Rubik's Cube Algorithms

A Rubik's Cube algorithm is an operation on the puzzle which reorients its pieces in a certain way. Mathematically the Rubik's Cube is a permutation group: an ordered list, with 54 fields with 6*9 values (colours) on which we can apply operations (basic face rotations, cube turns and the combinations of these) which reorient the permutation group according to a pattern.

To describe operations on the Rubik's Cube we use the notation: we mark every face of the puzzle with a letter F (Front), U (Up), R (Right), B (Back), L (Left), D (Down). A letter by itself means a 90 degree clockwise rotation of the face. A letter followed by an apostrophe is a counterclockwise turn.

F U R B L D | F' U' R' B' L' D'

Play the demonstration below:

For example: F R' U2 D means front face clockwise, right counterclockwise, a half turn of the upper face and then down clockwise. To read about slice turns, double layer turns, whole cube reorientation etc. go to the advanced Rubik's Cube notation page.

Usually we use sequences of these basic rotations to describe an algorithm. A Rubik's Cube algorithm presented in the Beginner's method is U R U’ L’ U R’ U’ L, used to cycle the three corner pieces on the upper layer, when the first two layers (F2L) are solved.

Degree of a Rubik's Cube algorithm

Every algorithm or permutation has a degree which is a finite number that shows how many times we have to execute the operation to return to the initial state.
Some examples:
F - degree is 4 because F F F F = 1.
R' D' R D - degree is 6 because we have to repeat the algorithm to return to the initial configuration.

Mathematical properties of the algorithms

In the introduction I have presented the Rubik's Cube as a permutation group. Below are the properties of the operations of this mathematical structure.

  • Associative - the permutations in the row can be grouped together: ex. (RB')L = R(B'L)
  • Neutral element - there is a permutation which doesn't rearrange the set: ex. RR'
  • Inverse element - every permutation has an inverse permutation: ex. R - R'
  • Commutative - it's not a necessary condition of the permutation group but notice that FB = BF but FR != RF
  • Degree of permutations - see above. ex: 4 x F = 1, 6 x (R'D'RD) = 1, 336 x (UUR'LLDDB'R'U'B'R'U'B'R'U) = 1

Rubik's Cube notation

We use letters to mark rotations on the cube. See the 3D widget will let you can the turns used in speedcubing. The widget renders without problems only in the latest web browsers.

Test algorithms pressing the buttons: