From cube-lovers-errors@curry.epilogue.com Thu Jun 6 23:32:00 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 XAA11082 for ; Thu, 6 Jun 1996 23:32:00 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 5 Jun 1996 23:07:24 -0400 From: Jim Mahoney Message-Id: <199606060307.XAA14353@ marlboro.edu> To: Jerry Bryan Cc: cube-lovers@ai.mit.edu In-Reply-To: (message from Jerry Bryan on Wed, 05 Jun 1996 10:22:31 -0500 (EST)) Subject: Re: A essay on the NxNxN Cube : counting positions and solving it >>>>> "Jerry" == Jerry Bryan writes: Jerry> This has been discussed before on Cube-Lovers, but I am Jerry> still puzzled or curious about the usage of the word Jerry> "orbit". So am I, actually. Dan Hoey just replied with a much more detailed understanding of the group theory aspects of this word than I have at present, which I'll have to think about some more. For myself, I mean no more and no less than a set of cubies which can move into each other's positions. For a 3x3x3 cube which I imagine to be made of of 3^3=27 smaller cubes (cubies), what I call "orbits" are exactly those cubies at the 8 corners, 12 edges, 6 faces, and 1 (unseen) at the center. Jerry> Secondly, if my understanding of your model is correct, you Jerry> are treating positions as distinct which cannot be Jerry> distinguished with normal coloring of a physical cube (even Jerry> an imaginary physical cube for large N). Yes, exactly. As Dan just said, he has discussed this vision of the cube in earlier notes, and called it the "theoretical invisible cube". When I started thinking about these larger cubes, I built them by making piles of dice. All the inner cubies were there, and all had definite orientations, and I could see them every time I tried to rotate a slice - which required carefully seperating out the layers, turning one, and putting everything back together. So perhaps that's why I liked those "invisible" inside pieces. But it also seemed more elegant. The restricted versions (only the outside, only the orientations of the corners and edges, etc.) are all special cases. Jerry> There are several implications of how you treat visibly Jerry> indistinguishable positions. For example, it impacts your Jerry> counts of how many positions there are. For another Jerry> example, it impacts your solutions (e.g., "invisible" Jerry> incorrect parity on the 4x4x4. "Invisible" bad parity can Jerry> also occur on the 3x3x3 if you remove the face center color Jerry> tabs. A slice move will give the edges and corners Jerry> opposite parity that is not visible.) Perhaps you could Jerry> discuss these issues with respect to your model. I'm not sure what there is to say; you seem to understand the issues. Yes, I am counting "visibly indistinguishable" positions as different, especially on the larger cubes, if by "visibly" you mean to only look at the outside. I'm assuming that either the whole thing is transparent, or that you can take it apart, and see the inside cubies if you like. There are parity constraints between the different orbits, including the ones on the inside that are "invisible," but they turn out to be fairly simple: the parity of each orbit of corners and the central cubie, from the outer layer all the way down to the inside, are independent, and can be chosen arbitrarily. And once they're fixed, the parity of all the other orbits is given. By "bad" parity I assume you mean a case when the edges and corners have different parities. Starting from the solved (even parity) 3x3x3 Cube, a slice move definitely does this; four outside edges cycle, and the corners don't move. However, on the 3x3x3, this *is* visible, since the face centers will also have odd parity. Moreover, the central cube (which you can't see, of course, and isn't really there on a real cube) also has odd parity, in a way: it has undergone an odd number of quarter turns. On a 4x4x4, a slice move on a solved cube changes the parity of the inside 2x2x2 corners (which you can't see) and the edges (which you can). The parity of the outer corners is left unchanged, since they didn't move, and the parity of the face centers is also unchanged, since 8 of them move in two cycles of four cubies. Then the fact that the outside edges are odd while the outside corners are even simply means that the inside 2x2x2 corners are also odd. That's all. Hope that helps, Dr. Jim Mahoney mahoney@marlboro.edu Physics & Astronomy Marlboro College, Marlboro, VT 05344