Date: Thu, 01 Sep 94 22:56:00 -0400
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: < U, R > revisited
```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!