NOTES ON THE 4x4x4 RUBIK'S CUBE
                                            - by D Joyner

    Consider the group of transformations of the 4x4x4 Rubik's cube. If
  we number the faces of this cube as follows

                    +-----------------+
                    |                 |
                    | 49  50  51  52  |
                    |                 |
                    | 61  62  63  64  |
                    |        U        |
                    | 73  74  75  76  |
                    |                 |
                    | 85  86  87  88  |
                    |                 |
 +------------------+-----------------+-----------------+-----------------+
 |                  |                 |                 |                 |
 |  53  54  55  56  |  1   2   3   4  |  5   6   7   8  |   9  10  11  12 |
 |                  |                 |                 |                 |
 |  65  66  67  68  | 13  14  15  16  | 17  18  19  20  |  21  22  23  24 |
 |         L        |         F       |         R       |          B      |
 |  77  78  79  80  | 25  26  27  28  | 29  30  31  32  |  33  34  35  36 |
 |                  |                 |                 |                 |
 |  89  90  91  92  | 37  38  39  40  | 41  42  43  44  |  45  46  47  48 |
 |                  |                 |                 |                 |
 +------------------+-----------------+-----------------+-----------------+
                    |                 |
                    | 57  58  59  60  |
                    |                 |
                    | 69  70  71  72  |
                    |        D        |
                    | 81  82  83  84  |
                    |                 |
                    | 93  94  95  96  |
                    |                 |
                    +-----------------+

 then the group is generated by the following 12 generators (written in
 disjoint cycle notation), corresponding  2 each to the six faces of the
 cube:

  U1:=(49, 52, 88, 85)( 62, 63, 75, 74)( 50,64,87,73)
       (51,76,86,61)(5,1,53,9)(6,2,54,10)(7,3,55,11)(8,4,56,12);
  U2:=(17, 13, 65, 21)( 18, 14, 66, 22)( 19,15,67,23)(20,16,68,24);
  L1:=(57,48,49,1)(69,36,61,13)(81,24,73,25)(93,12,85,37)
       (89,53,56,92)(90,65,55,80)(91,77,54,68)(66,67,79,78);
  L2:=(94,11,86,38)(82,23,74,26)(70,35,62,14)(58,47,50,2);
  F1:=(85,5,60,92)(86,17,59,80)(87,29,58,68)(88,41,57,56)
       (1,4,40,37)(2,16,39,25)(3,28,38,13)(14,15,27,26);
  F2:=(73,6,72,91)(74,18,71,79)(75,30,70,67)(76,42,69,55);
  R1:=(40,88,9,96)(28,76,21,84)(16,64,33,72)(4,52,45,60)
       (41,5,8,44)(42,17,7,32)(43,29,6,20)(18,19,31,30);
  R2:=(39,87,10,95)(27,75,22,83)(15,63,34,71)(3,51,46,59);
  B1:=(52,53,93,44)(51,65,94,32)(50,77,95,20)(49,89,96,8)
         (9,12,48,45)(10,24,47,33)(11,36,46,21)(22,23,35,34);
  B2:=(54,81,43,64)(66,82,31,63)(78,83,19,62)(90,84,7,61);
  D1:=(57, 60, 96, 93)( 58, 72, 95, 81)(59, 84, 94, 69)
       (70,71,83,82)(45,89,37,41)(46,90,38,42)(47,91,39,43)(48,92,40,44);
  D2:=(33, 77, 25, 29)( 34, 78, 26, 30)(35, 79, 27, 31)(36,80,28,32);

 (To check these are correct, the reader may want to print out a hard
 copy of this page and cut-fold-tape the above diagram into a cube.)


       STRATEGY

   The solution strategy is composed of 3 stages:

  Stage 1: Solve the corners. For this moves for the 3x3 Rubik's
  cube.

  Stage 2: "Pair" the edges so that the neighboring facets on
  neighboring middle edges have the same color. For this the
  following "clean edge moves" are useful:


  flipedge:=L2^2*D1^2*U2*F1^3*U2^3*F1*D1^2*L2^2*L1*U1*L1^3*U2^3*L1*U1^3*L1^3;

   (due to J. Adams who calls it "mpve 8"). This flips and swaps the two
   middle edge facets on the UF boundary. It affects some centers,
   by no other edges or corners.

  upedgeswap:=R2*B1^2*D1^2*B1^3*R2^3*B1*D1^2*B1^3*R2*B1^3*R2^2;

  (due to Thai, who calls it an "11 gram"). This move affects some
  centers but no corners and only 4 edge facets. It swaps and
  flips the right-most UF edge cubie with the right-most UB edge
  cubie, sending the U facet of the right-most UF edge cubie to
  the B facet of the right-most UB edge cubie.

  Stage 3: Solve the edges. For this the "clean edge moves" for
  the 3x3 Rubik's cube (see my "Notes on the 3x3 Rubik's cube").

  Stage 4: Solve the centers. For this, use the following
  "clean center moves":

  center3cycle:=R1^(-1)*F2*R2^(-1)*F2^(-1)*R1*F2*R2*F2^(-1);

  (also called "move 9", due to J. Adams). This move is a 3-cycle
  on centers facets, affecting no edges, no corners, and no other center
  facets. It is the 3-cycle (15,19,18) in the above notation.

  Some similar clean center moves:

  center1:=B1^2*R2^3*F2*R2*B1^2*R2^3*F2^3*R2;

  center2:=R2^2*B1^2*R2^3*F2*R2*B1^2*R2^3*F2^3*R2^3;

  These aren't really necessary since the the center3cycle can always
  be applied after a suitable set-up move (i.e., in combination with
  a suitable conjugation).
   The following move is occasionally useful:

                  centerswap:=(R2^2*U2^2)^4  ;

  This affects only 6 center facets (on the front and back faces) and
  no others. It is the product of 2 3-cycles: (15,34,23)(27,14,22)
  in the above notation.


     REFERENCES

  J. Adams, "How to solve Rubik's Revenge", Dial Press, NY, 1982

  Thai, ``The winning solution to Rubik's Revenge'', Banbury Books, 1982