The following is one of many solutions to the Rubik's Cube.
This solution was developed by David Singmaster, a British mathematician.
Before starting, a method is required for describing the various moves that will be made. There are six faces, with the following notations:
Also, note that during any sequence of moves the position of the center pieces with respect to one another is unchanged.
The cube is assumed to be scrambled. The first thing to do is to chose a color, say white (it tends to stand out from the other colors on the cube).
It's also a very good idea to always do a specific color first, since it will soon be learnt which colors are adjacent, which speeds things up considerably.
The first step is to form a cross on the top face of the cube.
Orient the cube so that the white center piece is on top.
The aim is to get the correct pieces in the UL, UB, UR and UF locations.
So, some of the following moves are needed: (be sure to do those in the first step first).
The second step is to correctly position three of the U face corner pieces.
The reason that only three of them, and not four, will be put into place is that this method uses a "working space" which greatly simplifies the later steps.
There are three basic possibilities for putting corner pieces into place:
This step involves correctly placing three of the four edge pieces on the "middle" layer of the cube.
For these moves hold the cube so that the white face is on the bottom.
The only middle layer edge piece that is not to be positioned is the one right above the corner piece that was not positioned correctly in step 2.
First of all, make sure the white side is on the bottom, and the "empty" (i.e. incorrect) corner piece on the white side is in the DRF location.
The middle layer edge pieces will all be positioned in this step, except for the FR one.
To move a piece into position, rotate the cube about its vertical axis, so that the intended location is the FR location. (For example, the FL piece is to be put in place. Rotate the cube a quarter turn counter-clockwise).
Now rotate the bottom slice so that the incorrect corner piece is in the DRF location. (So in the previous example -- for the FL piece -- first turn the cube, then apply D').
Now all is prepared for the move. The move to put the new edge piece into place can only be done if it is on the U slice.
If it is, note which side is NOT on the U face.
Either F' or R will need to be applied, depending on the orientation of the edge piece to be moved.
Now, apply U until the piece to be moved is in the UF or UR (depending on the previous move) location, and then F or R, to get it back to normal.
Here is an example...
To continue, simply keep rotating D or D' and moving the cube to set up the same position, with an "empty" corner in DRF, the intended location at RF, and the piece to move in the U slice.
Note that in some cases the piece may already be in the correct location, but orientated incorrectly. In this case take it out first (i.e. put any edge piece with the color whose center is opposite white on the cube into that location) and then put it back in that spot. In other words, with the DRF corner "empty" and the offending piece in the FR spot, apply F' U' F U R U' R'.
Now two thirds of the cube should be done, less two pieces: a middle layer edge piece and its adjacent corner piece, that appears to take a chunk out of the bottom (white) layer.
Note that it is possible for the "empty" corner piece on the bottom layer to get solved by accident. If so, just ignore it, and pretend that it is unsolved.
This is the only step that requires any actual memorization. The moves from the other steps should become very natural after a short time.
There are two basic parts to this step, as follows:
The goal of the whole step is to solve all of the 5 remaining edge pieces.
The first part is to solve three of these (UF, UL, UB), and the second part is to solve the other two together.
First of all, hold the cube so that the "empty" edge piece is in the BR position, and thus the "empty" corner piece is in the RDB position. To do moves in this part, first of all move a piece into the BR location, then move it to the U face, to one of those UF, UL, or UB positions.
The move is as follows. First, optionally rotate U. Then, apply R' or B.
Then rotate U the desired amount. Then do R or B' (to undo the first part of this move).
Now, there are four possibilities.
The remaining edge pieces are the BR piece and the UR piece.
Do the following:
Background
Each face can be turned either clockwise or counter-clockwise, with respect to the center (i.e. a move that may be clockwise to the viewer when looking at the cube, may not be clockwise for that face, in relation to the middle of the cube).
The names for the different kind of moves (the U face will be used as an example) are:
The individual pieces can be referred to by a two-letter (for edges) or three-letter (for corners) combination. For example, the piece in the upper right front corner is called URF, and the edge piece to the down and left of the cube is called DL.
Also, these notations refer to the piece that is in that place at that time, not the piece that should go there.Step 1 - Top edge pieces
There should now be a white cross formed on the top of the cube.
By now, it will be possible to think how the edge pieces are located relative to one another, which should speed things up.Step 2 - U face corner pieces
So now one side should be done, except for one corner piece.
This location will be used to swap corner pieces in and out, greatly simplifying later processes.
The moves in the first two steps are really quite intuitive.
After only a few repetitions, they will be simple and natural to do.Step 3 - Middle edge pieces
Yellow is the F center. Orange is the R center.
The Yellow-Orange edge piece is to be positioned, to the FR position.
The D face has already been rotated so that the DRF location does not contain a white corner piece.
The Yellow-Orange piece is in the UB location.
The Orange is the U side, and Yellow is the B side.
Thus, apply F' U2 F. Step 4 - Solve remaining edge pieces
An example:
Say the Blue-Yellow piece is in the BR location.
Furthermore, Blue is the U color, and Yellow is the L color.
The order would be: U [to put the UL location (the destination) in the right spot] B U' B'.
However, when actually trying to solve the cube quickly, before applying U' in the previous move, look to find the next edge piece that is required to put in the right location.
So rotate U until it is in the UB location, and then by applying B' return the cube to a stable position.
Then, rotate U some amount to get the UL piece (Blue-Yellow in the example) back to the right place.
There is a tremendous amount of freedom in this sequence of moves.
In fact, there is no need to return the edge pieces to the correct spots in between repetitions of this move.
Simply realize how the pieces go with respect to one another, and then finally align them, when all three (UF, UL, UB) are done.
To reduce memorization at the expense of some speed, two of these moves suffice.
In other words, apply all three of these moves in any sequence to an all-edges correct cube, and the result will be an all-edges correct cube.