This is one way to solve the cube from any scrambled position. It is certainly not the fastest way to do it. Rather, my idea was to do it using the smallest number of memorized move sequences (the reasons for seeking such a solution are, alas, obvious, at least in my case!). The present method uses three (see (1-3) below), plus a few variations and much repetition, and assumes that the first layer is "intuitive". Each of the moves are standard moves, see for example,Mark Jeay's Rubik page. These moves are also either commutators or close to same, and hence from a theorem in group theory correspond to 3-cycles, i.e. three cubies permuting while the other cubies remain fixed; this is essentially why they work.

Some Notation

For a given orientation of the cube, as is commonly done, we use "F" to denote a 90 degree clockwise rotation of the front face of the cube; in like manner, B,L,R,U,D respectively represent a 90 degree clockwise rotation of the back, left, right, up, and down faces of the cube. "F'" represents a 90 degree anti-clockwise rotation of the front face, etc; "F2" represents rotation of the front face through 180 degrees, etc. Operations are performed left to right, so that DL means first D, then L, etc.

The centre colour of each face never changes, so solving a scrambled cube consists in performing rotations to ensure that the colours of the surrounding cubies on each face match those of the centre cubie; the surrounding cubies come in two flavours: "corner" cubies and "edge" cubies. It is common to view the cube as consisting of three layers: Layer One (the top layer), Layer Two (the middle layer) and Layer Three (the bottom layer); for definiteness we orient the cube so that Layer One contains the white face; on the more recent cubes, this means that Layer Three contains the yellow face, and we assume this to be so in the rest of this discussion (earlier cubes may have a different colour opposite the white face).

Solving Layer One

Orient the cube so that the white center cubie is in the top face. Layer One is solved when the top face is white and the colours in the sides of the first layer all match the colour of the center cubie in each of the side faces. One standard way to do this is as follows:
1.  Do the edge cubies first. 
  The basic idea is to move each white edge cubie so that the white side is in the D face, rotate the D face so that the side colour of the white edge cubie and the side face centre colour match, and then use F2, L2, R2, or B2, as appropriate. 
2.  Do the corners. 
  The basic idea is to move each white sided corner cubie to the bottom layer, so that the white side of the cubie lies in a side face (i.e. F, B, L, or R ). Then, rotate the bottom layer so that the non-white, non-D side of the corner cubie matches one of the side centres and orient the cube so that the white side of the corner cubie is in the F face and do a "winkle" move, i.e. a move that pushes a white sided corner cubie in the bottom layer to a correctly oriented corner cubie in the top layer. For example, if the white side is on the right use (the right face winkle) D'R'DR or if the white side is on the left use (the left face winkle) DLD'L'

Solving Layer Two

The idea is to move edge cubies from Layer Three to Layer Two. The following standard move sequence achieves this. Rotate Layer Three so that the D face of the DB edge cubie matches the F centre. Then look at the B face of the edge cubie. If this edge cubie must move to the RF spot use
FDF'D' and a (right face) winkle move R'D'R   (1)

Otherwise, to move to the LF spot use
F'D'FD and a (left face) winkle move LDL'   (1)'

You may need to repeat this four or so times, to correctly place the four middle layer edge cubies.

Solving Layer Three

Turn the cube over so that the side opposite the white face (for us the yellow face) is now the "U" side. This step has three stages.
1.  Do the edge cubies first. 
  Here, ignoring the corner cubies for now, we wish to form a "yellow cross" on the U face with the side colours matching their centres. Typically, you will see a "yellow ell" on the U face (i.e. two adjacent yellow cubies on the U face, with the other edge cubies being non-yellow on the U face). Check first to see if you have a "good" ell or a "bad" ell. Rotate the U face so that the "yellow ell" is in the UF/UR position, with the side colour of the cubie in the UR position matching the R center colour. The move sequence 
B   ULU'L'   B'    (2)

is a 3-cycle which leaves the UR edge position fixed and clockwise rotates the UF, UL, and UB edge positions. It also converts a yellow ell in the UF/UR position into a yellow cross. So a "good ell" is one in which one can apply the move (2) to the ell in the UF/UR position to obtain a correctly oriented yellow cross. 

There are two possibilities for a "bad ell". If the UF, UL, UB cubies either have the correct colours or need to be rotated anti-clockwise to the correct positions, rotate the "yellow ell" to the UF/UL position apply (2) and check for a "good ell". Otherwise, there is a parity problem, and one must rotate the "yellow ell" to the UB/UR position, apply (2) and check again for a "good ell". Sometimes three or four iterations of this type are needed to obtain a "good ell". 

Occasionally, one is faced with a "yellow straight", i.e. two opposite cubies are yellow. To obtain a "yellow ell", apply (2). If one has just the yellow centre and no other yellow edges on the upper face showing (or one has a yellow cross, but the orientation is incorrect), apply (2) twice, obtaining first a yellow straight and then a yellow ell. 

2.  Position corner cubies. 
  For now we ignore the local orientation of the corner cubies. 

If exactly one corner cubie is correctly located, orient the cube so that this correct corner cubie is in the UFR position and apply 
URU'   L'   UR'U'   L    (3)

(repeat once if necessary). 

If no cubie is correcly positioned, first apply (3), and then do the above step. 

3.  Fix local orientation of corner cubies. 
  If any corner cubies need to be "rotated in place", proceed as follows. Orient the cube so that the candidate is in the UFR position. The current F face will remain the F face throughout all of this step (MOST IMPORTANT!)

If the yellow side of the candidate corner cubie is in the F face apply 
FDF'D'    FDF'D'     

(i.e. FDF'D' squared; notice that this is the square of (1) without the winkle) 

If the yellow side of the candidate corner cubie is in the R face apply 
DFD'F'    DFD'F'     

(i.e. DFD'F' squared; this is the inverse of the one above, just remember "DF" instead of "FD"). 

Then if other corner cubies need re-orientation, rotate each into the UFR position and repeat the above step, KEEPING THE F FACE THE SAME FOR EACH CANDIDATE (if you ignore this point, you will be left with a wonderfully scrambled cube!). At the end, you may have to turn the U layer to complete the task. 


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