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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 <-