# Results of two more cosets of the H group, this time face turn metric.

Submitted by rokicki on Fri, 03/24/2006 - 23:35.

After seeing Silviu have such success with H group (U, D, F2, B2, R2, L2)
cosets, I decided to give it a shot in the face turn metric. So far
I've completed the identity coset and the flip8 (upper and lower edges)
cosets; the superflip coset and flip4 (middle edges) are still running.
The identity, flip4, flip8, and superflip are the four centers of the
H group. I've also shown that of the approximately 234,101,145,600
positions represented by these four cosets, none have a depth greater
than 21. This exploration covers more than 5/1,000,000,000 of the total
cube space.
The identity coset has the following depths. For comparison on the right
I've also listed the depths using only the ten moves that are in H
because the two lists are so very close (the second column is also
my confirmation of the numbers originally posted by Michael Reid back
in January of 1995):

d all moves moves in H 0 1 1 1 10 10 2 67 67 3 456 456 4 3,079 3,079 5 20,076 19,948 6 125,218 123,074 7 756,092 736,850 8 4,331,124 4,185,118 9 23,639,531 22,630,733 10 122,749,840 116,767,872 11 582,017,108 552,538,680 12 2,278,215,506 2,176,344,160 13 5,790,841,966 5,627,785,188 14 7,240,785,011 7,172,925,794 15 3,319,565,322 3,608,731,814 16 145,107,245 224,058,996 17 271,112 1,575,608 18 36 1,352 -------------- -------------- 19,508,428,800 19,508,428,800The average depth using all moves is 13.5323; using only the ten moves in H is 13.5803, so restricting the second phase of the Kociemba algorithm to moves in H probably doesn't have a large impact on the final solution length. For the flip8 position, the depths are:

6 128 7 2,480 8 29,616 9 275,072 10 2,025,588 11 12,857,588 12 77,246,640 13 420,070,470 14 1,878,234,188 15 5,160,105,314 16 7,283,921,362 17 4,500,551,722 18 173,096,184 19 12,448 -------------- 19,508,428,800I anticipate the other two cosets will complete shortly; they both have fewer than 1000 positions remaining. Thank you to Silviu for both proving it could be done and discussing his approach with me. This is a much more effective coset to search (in terms of optimal solutions per second) than the edge cosets I had been using.