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Subject: More corrections
From: mark.longridge@canrem.com (Mark Longridge)
Message-Id: <60.601.5834.0C18D8D1@canrem.com>
Date: 	Wed, 8 Dec 1993 13:52:00 -0500
Organization: CRS Online  (Toronto, Ontario)


Mark gaffs again:
>          I'm now sure (I think) that it is really:
>
>       all edges flipped + 4 X
> (with the 4 X on sides F, R, B, L which should match Dan's diagram)

* sign *  No, I see I entered the position into my program wrong.
A central reflection of the edges with respect to the faces is
simply 6 X or checkerboard order 2, solvable in 12 qtw or 6 htw.
So the edges-only antipode is:  all-edges-flipped + 6 X.

Jerry Byran quote:
>Dan Hoey is correct.  Mirror-Image-of-Start is at level 12.
>Edges-Flipped is at level 9.  Mirror-Image-of-Start-and-Edges-Flipped
>is at level 15.  And, of course, Start is at Level 0.  This exhausts
>the list of configurations with order-24 symmetry.

Ok, only 9 qtw.... it's got to play havoc with corners. I got it now.

* Hmmm, what are all the possible orders of symmetry? *

Also I note my "Symmetry Level" is the opposite of Jerry's Order-N
symmetry:

>   If we define "symmetry level" as the number of distinct patterns
>generated by rotating the cube through it's 24 different orientations
in
>space then most known antipodes are symmetry level 6. Thus the lower
the
>number the higher the level of symmetry. The least symmetric positions
>have level 24, and this is very common. The most symmetric positions
>have level 1, the two positions START and 6 X order 2.

Of course all-edges-flipped I never included, as at the time I was
looking at the square's group.

-------------------------------

As a small postfix to my cyclic decomposition article, I found the
following patterns. I'm fond of pattern 16 myself. I am looking for
CD-type processes for 6 X order 3 and 6 X order 6. I find when I
am physical cubing (as opposed to computer cubing or old fashioned
mental cubing!) it really helps having a CD-type process memory-wise.
Memorizing the computer generated processes is like memorizing
prime numbers.

p161 Mark's Pattern 16  (F1 R1 L1 B1) ^3 + F2 B2 D2 F2 B2 T2    (18)
p162 2 X, 4 H full      (F1 T2 B1) ^4                           (12)
p163 4 ARM Full         (F2 T1 B2) ^4 + T1 D3                   (14)
p164 4 Y's Rotated      (F1 T2 D2) ^6 + F1                      (19)
p165 2 Swap, 4 H full   (F1 L2 T2 R2 B1) ^2 + L2 R2 T2 D2       (14)
p166 2 H adj swap       (F1 L2 T2 R2 B1) ^2 + L2 T2 R2 D2 L2 T2 (16)

No doubt these are compressible and hence not as efficient, but if
you consider ease of execution....

-> Mark <-