Number of N+k Amazons Solutions

for Some Values of N and k

 

 

 

N

k=1

k=2

k=3

k=4

k=5
11 or less 0 0 0 0 0
12 72 0 0 0 0
13 412 120 0 0 0
14 10,320 1,664 176 16 0
15 71,212 52,804 7,896 696
16 678,656 681,220
17 6,122,160
18

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


N

k=1

k=2

k=3

k=4

k=5
11 or less 0 0 0 0 0
12 9 0 0 0 0
13 53 17 0 0 0
14 1,290 209 22 2 0
15 8,920 6,624 996 89
16 84,832 85,204
17 765,446
18

Table 2. Number of fundamental solutions to the N+k Amazons 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
11 or less 0 0 0 0 0
12 0 0 0 0 0
13 12 16 0 0 0
14 0 8 0 0 0
15 148 188 72 16
16 0 412 0 0
17 1,408
18 0 0 0

Table 3. Number of centrosymmetric solutions to the N+k Amazons 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 Amazons Problem are changed by flips.)


[Back to N+k Queens page]

Last update: April 16, 2008
Contact: Doug Chatham at d.chatham@moreheadstate.edu
The graphics on this page were generated by CVP Game Courier.