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I will be using the official WCA notation from Article 12 of the WCA Regulations. To make things easier to explain, it is useful to develop a notation that easily describes certain moves. For the purposes of the diagrams below, F is the red face, B is the orange face, L is the blue face, R is the green face, U is the yellow face, and D is the white face.

Face Moves

Thus, I will use this notation in which each face is represented by a letter (F (front), B (back), L (left), R (right), U (up), and D (down)) and moves can be described by the letter of the face that is being turned. An upper-case letter means to turn that face in the clockwise direction. If that letter is followed by a prime ('), then that face should instead be turned counter-clockwise. Similarly, if that letter is followed by the number two (2), then that face should be turned twice. Sometimes, I will follow a 2 with a prime (2'), indicating that the move should be made in the counter-clockwise direction (for speed reasons).

F F2 F2' F'
B B2 B2' B'
L L2 L2' L'
R R2 R2' R'
U U2 U2' U'
D D2 D2' D'

Double Outer Face Moves

A capital letterfollowed by a lower case "w" works the same way, except instead of just turning the outside face, the middle layer adjacent to it should be moved as well (in the same direction). For example, "Dw" is equivalent to performing "D" and "E" simultaneously.

Fw Fw2 Fw2' Fw'
Bw Bw2 Bw2' Bw'
Lw Lw2 Lw2' Lw'
Rw Rw2 Rw2' Rw'
Uw Uw2 Uw2' Uw'
Dw Dw2 Dw2' Dw'

Slice Moves

Slice turns (M, E, and S) are sometimes useful for certain algorithms. In a slice move, the "outside" layers remain stable, and a middle layer will slice through them. They can have the same suffixes as the other face turns.

M M2 M2' M'
E E2 E2' E'
S S2 S2' S'

Cube Rotations

Cube rotations (x, y, and z) do not involve the turning of any layers of the cube. Instead, the entire cube is rotated (imagine an x, y, and z axis going through the cube). These are often used to move the cube into a position that makes an algorithm easier to perform.

x x2 x2' x'
y y2 y2' y'
z z2 z2' z'