	1
...	...
32	181254386312
33	533869114600	split into subtasks by 1 preset column
34	2419751731604	split into subtasks by 2 preset columns
35	6905549827664	split into subtasks by 2 preset columns
36	33686204023352	split into subtasks by 2 preset columns
37	97762487422768	split into subtasks by 4 preset columns
38	496654774712700	split into subtasks by 4 preset columns
39	1472761266426976	split into subtasks by 4 preset columns

Notes: The values of columns preset0, preset1, ... are zero based.
"Preset0" is the first column from the middle of the board, "preset1" is the next column from the middle of the board.

Number of ways of placing n nonattacking semi-queens on an NxN board.
See OEIS sequence A099152.
A semi-queen (a fairy chess piece) moves (and attacks or captures) on one of the diagonals only.
The name "semi-queen" was taken from OEIS sequence A006717 (there on a different board topology).

N	total solutions	subtask results
1	1
...	...
18	5006620130753
19	62395131973755	split into subtasks by 4 preset columns
20	818456924866815	split into subtasks by 4 preset columns

Note: The values of columns preset0, preset1, ... are zero based.

Number of ways of placing n nonattacking superqueens (amazons) on an NxN board.
See OEIS sequences A051223 and A051224.
An amazon/superqueen (a fairy chess piece) moves (and attacks or captures) like a regular queen and a knight

N	total solutions A051223	subtask results	symmetric solutions (180°) (no known OEIS sequence)	symmetric solutions (90°) (no known OEIS sequence)	symmetric solutions count only once A051224 = (A051223 + 180degSymmSol + 2 * 90degSymmSol)/8 (for N>1)
1	1		1	1	1
...	...		...	...	...
23	306819842212	split into subtasks by 3 preset columns	417684	0	38352532487
24	2883202816808	split into subtasks by 4 preset columns	1220024	920	360400504834
25	28144109776812	split into subtasks by 5 preset columns	3904588	908	3518014210402
26	286022102245804	split into subtasks by 5 preset columns	12040500	0	35752764285788

Note: The values of columns preset0, preset1, ... are zero based.

Number of ways of placing n nonattacking chancellors/empresses/marshals on an NxN board.
See OEIS sequences A137774.
A chancellor or empress or marshal (a fairy chess piece) moves (and attacks or captures) like a rook and a knight.

N	total solutions	subtask results
1	1
...	...
17	9248221393974
18	161670971937362	split into subtasks by 3 preset columns
19	2996936692836754	split into subtasks by 4 preset columns

Note: The values of columns preset0, preset1, ... are zero based.

Number of ways of placing n nonattacking queens of the night on an NxN board.
See OEIS sequences A102388.
A queen of the night (a fairy chess piece) moves (and attacks or captures) like a queen and a nightrider.
A nightrider is a rider along straight lines of knight moves (a fairy chess piece too).

N	total solutions	subtask results
1	1
...	...
28	3698242
29	14120996	split into subtasks by 3 preset columns
30	59531852	split into subtasks by 3 preset columns
31	252272512	split into subtasks by 4 preset columns
32	1163430462	split into subtasks by 4 preset columns

Note: The values of columns preset0, preset1, ... are zero based.

Lexicographically first/smallest solutions of the N-queens problem.
See OEIS sequence A141843. OEIS b-file.

N	first/smallest solution
1	1
...	...
40	1 3 5 2 4 9 11 13 15 6 8 19 21 23 30 32 34 36 38 40 31 33 35 37 39 20 22 14 18 10 12 24 17 7 28 26 16 29 27 25
41	1 3 5 2 4 9 11 13 15 6 8 19 7 22 25 30 32 37 39 33 38 40 34 36 41 35 18 23 10 12 16 20 17 14 21 28 26 31 29 27 24
42	1 3 5 2 4 9 11 13 15 6 8 19 21 23 25 32 34 36 38 40 42 33 35 37 39 41 16 7 24 20 14 17 26 10 12 30 28 18 31 22 27 29
43	1 3 5 2 4 9 11 13 15 6 8 19 7 22 24 26 37 39 34 38 35 33 42 36 41 43 40 20 14 17 10 12 28 18 16 27 25 21 32 30 23 31 29
44	1 3 5 2 4 9 11 13 15 6 8 19 7 22 24 32 35 38 41 33 42 37 34 43 39 36 44 40 21 25 18 14 12 20 16 29 10 23 27 30 17 31 26 28
45	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 29 31 36 40 42 32 38 43 39 44 35 37 45 41 21 18 24 14 12 23 16 20 17 28 25 34 26 33 30 27
46	1 3 5 2 4 9 11 13 15 6 8 19 7 22 24 26 34 37 39 41 36 44 46 35 43 38 40 42 45 23 21 16 14 10 27 18 12 25 17 20 30 33 29 32 28 31
47	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 29 36 39 42 44 33 38 43 46 40 37 41 45 47 21 17 24 26 12 14 16 18 20 23 34 28 30 32 27 35 31
48	1 3 5 2 4 9 11 13 15 6 8 19 7 22 24 26 28 36 38 40 37 44 47 45 48 39 41 43 46 42 10 16 20 25 12 31 27 14 17 23 21 34 18 30 33 35 32 29
49	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 37 40 45 41 44 38 48 39 49 43 46 42 47 16 12 24 20 28 17 14 30 21 18 35 31 26 23 33 36 34 32
50	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 36 38 40 45 43 48 46 39 37 50 41 44 42 47 49 20 18 24 26 30 12 16 21 17 14 31 35 23 29 33 28 34 32
51	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 39 41 43 40 47 50 48 51 42 44 46 49 45 16 26 23 12 30 18 14 17 20 24 21 28 34 36 38 33 35 37 32
52	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 38 40 42 46 39 50 48 51 45 43 41 52 44 49 47 21 26 28 30 18 14 12 17 23 16 32 24 20 31 37 34 36 33 35
53	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 33 41 43 45 42 49 52 50 53 44 46 48 51 47 12 14 24 17 21 28 30 34 16 18 20 26 23 38 40 35 32 36 39 37
54	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 40 42 44 41 49 51 53 43 52 48 45 54 46 50 47 14 18 30 21 26 32 17 20 28 12 16 23 37 34 38 24 39 36 33 35
55	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 41 43 46 50 47 45 54 42 53 55 44 48 51 49 52 23 14 28 17 33 20 16 34 24 18 21 32 30 37 26 40 36 39 35 38

Note: The values of these queen positions in their columns (or rows) are 1-based (as in the OEIS b-file).

Please send questions, notes, hints, error corrections, flames a.s.o. to email account "wschubnq" at the mail domain "gmx.de".

Last change: 12/10/2011