Date: 9 August 1982 0737-EDT (Monday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Invisible Revenge Message-Id: <09Aug82 073740 DH51@CMU-10A> When you have seen Rubik's Revenge, have you seen everything? Maybe not. Supergroups There is some confusion about the meaning of the term Supergroup for the larger cubes. There are two issues at stake: (1) In the 4^3 and larger cubes, there are pieces that may be permuted but are colored the same. Are positions that differ only in the permutation of identically- colored facets to be considered distinct? (2) In all odd-sized cubes, each face has a center facet that may be twisted. Are positions that differ only in the twist of face centers to be considered distinct? If the answer to (1) is `no', the puzzle is not a group, but a collection of cosets. I find this interesting only in that it is understood by the masses, and then the answer to (2) is `no'. If the answer to both (1) and (2) is `yes', we have what I would call the Supergroup. I mark my cubes so that I can distinguish patterns in this group. I discussed a way of doing this for the 3^3 in my message of 9 January 81 0551-EST. For Rubik's Revenge, I use a similar procedure, except now two spots must be cut out on each face: +-----------------------+ | | | | | | | *|* | | |-----+-----+-----+-----| | | *|* | | | | | *|* | |-----+-----+-----+-----| | | | *|* | | | | | | |-----+-----+-----+-----| | | | | | | | | | | +-----------------------+ Also, I now arrange the faces T-symmetrically, with the spots toward the girdle. Jim Saxe first brought it to my attention that we may answer `yes' to (1) and `no' to (2). We get a group that ignores face center orientation. This is probably what people mean when they say there is no Supergroup for Rubik's Revenge: This is the group of the puzzle, and the Supergroup (as I have defined it) is the same for the 4^3. But in the 5^3 this group is distinct both from the Supergroup and from the Color Cosets. The 57th Piece Alan Bawden, in his message of 5 June 82, despaired of explaining the mechanical workings of Rubik's Revenge. I will now rush in. If you take your Rubik's Revenge apart (as described in Ronald B. Harvey's message of 19 May 82, not recommended by Alan Bawden), you will find that the cubies ride around on a sphere: the mysterious 57th piece. Only the face centers are connected to the sphere; they have flanges that hold the edges in place, and the centers and edges have flanges that hold the corners in place. The tricky part is the linkage between the sphere and the face center cubies. The four center cubies of a face have extensions that together form a mushroom-shaped plug. Each plug extends from the center of the face inward and is cut in quarters lengthwise. There are six sockets on the sphere in which these plugs will fit and may rotate, but may not be pulled out. This is sufficient to implement a puzzle isomorphic to the 3^3, namely the 4^3 where we allow only face twists. The other necessity for a 4^3 is to be able to twist the six center slices, which we name after their adjacent faces. To accomplish this, there are grooves in the sphere that form three orthogonal great circles. Each groove has the cross section of half of a socket, and includes the four half-sockets that correspond to a center slice. When we twist one of those center slices, the half-plugs formed by pairs of face centers in that slice ride around the sphere in the grooves. Of course there is an adjacent center slice; when it is moved, it takes the sphere with it. The reason the grooves cannot have the cross-section of a socket is that then, when we twisted half the cube with respect to the other, the sphere might turn forty-five degrees, preventing center slice twists along the other great circles. When you twist the U center-slice of your cube, does the sphere move or stay fixed? To find out without taking the cube apart, hold the cube by the D center slice and repeatedly twist the U center slice clockwise. Don't touch the U or D faces while doing this. Mostly, the U face will turn and the D face will stay fixed, because the cubie-cubie friction is greater than the cubie-sphere friction. Eventually, however, you will either see the U face lag behind the U center slice, or the D face move to follow the U center slice. If the U face lags, the sphere is not moving; if the D face moves, the sphere is moving. I will take this opportunity to mention another feature of the interior sphere: It has screws in it. I took my screws out, but the sphere didn't come apart. Then I put them back in, on the `don't screw with it' principle. Perhaps they are there so Rubik's Revenge won't float? This issue was raised by Tom Davis (12 August 80) back when people were interested in solving the 3^3 underwater. The last I heard, Richard Pavelle (25 July 1980) was able to solve the cube with only five gulps of air. I imagine some of the 30-second whiz kids can solve Rubik's Cube while completely submerged. But Rubik's Revenge? Don't hold your breath. The Mechanical Invisible Group Alan Bawden's message posed an interesting question about the 57th piece, which I will state somewhat differently. Suppose we paint the sockets of the sphere according to the colors of the face centers that inhabit them. Then we mix up Rubik's Revenge and solve it. Must the sockets still match their face centers? The answer is no. In fact, the sphere may be in any of the twenty-four positions consistent with it having once matched the face centers. To show this, we will show how to perform a ninety-degree whole-cube move of the outside without moving the sphere. This is equivalent to turning the sphere ninety degrees, and we can repeat the procedure and its conjugates to move the sphere to any of its twenty-four positions. In order to twist Rubik's Revenge without moving the sphere, we place the cube in a position such that the U, F, and R center slices do not move the sphere and restrict ourselves to the moves: U1 Clockwise quarter twist of the U face U2 Clockwise quarter twist of the U half (face and center slice together) D1 Clockwise quarter twist of the D face U1',U2',D1' Counterclockwise quarter twists R1,R2,L1,R1',R2',L1' Likewise for R F1,F2,B1,F1',F2',B1' Likewise for F The move is actually fairly simple. The tricky part is moving the L center slice cubies without moving the sphere. To do this, remember the eight-flip X = (R1 L1 U1 D1 F1 B1)^2 from the 3^3. This exchanges the L and R center slices in the 4^3, allowing us to cycle the L center slice cubies in the R center slice. R2 L1' X R2 R1' X is a sphere-fixing whole-cube move taking 24 qtw after cancellation. The Theoretical Invisible Group So much for tawdry reality. As Allan C. Wechsler pointed out on 6 August 82, we can imagine a 4^3 puzzle that contains a 2^3 on the inside. If we solve the outside, must the inside be solved? The process shown above for the 57th piece implies not, for that process performs the RL antislice on the 2^3 with respect to the outside. Can any move of the 2^3 be accomplished? Elementary group theory says no, for odd permutations of the Rubik's Revenge edge cubies are also odd permutations of the 2^3's cubies. I ran the Furst, Hopcroft, and Luks algorithm on the problem. It turns out that the permutation parity is the only restriction on what can be done with the 2^3. Thus if we solve the outside, the inside may be in any one of the (8! 3^7)/2 positions obtainable in an even number of quarter twists on a 2^3 puzzle. This in particular includes all whole-cube moves of the 2^3. Unfortunately, I don't know any simple processes for whole-cube moves or for turning two adjacent faces.