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Subject: Skewb thoughts
From: mark.longridge@canrem.com (Mark Longridge)
Message-Id: <60.1021.5834.0C1CC7AE@canrem.com>
Date: Sun, 29 Jan 1995 23:40:00 -0500
Organization: CRS Online  (Toronto, Ontario)

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 <-