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Analysis of the 3x3x3 <U, R> group (continued) ----------------------------------

branching Moves Deep arrangements (q only) factor 0 1 1 -- 1 4 5 4 2 10 15 2.5 3 24 39 2.4 4 58 97 2.416 5 140 237 2.413 6 338 575 2.414 7 816 1,391 2.414 8 1,970 3,361 2.414 9 4,756 8,117 2.414 10 11,448 19,565 2.407 11 27,448 47,013 2.401 ML's Conjecture: The < U, R > group is >=20 turns deep in qt metric

UR Reflective processes: (in the q metric)

A different sort of symmetry which I started to investigate, having

been inspired by my friend who solves his cube 2 adjacent faces

last!

These are the only UR reflective processes at 10 q turns: U3 R1 U1 R1 (U2) R3 U3 R3 U1 = R3 U1 R1 U1 (R2) U3 R3 U3 R1 (10) U1 R3 U3 R3 (U2) R1 U1 R1 U3 = R1 U3 R3 U3 (R2) U1 R1 U1 R3 (10) Here is the obvious one we all know: ( U2 R2 ) ^ 3 = ( R2 U2 ) ^ 3 (12) I liked this pattern in particular... U1 R1 U2 R3 U2 R3 U2 R1 U1 = R1 U1 R2 U3 R2 U3 R2 U1 R1 (12)

I hope to have an algorithm to plumb the depths of the < U, R > group

soon. Amusingly my friend complained about not been able solve

the cube completely as he was stuck in a position with 2 flipped

edges. After watching him squirm for a few weeks I did tell him

you can't flip edges in the < U, R > group! ;->

Congrats to Dan Hoey, Dik Winter, Jerry Bryan and Ludwig Plutonium

for making it into the 1994 Internet White Pages! I'm in good

company.

-> Mark <-

Email: mark.longridge@canrem.com

P.S. I just read the last J.B. post and see I've been somewhat

overshadowed. Ok let's see some antipodes! At least our

results are the same though. So, ummmm I guess ML's

conjecture is correct!