Date: Sun, 16 Apr 95 21:57:00 -0400
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: Antislice revisited
```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

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 <-
```