A Detailed Example of the Thistlethwaite Algorithm
              Also known as the 52-move Strategy
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  Morwen B. Thistlethwaite is the mathematician who had the original
insight in composing an algorithm based on nested subgroups. He is
currently teaching at the University of Tennessee.

  First decide on a coordinate system for the cube (i.e. decide which
colours are L,R,F,B,U,D); then get it into the following position:

           (UFL)+ (URF)- (UBR)+ (ULB)- (LF)* (FR)* (RB)* (BL)*

  It may help to visualize the above by noting the 4 middle edges are
flipped and the U colour on the upper corners faces towards the front
and the back, like so:

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Stage 1:
There are 4 bad edge pieces, namely in positions LF, FR, RB, BL.
Manoeuvre these to the U-face by F1 L1 R3 D2 B2. Then the move U
corrects them.

     Summary of Stage 1:   F1 L1 R3 D2 B2 U1 (6 moves).

Stage 2:
The LR-slice edge pieces are now in positions LD, FR, RB, BL.
Manoeuvre these to the UD-slice by F2 D2 L1 R3 F1. Now taking the
corner positions in order (as in the diagram in Stage 3 instructions),
the respective twists of the pieces in these positions are
0,2,0,2,0,0,1,1. This combinations of twists is not given in Stage
2 tables, but a 180 degree rotation about the LR-axis followed by
a reflection in the LR-slice transforms this to 2,2,0,0,0,1,0,1,
which is in the tables. The move given is F1 L1 F1 L2 F3 L1 F2 B1 L2,
and transforming this by the above (involutory) symmetry gives
B3 R3 B3 R2 B1 R3 B2 F3 R2. Therefore we perform the inverse of
this move, after which L and R faces have L and R colours on them
only.

     Summary of Stage 2:   F2 D2 L1 R3 F1 R2 F1 B2 R1 B3 R2 B1 R1 B1
                           (14 moves)

Stage 3:
The positions where corners are out of orbit are numbers 1,2,5,8.
The preliminary instructions for this stage instruct us to perform
L3 U2. For the remainder of this stage alter the coordinate system
so that the original D-face faces you and the original F-face faces
upwards. In this new coordinate system the positions where corners
are out of orbit are 1,5. The permutation of corners is (1357)(24).
Multiplying this on the right by (15)(24) gives (13)(57) which is a
permutation of corners in G3. Therefore we must refer to page 7 of
the Stage 3 tables. The edge pieces of the FB-slice are in positions
3,4,5,8. The tables give us L1 F2 L3 U2 L1 F2 R2 F2 B2 R1, or
L1 D2 L3 F2 L1 D2 R2 D2 U2 R1 in the original coordinate system.
Perform the inverse of this move.

     Summary of Stage 3:   L3 U2 R3 U2 D2 R2 D2 L3 F2 L1 D2 L3
                           (12 moves)

Stage 4:
The corners are restored by L2 R2 (original coordinates). Looking
at edge pieces, as no slice is fixed and there is a 2-cycle and
a 4-cycle, we refer to page 7 of the Stage 4 tables. The only
entries where the correct arrangement of pieces is permuted and
the 4-cycle is of the correct type are (143)(1342)(34) and
(143)(1243)(34). If we hold the cube with the original R-face
facing the operator, and the original B-face uppermost, we find
we have the second of these permutations. Therefore perform the
inverse of the move give, i.e. perform B2 R2 F2 R2 D2 U2 B2 U2
R2 U2 R2 U2 in original coordinates.

     Summary of Stage 4: L2 R2 B2 R2 F2 R2 D2 U2 B2 U2 R2 U2 R2 U2

Total number of moves required: 6+14+12+14 = 46 (45 with canceling)

Mark's Note:
Stage 4 is reducible to L2 T2 L2 T2 F2 L2 B2 F2 D2 F2 L2 F2 (12 moves)