From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 6 11:10:06 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 LAA28883 for ; Fri, 6 Aug 1999 11:10:05 -0400 (EDT) Message-Id: <199908061510.LAA28883@mc.lcs.mit.edu> Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Tue, 3 Aug 1999 02:11:45 -0700 (PDT) From: Tim Browne To: Cube-Lovers@ai.mit.edu Subject: Notes on the Bandaged Cube... combinations and other musings. In-Reply-To: I just picked up a bandaged cube the other day, and I get the feeling that this might be one of the few puzzles which would beat even the Square-1 for solving difficulty. For those of you who haven't seen this nightmare of a cube, Hendrik Haak has a picture of it in the museum section of his Puzzle Shop (he calls it a "Bicube"). After playing around with it for a bit, I figured I'd try and work out the number of possible combinations. The cube is constructed from a standard Rubik's Cube mechanism, so all the standard Rubik's Cube restrictions apply here. The cube is made up of 13 pieces. One of them is created by fusing an edge piece between 2 centres, effectively turning it into a 2x2 piece. Because of this piece, 2 axes of motion are effectively cut off permanently, making a maximum of 4 axes of rotation. There are 4 pieces which are made up by fusing an edge to a centre, 7 pieces are a corner/edge fusion, leaving us with one standard corner piece. These fusions make the puzzle much more difficult than it first appears, as the contortions of all of these 1x2 pieces effectively block axes of rotation which were easily accessible only a quarter turn ago, sometimes getting so bad as to make the only available axis the one you just turned, in some extreme cases even forcing you to back out following exactly the same path you used to enter the current state. Needless to say, this makes solving the puzzle very frustrating indeed. Anyhoo, back to the combinations... Two of the centres are effectively locked into place for all time, leaving us with 4 edge/centre fusions which can be rotated one of 4 ways, giving us a factor of 4^4 combinations. Piece rotations in place are impossible. The first potential restriction would be creating more than one area where only the corner cube could fit, but the 2x2 piece makes this an impossibility. The second would be potential collisions between pieces. There are 5 pairs of adjacent sides where you have a 1:16 chance of a conflict, and 2 ways of creating a double conflict, so we reduce this amount by 5*256/16-2=82, leaving us with 174 possible arrangements of the edge/centre fusions. The edge/corner fusions and the lone corner piece will be handled together. The corner piece can be placed into any one of 8 corner slots, while the edge/corner fusions fit tightly into the remaining slots surrounding it, giving us a factor of 8!=40,320 combinations. Now for the restrictions... let's start with a simple swap first. Let's take an example side. Like numbers in the table are bandaged together. 112 345 345 Would it be possible to swap piece 1 and piece 3? Given the restrictions carried over from the Rubik's Cube, the only way you can swap a pair of edges or a pair of corners is if you also swap either a second pair of the same type, or a pair of its opposite. To see this latter case, rotate a slice of a solved cube 90 degrees either way and then "reconstruct" it using your favourite patterns. God's Algorithm in this case is expressly prohibited. Since the corners and edges are fused, a swap of one expressly implies a swap of the other, so this is OK. How about swapping pieces 3 and 5? This one's a bit more difficult. Not only are you swapping the pieces, but you're rotating them as well. When you swap these 2, both edge pieces are inverted, maintaining the even parity, so that's OK... the corner half of piece 3 is given a positive rotation, while the corner half of piece 5 is given a negative rotation. Modulo 3 parity is maintained, so this is also OK, meaning the whole swap is OK. Swapping any 2 edge/corner fusions on the cube can be broken down into a compound movement of either of these, so any of these pieces can be swapped with any other similar piece without restriction. The second potential restriction happens with edge/centre fusions. In certain special cases, the centres can rotate in such a way they they box in a 1x1x1 area, limiting it to 1 possible place. This can happen in one of two possible ways, bringing our combinations from 174x40,320 down to 172x40,320 + 2x5,040. The final possible restriction happens when we swap an edge/centre fusion with an edge/corner fusion. What happpens when we swap pieces 4 and 5? So you don't have to scroll back, here it is again... 122 122 345 --> 344 ? 345 355 There are 3 problems with this: 1) you're swapping a single pair of edge pieces, 2) you're flipping a single edge piece, and 3) You're rotating a single corner in place, *all* of which are definite no-no's on the Rubik's Cube, either solo or in any combination, so you definitely can't do it here. The corner rotation can be accounted for easily enough by rotating the 1x1x1 piece in place to compensate, which you've probably noticed that I conveniently left out of all calculations to this point, to save multiplying and dividing by 3 unnecessarily. ALL of these problems can be compensated for by either simply not doing it, or by doing the same thing with another set, giving us 2 pairs of swapped edges and 2 edges flipped, compensating for the corner rotation again with the 1x1x1. This effectively slashes the potential combinations in half, bringing us down to 86x40,320 + 1x5,040 = (86x8+1)x5,040 = 689 x 5,040 = 5040 689 ---- 45360 403200 3024000 -------- 3472560 possible combinations. Most puzzles of this type are difficult because of the sheer number of combinations. This is perhaps the only one of this type which is so difficult because of its limits. If you want to take it apart and put it back together randomly, you've got a 1 in 6 1/1920th chance of doing so correctly, perhaps this puzzle's one advantage over the cube. However, if you want to tough it out and devise a pattern for it, then you'll need to work out 172x8+2 = 1,378 patterns to get it back to its default shape, followed by at least 6 more patterns to restore the edge/corner fusions to their proper positions, making at least 1,384 patterns to work out for a general solution. I've graciously decided to leave this as an exercise for the reader. ;-) L8r.