   Date: Sun, 29 Jan 95 23:40:00 -0500   From: Mark Longridge <mark.longridge@canrem.com > ~~~ Subject: Skewb thoughts

Extract from Martin's very detailed skewb analysis:

Then the group CG = < C, G > is the set of all positions a puzzler
could observe. There are 24 solved position in CG (solved up to
rotations).

```The group CG has size 2 * 6!/2 * ((3^4*4!/2) * (3^4*4!/2) / 3^2)
|CG|     = 75,582,720
```
```Note that:      |CG| /24 =  3,149,280
```
```The group G has size 6!/2 * ((3^4*4!/2) * (3^4*4!/2) / 3^2)
|G|      = 37,791,360
```
```Note that:      |G|  /12 =  3,149,280
```

The number of positions both David Singmaster and Tony Durham
(the inventor) find for the skewb is 3,149,280.

If we use only one tetrad of the skewb then GAP also finds this
number:

```      corners                                  centers
(each turn permutes 4)           (each turn permutes 3)
skewb := Group(
( 1,11,17) ( 2,12,20)( 4,10,18)(22, 6,14) (25,27,29),
( 2,10,22) ( 1, 9,23)( 3,11,21)(17, 5,15) (25,27,30),
( 4,14,20) ( 1,15,19)( 3,13,17)( 7,11,23) (25,28,29),
( 6,12,18) ( 5,11,19)( 7, 9,17)(21, 1,13) (26,27,29)
);;

Size (skewb);
>   3149280
```

Mr. Singmaster had indicated in his last Cubic Circular that we may
determine the skewb's orientation if only one of the tetrads are
moved.

By moving first one tetrad and then the other we can easily
change the skewb's orientation in space.

Martin finds that the diameter of the skewb is 11 moves, with
perhaps 90 antipodes. The idea that the skewb has 2 positions
at 0 moves is rather odd, but I think if we divide Martin's
table by 2 we should get the answer for visually distinguishable
states for a skewb fixed in orientation.

```------------------------------------------------------------
```

I'm still trying to tame the dodecahedron.
-> Mark <-     