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Mike Reid writes: > 90 or 180 degree number of number of strong number of weak > slice turns positions local maxima local maxma > > 0 1 0 0 > 1 9 0 0 > 2 51 0 0 > 3 247 0 7 > 4 428 0 212 > 5 32 8 32 > > the strict local maxima are all equivalent under symmetries of > the cube. they are the composition of pons asinorum with any > of the eight positions called "six dots".

Very Interesting.

Here's a 12q process for pons asinorum + 6 dots:

(F2 B2) (T1 D3) (F1 B3) (L3 R1) (T1 D3)

I have some thoughts on the relationship between the groups of the

slice, antislice and squares, although some of it is old news,

discovered by Mr. Singmaster back in 1980. The fact that, at most,

9 anti-slice moves always suffice to restore any position in

this group I think is new.

If we define Combo = < antislice , slice >

I.e. Combo is the group generated by antislice and slice moves.

Then....

Size (Combo) = 15,925,248

The Combo group fully includes the square's group and each group has

some elements in common:

Size (Intersection (antisl, slice, sq)) = 8

Also Size(Combo) / Size Sq) = 24

The Combo group has trivial centre.

I should also note that....

Size (Intersection (sq, antisl)) = 256.

Mike continues:

> in the same way, local maxima (within the antislice group) in the

> 90 degree antislice metric are local maxima in the full cube group

> (quarter turn metric). perhaps mark will tell us more about this.

Well, I didn't calculate local maxima for the anti-slice group,

but I will look at it. I did create a file of all the processes

for each anti-slice position, and most of the anti-slice group

antipodes are quite ugly looking!

One of the "not quite so ugly" antipodes:

( F1 B1 L1 R1)^3 + T1 D1 F1 B1 T1 D1 = 18 q The Centre elements of the Antislice group ------------------------------------------

There are 4 elements which commute with all elements of the

<AS> group, the identity and the three 4 cross order 2 patterns.

(F1 B1) (T1 D1) (L2 R2) (T1 D1) (F1 B1) (T2 D2) = 16 q TTT TTT TTT RLR BFB LRL FBF LLL FFF RRR BBB RLR BFB LRR FBF DDD DDD DDD Cheers! -> Mark <-