# All 164,604,041,664 Symmetric Positions Solved, QTM

Finally, some eight years later, the same thing has been accomplished in the quarter-turn metric. The table below gives the overall distance distribution of symmetric positions in the quarter-turn metric:

dist positions mod M mod M+inv mod 48 0 1 1 1 1 1 12 1 1 12 2 18 3 3 18 3 108 5 3 12 4 411 19 14 27 5 1,104 46 25 0 6 3,744 163 102 0 7 11,760 490 258 0 8 36,731 1,582 902 11 9 111,144 4,632 2,370 24 10 358,138 15,054 7,908 10 11 1,028,848 42,895 21,647 16 12 3,266,949 136,691 70,284 21 13 9,443,588 393,602 198,149 20 14 29,201,318 1,218,544 618,945 38 15 83,765,676 3,490,523 1,752,632 12 16 268,136,523 11,177,948 5,644,771 27 17 819,440,112 34,146,994 17,140,681 0 18 3,083,699,868 128,516,587 64,687,074 12 19 11,628,867,276 484,563,973 243,091,188 12 20 41,538,350,563 1,730,948,894 869,224,590 19 21 67,617,360,740 2,817,563,628 1,413,565,363 20 22 37,373,063,137 1,557,455,774 783,700,221 1 23 2,147,815,036 89,510,797 45,330,245 28 24 78,820 3,635 3,324 4 25 36 2 2 36 26 3 1 1 3 --------------- ------------- ------------- -- tot 164,604,041,664 6,859,192,484 3,445,060,704 0Like Radu and Kociemba's result did for the count of distance-20 positions in the half-turn metric, this exploration did for the count of distance-24 positions in the quarter-turn metric: it vastly expanded the set of known positions. Indeed, we believe that most positions that are distance 24 or greater in the quarter turn metric are symmetric. We have shown that the number of asymmetric distance-20 positions in the half-turn metric vastly outnumber the set of distance-20 symmetric positions.

Further, no position of distance 25 or greater was found, other than those known; this supports our belief that the only positions at distance 25 or greater are those already know. These positions are the single distance-26 position in three distinct orientations, and the two immediate neighbors of this position in 12 and 24 distinct orientations, respectively.

Almost all the symmetric distance-24 positions are also antisymmetric; there are only 311 distinct distance-24 positions that have symmetry but not antisymmetry (modulo symmetry and antisymmetry).

As part of this work, we have duplicated Radu and Kociemba's results, and validated every number in every table on Kociemba's site with respect to symmetric positions in the half-turn metric. We did this in part to validate our programming, since the vast majority of the code is in common between the quarter-turn and half-turn metrics.

We are presently searching for distance-24 positions in the quarter-turn metric using a technique similar to those we have been using for distance-20 positions in the half-turn metric. We will post results on this search soon.

More details are available at http://cube20.org/symmetry/ and there is a short paper at http://tomas.rokicki.com/qtmsymmetry.pdf summarizing these results.