From cube-lovers-errors@mc.lcs.mit.edu Fri Mar 12 14:57:16 1999 Return-Path: Received: from sun28.aic.nrl.navy.mil (sun28.aic.nrl.navy.mil [132.250.84.38]) by mc.lcs.mit.edu (8.9.1a/8.9.1-mod) with SMTP id OAA18330 for ; Fri, 12 Mar 1999 14:57:15 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Message-Id: <36E7D039.265049BA@marlboro.edu> Date: Thu, 11 Mar 1999 09:16:25 -0500 From: Jim Mahoney Organization: Marlboro College To: Martin Moller Pedersen Cc: cube-lovers@ai.mit.edu Subject: Re: help on 5x5x5 wings References: <199902270400.FAA482087@bonestell.daimi.au.dk> Martin Moller Pedersen wrote: > > I am trying to solve my new cube the 5x5x5 cube. > I have managed to solve all of it except the wings. > The wings are the y's in the following diagram: > > ZyZyZ > yZZZy > ZZZZZ > yZZZy > ZyZyZ > > I have many moves for the 3x3x3 but I can't figure out how to apply these moves > to the wings. Hi Martin. Here's an excerpt from a longer disucussion of the NxNxN cube which I posted to cube-lovers some time ago. Other folks have done similar work, and published consistent results. In what follows a "cubie" is one of the small, colored cubes that make up the NxNxN, a "slice" is an NxN plane of the cube (even if the inside cubies don't exist), and an "orbit" is a set of cubies which can be moved into each other's places, like the corners or edges. The method below can be made to work for any kind of orbit, including the "wings" you ask about. Good luck, Jim Mahoney -- excerpt from http://www.marlboro.edu/~mahoney/cube/NxN.txt -- ===================================================================== (VI) How to Cycle Three Cubies ===================================== ===================================================================== The basic idea is to find a move sequence that will (1) take a chosen cubie off from its "hot seat" on a chosen slice *without* (here's the trick) disturbing any other cubie on that slice. The rest of the cube can be completely scrambled by this operation. Then (2) rotate the chosen slice, (3) undo step (1), putting the original cubie back into its original slice and undo whatever changes were made to the other cubies, and (4) undo step 2. The sequence always of the form A R A' R' where "A" is step 1, "R" is a rotation of a single slice, and the ' mark means, as usual, the inverse operation. Here's a detailed example, using the Corner orbit of a 3x3x3 cube, with the top layer as the "chosen slice" and the cubie marked "1" in the unfolded sketch of a cube below as the focus of attention. In eight moves the cubies in locations 1, 2, and 3 will trade places. The starting position: U a - 1 - 2 - d - (a,1,2,d,e,3,g,h) are a Corner orbit. | L | F | R | B e - 3 - g - h - (U, D, L, R, F, B) are the possible D clockwise rotations. (1) Get "b" off the chosen slice, without disturbing any other cubie on that slice. Replace it with the cubie that you want to put in its place. e - a - 2 - d - -> L -> | | | | 3 - 1 - g - h - e - a - 2 - d - -> D -> | | | | h - 3 - 1 - g - a - 3 - 2 - d - -> L' -> | | | | After L D L' e - h - 1 - g - The top layer was (a,b,c,d); now it is (a,f,c,d). "b" has been taken off the top slice, and "f" is in its place. (2) Rotate the chosen slice to place a new cubie in the hot seat. 3 - 2 - d - a - -> U -> | | | | After (L D L') U e - h - 1 - g - (3) Undo step 1, which pops the chosen cubie "b" back to its original slice, *and* (here's the key part), restore (nearly) all other cubies to their original locations, since none of the disturbed ones were on the slice that rotated in step (2). 3 - 1 - d - a - -> L D' L' -> | | | | After (L D L') U (L D' L') e - 2 - g - h - (4) Undo step 2, restoring the chosen slice back to its original position. a - 3 - 1 - d - -> U' -> | | | | After (L D L') U (L D' L') U' e - 2 - g - h - So the move sequence to cycle corners (1,2,3) is simply (L D L') U (L D' L') U' (reading left to right). With a few extra moves before this sequence (which should be undone afterwards) to arrange the cubies which should be moved into the places which are actually modified by this operation (or a similar one), this trick and its variations can be used to put back all 8 corners into their proper places. And with a bit of exploration, this same idea can be used to cycle three cubies of any type, in any orbit, on any layer, without disturbing anything else. For the Edge-Singles on the 3x3x3, for example, to bring an edge off the top slice without disturbing anything else on top, step (1) can be S D S', where "S" vertical is a rotation of a center slice.