
FactsKnown Solution CountsFor reference, the column headings of the individual sequences are linked with their corresponding entries in the OnLine Encyclopedia of Integer Sequences.
Symmetry InsightsThe 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 selfsymmetry. 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 selfsymmetric 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 selfsymmetric, 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 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 crossmerges 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 #1Define the four counts of above table:
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 ◊ Property #2Finally, 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 selfsymmetric, 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. ◊ 
SponsorsContactProf. Rainer G. Spallek Thomas B. Preußer Bernd Nägel
