On 08/09/94 at 01:48:00 mark.longridge@canrem.com said:
Analysis of the 3x3x3 <U, R> group ----------------------------------branching Moves Deep arrangements (q only) factor0 1 -- 1 4 4 2 10 2.5 3 24 2.4 4 58 2.416 5 140 2.413 6 338 2.414 7 816 2.414 8 1,970 2.414 program starts to really bog down after this...I leave it to Jerry or Dan to check my results. I checked up to 2
moves deep by hand and verified 10 different positions.
Ok, here it is. This search is narrower and deeper than any
I have ever done before. Frey and Singmaster give <U,R>
a good bit of attention in their book, pointing out that it
is trickier than it might first appear.
It is called the Two-Generator Group. The size of the group
can be calculated as (7!5!/2)(3^6/3) = 73,483,200. The
3^6 factor accounts for twisting the corners, but there is no
2^n factor as the edges cannot be flipped.
These results are in terms of Q turns without any conjugate
class checking. I would regard the following as open problems:
local maxima, results with Q+H turns, and results in terms of
conjugate classes. In this particular case, it would not be
M-conjugates. I would have to look at Dan Hoey's 98 subgroups
of M to see which subgroup applies to <U,R>.
0 1 1 4 4 2 10 2.5 3 24 2.4 4 58 2.416 5 140 2.413 6 338 2.414 7 816 2.414 8 1,970 2.414 9 4,756 2.414 10 11,448 2.407 11 27,448 2.398 12 65,260 2.378 13 154,192 2.363 14 360,692 2.339 15 827,540 2.294 16 1,851,345 2.237 17 3,968,840 2.144 18 7,891,990 1.988 19 13,659,821 1.755 20 18,471,682 1.352 21 16,586,822 0.898 22 8,039,455 0.485 23 1,511,110 0.188 24 47,351 0.031 25 87 0.002
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU
If you don't have time to do it right today, what makes you think you are
going to have time to do it over again tomorrow?