A while back Jerry asked....

                  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>?

Well, we basically know the answer is yes. There are elements in
<U,R> which require less moves if we use all the generators of G.

To be more specific, look the 6 twist pattern in <U,R> which
requires 22 q turns:
         ^^^^^^^^^^
>>  Equivalent to (U1 R1)^35= (R1 U1)^35 & Shift Invariant
>>  UR11 = U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 U2 R1 U1 R1 U1 R3 U1 R3 U1 R3

After a bit of computer cubing I found:

p183 6 Twist      R1 U3 R2 U3 R1 D3 U3 R1 U3 R3 D2 R3 U3 R1 D3 U3
                  (18 q, 16 q+h  moves)
                  ^^^^^
I'll spare everyone all the gory details. I'm certain there are
all sorts of other examples, but here is one case where we can
save 4 q turns. It may be of some small interest to see which
of the two processes can be executed more rapidly by the human hand.

-> Mark <-