There are a total of 1,130,184 scrambles given. Of these, 1,020,618 scrambles have no symmetry or antisymmetry, and thus represent 96 distinct positions each (48 rotations and reflections, plus the 48 rotations and reflections of the inverse scramble). Another 76,941 scrambles have no symmetry but have antisymmetry, and represent 48 distinct positions each (the rotations and reflections). Of these 76,941 scrambles, 8,326 are self-inverse. The remaining 32,625 scrambles exhibit various symmetries; a breakdown of their symmetry is given by this page on symmetric positions.

Thanks to Andrew Skalski for solving most of the involutions.

Calculating the symmetry of each scramble, and the various distinct positions represented, is left as an exercise.

Before you download the large (11MB) file you might try downloading random10.txt containing 10 randomly selected, positions, or random100.txt or random1000.txt containing 100 and 1000 randomly selected positions, respectively, to ensure you can handle the format of the scrambles.

The data file htm2.zip is presented in zip format; this does not give the greatest compression achievable but should be readable on all systems. The enclosed htm2.txt file has newlines for line terminators.

I am sporadically searching for and finding new distance-20 positions. I may update these files and numbers as time permits.

If you attempt to analyze this data for patterns, be aware that the process I am using to find the distance-20 positions (described here) finds them in related cosets, so there may be some apparent relationships or characteristics due just to the order of the search.

U1U1F1U1U1R3L1F1F1U1F3B3R1L1U1U1R1U1D3R1L3D1R3L3D1D1There are essentially only two known distance-25 positions, both immediate neighbors of the above distance-25 positions; scrambles for them are

U1U1F1U1U1R3L1F1F1U1F3B3R1L1U1U1R1U1D3R1L3D1R3L3D3 U1F1U1U1R3L1F1F1U1F3B3R1L1U1U1L1U1D3R3L1D1R3L3U1U1There are only 79,780 known distance-24 positions in the quarter-turn metric. Of these, 78,820 have some form of symmetry; there are 3,324 distinct positions with respect to symmetry and antisymmetry. There are only 960 remaining positions, which are generated by orientations of the 14 non-symmetric positions with antisymmetry and the only three known non-symmetric positions without antisymmetry. These final three positions have the following scrambles:

U1U1F1B1R3L3U1R1R1F3B1U1U1R1U1U1D1D1F3B3D1F3B1D3 U1U1F1B1B1U1U1D1B1R3U1L3B3U1L1U3L1F3U1R3U3D1R3D3 U1U1F1R1F1R1D3R1L3B3F3R1D3B1D3U3U3L1U1F3D1F3L3B1Distance-24 positions in the quarter-turn metric lacking both symmetry and antisymmetry may be the hardest types of positions to find, even though I'm fairly certain many exist.

The data file qtm.zip contains scrambles for all known distance-24 or greater positions in the quarter-turn metric.