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1. (UU) (F'B')(RL)(RL)(FB) 2. (UD') (F'B')(RL)(RL)(FB) 3. (UD) (F'B')(RL)(RL)(FB) 4. (UD')(FB')(LR')(FB')(FB')

#4 is the one Mike Reid already found in the slice group. #3 is the

one I already found in the antislice group. #1, #2, and #3 are

obviously closely related. #1 and #2 appear not to be in either

slice or antislice, but I have been fooled before by alternative

sequences which yield the same position.

#1, #2, and #3 all have the property that |{m'Xm}|=6 and |Symm(X)|=8. As has already been discussed, #4 has the property that |{m'Xm}|=24 and |Symm(X)|=2.

The symmetry groups for #1, #2, and #3 are of a type Dan Hoey's

taxonomy calls class P, class S, and class AX, respectively.

These particular classes are hard to describe succinctly without

introducing a lot of notation. But in all three cases, the symmetry

groups (subgroups of M such that X=m'Xm} consist of four rotations

and four reflections, and have as an axis of symmetry one of

the three major axes of the cube (U-D, F-B, or R-L). There

are three groups P1, P2, P3 with axis of symmetry U-D, F-B,

and R-L, respectively, and similarly for S1, S2, and S3, and

for AX1, AX2, and AX3.

For #4, we have Symm(X)=HV in Dan's taxonomy, where HV={i,v}, and

where i is the identity in M and v is the central inversion in M.

If proper typography were available, the i and the v would be

upper case script letters to follow Frey and Singmaster.

There are relatively few positions in all of cube space

for which Symm(X)=Pi or Symm(X)=Si or Symm(X)=AXi (i in 1..3).

There are only 10 P positions through level 10 in the search tree (of which

just one is a local maximum). There is only one S position through

level 10, and only one AX position through level 10, both of

which are of course local maxima. The positions

are not Q-transitive, but the positions look "symmetric", and they

fulfill the (incorrect) intuition that "symmetric" positions must

be local maxima. We have no reason to say that other P or S or AX

positions further down the search will be local maxima.

I find position #4 extremely intriguing. In general, HV is not

very strong symmetry, and there are relatively speaking, quite

a few HV positions in cube space.

We could create an HV position as follows.

Put any edge cubie anywhere (say UF in RD). Put the "opposite"

cubie in the "opposite" cubicle (DB in LU in this case). Continue

for the remaining edge cubies, and then do the same thing for the

corners, remembering only to make sure the edges and corners have

the same parity. You can easily make an HV position that looks

quite "random" to the casual glance, and in fact most HV positions

don't look very "symmetric".

But Mike's position looks very "symmetric" at a casual glance, as

if its symmetry must be much stronger than HV. I certainly would

not have expected to find an HV position as a local maximum close

to Start. I think the "look" of Mike's position as "symmetric",

and the fact that it is a local maximum close to Start are

related. Without getting too long winded, I think the reasons

are two-fold. First, the corners and edges have much stronger

symmetry separately than they do collectively. Second, the

symmetry looks much stronger if you ignore the centers (i.e.,

if you ignore the rotational positioning of the cubies), perhaps

in the sense of Dan's CSymm function. For example, the corners

are properly positioned with respect to each other, even though

they are not properly positioned with respect to the fixed face

centers.

In the next few days, I intend to calculate Symm(X)

for the corners and edges separately for Mike's position, as

well as calculating CSymm(X) for the corners and edges

separately and combined. I think the results will be

enlightening.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU