NOTES ON THE 3x3 RUBIK'S CUBE - D Joyner Consider the group of transformations of Rubik's magic cube. If we number the faces of this cube as follows +--------------+ | 1 2 3 | | 4 U 5 | | 6 7 8 | +--------------+--------------+--------------+--------------+ | 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 | | 12 L 13 | 20 F 21 | 28 R 29 | 36 B 37 | | 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 | +--------------+--------------+--------------+--------------+ | 41 42 43 | | 44 D 45 | | 46 47 48 | +--------------+ then the group is generated by the following generators, corresponding to the six faces of the cube: U:= ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19), L:= ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35), F:= (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11), R:= (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24), B:= (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27), D:= (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40). The reader may want to verify this by printing out a hard copy of the this page and cut+fold+tape the above diagram into a cube. The size of the group generated by these permutations is 43252003274489856000. STRATEGY Let x^y=y^(-1)*x*y denote conjugation and [x,y]=x^(-1)*y^(-1)*x*y denote the commutator, for x,y group elements. If x,y,z denote 3 group elements, let [x,y,z]=x^(-1)*y^(-1)*z^(-1)*x*y*z. If x is a group element and n>0 is an integer then x^n=x*x*...*x (n times). The solution strategy is composed of 3 stages: Stage 1: Solve the top face and top edges. For this the following moves are useful: "monotwist":[F,R^(-1)]^2 "monoswap":D^(F^(-1))*(D^2)^F*(D^(-1))^(F^(-1)) "monoflip":(epsilon R)^4, where epsilon is the counterclockwise middle slice quarter turn "edgeswap":(U^2)^(R^2*L^2) Stage 2: Solve the middle edges (and bottom edges as best as possible). For this the following "clean edge moves" are useful: R^2*U*F*B^(-1)*R^2*F^(-1)*B*U*R^2 is the top edge 3-cycle (uf,ub,ur), \$M_R^2U^{-1}M_R^{-1}U^2M_RU^{-1}M_R^2\$ (I call this the ``\$2332132\$ move''), is a 3-cycle on edges: (ul,ur,uf). [U,F^(-1),R]*[U^(-1),B,R^(-1)]^L This flips, but dos not permute, the top edges uf, ub (R^2U^2)^3 permutes 2 pairs or edges (uf,ub)(fr,br) (L^2*F^2*B^2*R^2*F^2*B^2)^(D*B^2*F^2) permutes 2 pairs of top edges (uf,ul)(ur,ub) Stage 3: Solve the bottom corners (and bottom edges if necessary). For this the following "clean corner moves" and "clean corner-edge moves" are useful: ((D^2)^R*(U^2)^B)^2 twists the ufr corner clockwise and the bld corner counterclockwise The move ((U^2(D^2)^(F*R^(-1)))^2)^(R^(-1)) has the same 2-corner-twist effect as the one above. \$(R^{-1}D^2RB^{-1}U^2B)^2\$ is a two corner twist: urf+, bld++ \$(M_RU)^3U(M_R^{-1}U)^3U\$ flips 2 edges: uf+, ub+ \$U^{-1}FR^{-1}UF^{-1}RL^{-1}UB^{-1}RU^{-1}BR^{-1}L\$ is another edge flip, but doesn't involve slice moves: uf+, ub+ ((D^2)^(F*D^(-1)*R)*U^2)^2 permutes 2 pairs of corners (ufr,ufl)(ubr,ubl) [(D^(-1))^R,U^(-1)] corner 3-cycle (bru,blu,brd) \$URU^{-1}L^{-1}URU^{-1}L\$ is a corner 3-cycle: (ufl,ubr,ubl) B^(U^(-1)*F)*U^2*U^B*U^2*B^(-1) permutes two top edges and 2 top corners (ulb,urb)(ub,ur) REFERENCES These moves were compiled from the books by Singmaster and Berlekamp, Conway, and Guy, and Bandelow: C. Bandelow, Inside Rubik's cube and beyond, Birkhauser, Boston, 1980 D. Singmaster, Notes on Rubik's magic cube, .... Berlekamp, Conway, Guy, Winnning Ways, II, ....