An application of Group Theory to Rubiks Cube

An application of Group Theory to Rubik’s Cube Duran Cesur Swarthmore College, Department of Mathematics & Statistics “The object is a wonderful example of rigourous beauty, the big wealth of natural laws: it is a perfect example of the human mind possibilities to test their scientific rigour and to dominate them. It represents the unity of real and beautiful, which means for me the same thing. ” - Ernö Rubik Notation: Name the faces of the cube according to their relative orientation to you. So they can be called Front, Back, Right, Left, Up and Down (abbreviate as F, B, R, L, U, D). Furthermore, use the same abbreviation to symbolise the process of rotating that face. Thus rotating the front face clockwise by 90° is described with F (refer to the figure below). Thus F-1 corresponds to a counter clock-wise rotation. The same notation applies to the other moves as well. This means that if the cube was disassembled mechanically, the possible reconstructions are limited, because if the cubelets are assembled with an odd permutation, it is impossible to solve the cube. After a careful consideration of the facelets of the corner cubelets and the edge cubelets, it is easy to derive the following: 1. There is no process that can create an odd permutation of the edge facelets. 2. An odd permutation of the corner facelets requires an odd number of moves and an even permutation of the corner facelets requires an even number of moves. The F-subgroup: All the possible processes that are the result of turning the F face only. Its only elements are E (the identity), F, F-1 and F 2. The SLICE SQUARED subgroup: The name comes from the fact that the moves in this group are equivalent to rotating one of the central slices by 180 °. The group has the following moves: X=(R 2 L 2), Y=(U 2 D 2) and Z=(F 2 B 2). This group is of order 8. Applying XYZ to a solved cube results in a “checkerboard” pattern (refer to the figure below). Now notice that any move permutes four corner cubelets and four edge cubelets. So when F is applied to the cube, the Up-Front-Left (ufl) corner goes to the place of the former Up-Front-Right (ufr) and so on. This defines a four-cycle and we can symbolise this cyclic permutation by: (ufl, rfu, dfr, lfd). The move F brings about not only a four-cycle of four corner cubelets and a four-cycle of the edge cubelets, but also three four-cycles of the facelets of the four corner cubelets and two four-cycles of the edge cubelets. This leads to some interesting results as to what is and is not possible with the cube. Theorem 1: There is no process whose only effect is to switch to corner cubelets. Theorem 2: There is no process whose only effect is to switch two edge cubelets. Theorem 3: The only possible processes on the cube are those whose effect is an even permutation. RUBIK’S CUBE GROUPS: Notice that every sequence of moves is a group because: -one sequence followed by is another sequence (closure) - followed by is associative - do nothing sequence does nothing (identity) - every sequence of moves can be done backwards (inverses) 2. Place the UP corner cubelets. 3. Place the fr, rb bl and fl cubelets in place with proper orientation. 6. Place the DOWN edge cubelets in place without regard to orientation 7. Put the DOWN edge cubelets into proper orientation. The SLICE subgroup: Define the moves as X=(RL-1), Y=(UD-1) and Z=(FB-1). Each one of these moves is equivalent to a move of a middle slice; hence the name. The (F 2 R 2) subgroup: Repeating this group gives an abelian cyclic subgroup of order 6. Applying this move three times in a row will exchange the positions of the uf edge cubelet and df edge cubelet, as well as ur with dr. Variations of this move can be applied to swap different edge cubelets around, which can be very useful when solving the cube. For example, by applying conjugates to processes such as the one described above, instead of swapping two edge cubelets that are opposite from each other, one can swap others around, which can be very useful when solving the cube (try D(F 2 R 2)3 D-1 for example). The FRBL [“furball”] subgroup: It turns out, after some experimenting, that it takes 315 repetitions (1260 moves) to restore the cube. Therefore this subgroup has order 315. Now we can find the order of the solvable Cube Group. There are 8 corner cubelets (each with 3 different possible states), and 12 edge cubelets (each with 2 different possible states) but only 1 out of 12 different cubes is solvable. This gives us the upper bound of: (8!)(12!)(38)(212)/(12) which is equivalent to 43, 252, 003, 274, 489, 856, 000 (43 quintillion). One thing to notice is that the cube group is not Abelian; F followed by R is not the same as R followed by F. 1. Place the UP edge cubelets 5. Put the DOWN corner cubelets in proper orientation Theorem 5: The total twist parity of the cube cannot change. Thus there are 12 (=2 x 3 x 2) independent ways to reassemble the cube and only one of them is solvable. There are many methods of solving the cube, including speed cubing. However, the method I am describing below is a more ‘group theoretical’ approach to solving the cube. 4. Place the DOWN corner cubelets in place without regard to orientation Theorem 4: There is no process that will flip exactly an odd number of edge cubelets without disturbing any other edge cubelets. These results lead to some significant choices, were we to disassemble the cube and reassemble it. 1. The edge cubelet could be flipped or not. 2. A corner cubelet could be given +1, -1 or 0 twist. 3. A pair of corner cubelets (or edge cubelets) could be exchanged. SOLVING THE CUBE: Notice that the method described above corresponds to solving the cube layer by layer. In ‘speed cubing’, multiple layers are solved simultaneously, however the method described above entails most of the group theory explained in this poster. REFERENCES: 1. D. Joy, “Adventures in group theory : Rubik's Cube, Merlin's machine, and other mathematical toys”, Baltimore, 2002. 2. 2. E. Rubik [et al. ] ; translated & edited by David Singmaster, “Rubik's cubic compendium”, New York, 1987. 3. 3. D. Singmaster, “Notes on Rubik's ‘Magic cube’” , Hillside, 1981. 4. 4. P. Neumann, G. Stoy, E. Thompson, “Groups and Geometry”, Oxford”, New York, 1994. Next… FROM THIS. . . TO THIS. . . Acknowledgements: Special thanks to Mustafa Paksoy ’ 07, Dimitar Enchev ’ 07 and Rachel Corballis ‘ 07 for their technical and moral support.