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)     factor

 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
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?