Number of N+k Queens Solutions

for Some Values of N and k


N

k=1

k=2

k=3

k=4

k=5
5 or less 0 0 0 0 0
6 16 0 0 0 0
7 20 4 0 0 0
8 128 44 8 0 0
9 396 280 44 8 0
10 2,288 1,304 528 88 0
11 11,152 12,452 5,976 1,688 196
12 65,712 105,012 77,896 30,936 7,032
13 437,848 977,664 1,052,884 627,916 225,884
14 3,118,664 9,239,816 13,666,360 11,546,884 6,077,320
15 23,387,448 90,776,620 179,787,988
16 183,463,680 897,446,092
17 1,474,699,536
18

Table 1. Number of N+k Queens solutions.
i.e., number of ways to place N+k Queens and k Pawns
on an N x N board so that no Queens attack each other.


N

k=1

k=2

k=3

k=4

k=5
5 or less 0 0 0 0 0
6 2 0 0 0 0
7 3 1 0 0 0
8 16 6 1 0 0
9 52 37 6 1 0
10 286 164 66 11 0
11 1,403 1,572 751 215 29
12 8,214 13,133 9,737 3,871 879
13 54,756 122,279 131,672 78,560 28,268
14 389,833 1,155,103 1,708,295 1,443,461 759,665
15 2,923,757 11,347,863 22,474,269
16 22,932,960 112,182,378
17 184,339,572
18

Table 2. Number of fundamental solutions to the N+k Queens Problem.

A fundamental solution is an equivalence class of solutions, where rotations and reflections
of a solution are considered equivalent. (i.e. Two or more solutions that are rotations
and/or reflections of each other count as only one fundamental solution.)

 


N

k=1

k=2

k=3

k=4

k=5
5 or less 0 0 0 0 0
6 0 0 0 0 0
7 4 4 0 0 0
8 0 4 0 0 0
9 20 16 4 0 0
10 0 8 0 0 0
11 72 124 32 32 36
12 0 52 0 20(4) 0
13 200 568 492 564 260
14 0 1,008 0 804 0
15 2,608 6,284 6,164
16 0 12,932 0 0
17 17,040
18 0 0 0

Table 3. Number of centrosymmetric solutions to the N+k Queens Problem.

A centrosymmetric solution is one that is unchanged by a 180-degree rotation but not by a
90-degree rotation. A number in parentheses indicates the number of doubly centrosymmetric
solutions, which are solutions unchanged by a 90-degree rotation. (All solutions
to the N+k Queens Problem are changed by flips.)


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Last update: April 16, 2009
Contact: Doug Chatham at d.chatham@moreheadstate.edu
The graphics on this page were generated by CVP Game Courier.