Facts

Known Solution Counts

For reference, the column headings of the individual sequences are linked with their corresponding entries in the On-Line Encyclopedia of Integer Sequences.

Properties

1.	For all N > 1: 8U = S + W + 2V.	(Proof)
2.	For all N ≡ 2,3 mod 4: V = 0.	(Proof)

Note

Due to the far tighter constraints for self-symmetric solutions, their number can be determined easily for somewhat greater N than the number of all solutions.

Symmetry Insights

The properties given above can be derived directly from the analysis of the possible solution symmetries. It turns out that the interesting solutions are those that display some kind of self-symmetry. While all other solutions have eight distinguishable appearances, which may all be transformed into one another by reflecting or rotating the hosting board, some of these appearances collapse into a single instance if the backing fundamental solution has self-symmetric properties.

At first, observe that for all N > 1, no solution can ever be symmetric to itself with respect to reflection: All four possible reflection axis are attacking paths so that no more than a single queen can be placed directly upon it. Every queen not placed upon it would, however, result in a reflected image attacking the original so that both cannot be part of the same solution. Consequently, only solutions containing a only single queen can be self-symmetric, which are, in turn, are only complete for N = 1

Rotational symmetries, on the other hand, are very well possible and may be distinguished into two subcases:

Point Symmetry

about the board center, which is equivalent to a self-symmetry with respect to a rotation by 180°.

Rotational Symmetry

in the narrow sense for describing the self-symmetry with respect to a rotation by 90°. This case implies point symmetry as the appearance of the solution is not altered even by the second application of the 90°-rotation and is also equivalent to self-symmetry with respect to a 270°-rotation.

Point symmetric solutions only have four instead of eight distinct appearances as the original pairs of solutions, which equivalent with repect to a 180°-rotation, become indistinguishable. This can be verified in the figure above where one image of the top row cross-merges into another of the bottom row when point symmetry is assumed.

Rotational symmetric solutions even only have two distinct appearances, which capture the two reflective images, which cannot be equivalent for N > 1 as shown above. In the figure, these two classes are represented by the lightly and strongly shaded images. Assuming rotational symmetry, all members of either class become equivalent.

Property #1

Define the four counts of above table:

S

counts all the appearances of any solution.

W

counts all the appearances of point-symmetric solutions.

V

counts all the appearances of rotationally-symmetric solutions.

U

counts only the unique classes of fundamental solutions, which comprise all the appearances, which are equivalent through reflection or rotation.

The number of fundamental solutions U can then be partitioned into three groups:

Hence, we obtain:

V = 2x
W = 2x + 4y
S = 2x + 4y +8z
U =  x +  y + z

which can be transformed into the first property stated above:

 S = 8U - W - 2V
8U =  S + W + 2V

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Property #2

Finally, consider boards with N = 4k + r with r∈{2,3}. Its quadrants (excluding any center row or column) have sides of length:

D=⌊N/2⌋=2k+1

On a completely filled board, the first D columns must also host D queens. If the board was self-symmetric, none of these queens could be found in a potential center row as it would be in conflict with its own image after a 180°-rotation (or two 90°-rotations). Thus, they must be distributed among the top and bottom quadrants. Since D is odd, an even distribution and, hence, a rotational symmetry (by 90°) is impossible.

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