I performed a Local Maxima analysis. (This is for Q-turns only.) The Local
Maxima are in column 4. The shortest local maxima (six of them) are of
length 12. Interestingly, this is the same length which is suspected of
being the shortest Q-turn length for a local maximum in the full cube group.
Is there any connection? Also, the global maxima are of length 25. Does
this tell us anything about the Q-turn length of the global maxima for the
full cube group? Finally, pick any cube X in <U,R>. We know
|X| in G <= |X| in <U,R>. Can anybody find a cube X such that
|X| in G < |X| in <U,R>? Alternatively, can anybody prove
|X| in G = |X| in <U,R> for all X in <U,R>? (This assumes Q-turns only in
all cases. The questions would all have to be asked again for Q+H-turns.)

Number of Moves Which Go Closer to Start

Level       Total   0           1           2           3          4
            Cubes

 0           1   1           0           0           0          0
 1           4   0           4           0           0          0
 2          10   0           8           2           0          0
 3          24   0          20           4           0          0
 4          58   0          48          10           0          0
 5         140   0         116          24           0          0
 6         338   0         280          58           0          0
 7         816   0         676         140           0          0
 8       1,970   0       1,632         338           0          0
 9       4,756   0       3,940         816           0          0
10      11,448   0       9,448       1,996           4          0
11      27,448   0      22,584       4,836          28          0
12      65,260   0      53,236      11,862         156          6
13     154,192   0     125,196      28,616         360         20
14     360,692   0     289,908      69,196       1,472        116
15     827,540   0     652,792     168,008       6,180        560
16   1,851,345   0   1,428,560     398,634      21,860      2,291
17   3,968,840   0   2,938,808     934,908      84,312     10,812
18   7,891,990   0   5,422,844   2,109,480     309,916     49,750
19  13,659,821   0   8,065,268   4,288,796   1,068,480    237,277
20  18,471,682   0   7,948,748   6,625,644   2,947,320    949,970
21  16,586,822   0   3,485,748   5,507,066   4,831,060  2,762,948
22   8,039,455   0     286,176   1,165,888   2,665,080  3,922,311
23   1,511,110   0         740      15,202     156,432  1,338,736
24      47,351   0           0           8         332     47,011
25          87   0           0           0           0         87

73,483,200   1  30,736,780  21,331,532  12,092,992  9,321,895

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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?