From cube-lovers-errors@curry.epilogue.com Fri Oct 25 22:41:42 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id WAA03041; Fri, 25 Oct 1996 22:41:42 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Fri, 25 Oct 96 18:51:55 EDT Message-Id: <9610252251.AA14688@sun34.aic.nrl.navy.mil> 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