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