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, ....