   Date: Sun, 03 Dec 95 20:09:00 -0500   From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: <U,R> & G

```                  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 <-
```     