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Date: Fri, 25 Oct 96 18:51:55 EDT
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From: Dan Hoey
To: cube-lovers@ai.mit.edu
Subject: Re: Siamese Rubik's Cubes
Norman Diamond wrote:
...
> Siamese Rubik's cubes share an entire column of cubies, i.e. in the
> case of two 3x3x3's they share an edge cubie and two corner cubies.
...
> The idea of bandaging has been extended further by Dieter Gebhardt
> (publications in CFF) and others....
Most cases of bandaging create a puzzle whose transition graph is not
the Cayley graph of a group. For instance, if two disjoint
edge-corner pairs were taped together, you would have some positions
with ten possible QT neighbors and some with eight.
But the corner-edge-corner bandaging does create a group: Fix the
position of the bandaged part, and permute the other 46 facelets (six
corners, eleven edges, and six face centers) with two face moves and
two slice moves. The resulting group can have at most 5! corner
permutations, as in the two-generator group (see Singmaster or the
archives (21 July 1981, 31 Aug 1994)). There are at most 11! edge
permutations, and the face center permutations represent the rotation
group of the cube, with 24 elements. There can be at most 3^5 corner
orientations and 2^10 edge orientations. Finally, the total
permutation parity (corner, edge, and face center) must be even. Gap
tells me the group has
14302911135744000 = 5! 3^5 11! 2^10 24/2
elements, so all such positions are achievable. I haven't run the
Supergroup through Gap, so I'm not sure whether it 2048 times as many
positions.
Of course the regular Siamese cube has the square of this many
positions, because there are two cubes. A different kind of Siamese
cube would be one in which the three 17-cube slabs can rotate 180
degrees with respect to each other. It would certainly be difficult
to build. I think the interaction between the slab moves and the
Lucky Six group would make it hard to solve, as well.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil