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
21	11271715349614463	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.
Please also see Colin S. Pearson's N-queens page and his list of first solutions.
Please also see Martin S. Pearson's N-queens start page and his list of first solutions.

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
56	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 *)
57	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 *)
58	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 *)
59	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 *)
60	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 *)
...	...
1000	1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 20 51 53 21 56 58 60 23 63 24 66 28 26 70 72 74 76 78 30 32 82 84 86 33 89 34 92 38 36 96 98 100 102 40 105 107 42 110 43 113 44 116 118 120 46 123 47 50 127 49 130 132 134 136 52 54 140 142 144 55 57 148 150 152 59 155 157 159 61 162 62 65 166 64 169 171 173 67 176 178 68 181 69 184 186 71 75 190 192 73 195 197 77 200 202 204 206 79 81 210 212 80 83 216 218 85 221 223 225 227 87 230 88 91 234 90 237 239 241 243 9 3 95 247 94 250 97 253 255 257 99 260 262 101 265 267 269 103 272 104 275 108 106 279 281 283 285 109 111 289 291 *)

These results formatted as an OEIS b-file of sequence A141843. This is is not _the_ current official b-file of OEIS sequence A141843 (but it may become part of it in future).
Note: The values of these queen positions in their columns (or rows) are 1-based (as in the OEIS b-file).
*) These (partially) queen positions are known for sure, see the "small N-queens solutions" below.

Lexicographically small N-queens solutions (but probably not first/smallest solutions for N>55).
List of 1st/smallest or small solutions, "small" if 1st/smallest not (yet) known:

for board sizes / number of queens N=1..1000
contains one 1st/smallest or small N-queens solution per line
line format: "<N>: <pos1> <pos2> ... <posN>"
if a line contains no " " (3 blanks) marker: 1st/smallest solution for that N
if a line does contain a " " (3 blanks) marker between two <posX>: small solution (probably not the 1st/smallest).
The part left of the marker cannot be smaller because it matches the "minimal backtrack-free queens placement".
That's why these numbers of the left part are valid elements of the OEIS sequence A141843.
(But most of them cannot be added to the A141843's b-file, cause of b-files are not allowed to contain "unknown gaps".)
if no solutions exist (only for N=2 and N=3): all <posX> = "0"
sources for the lines in list nqueenSmall.txt:
- for N=1..55: from solving the first-N-queens problem (see above for that solutions)
- for N=56/57/59: from "probing runs" when trying to find small upper bounds before solving the first-N-queens problem
- for all other N: from a modified "genetic search algorithm" for a solution
at least for the lines originating from "genetic search algorithm" it's not too hard to find better (i.e. smaller) solutions cause of not too much cpu time was spent to find them

OEIS a-file (not yet posted to OEIS sequence A141843):

is an extract from the list of (above) "1st/smallest or small solutions"
"flattened" triangle of the 1st/smallest solutions or (if 1st/smallest is not known) of the initial known part of small solutions (1-based)
the non-initial (right) part of a small solution is replaced with "-1" to sign "unknown" (allowed in OEIS a-files)
limited arbitrarily to 10000 lines
note that A141843's b141843.txt is just the longest initial part of a141843.txt containing no "-1" term

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

Last change: 03/26/2012