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 6 times 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

Often used algorithms

There are many examples of iconic cubing things, but none are as omnipresent or as widely useful as algorithms. Below we will be going over the most famous algorithms, such as Sune, Sledgehammer, and many more. The majority of these will be CFOP algorithms, and some will be used in other methods such as Petrus, ZZ and Roux.

Sune

Sune is an OLL algorithm, which means it orients the last layer. It is part of a special subcategory called OCLL, which means that it only orients the corners (is used when all edges are oriented). It was proposed by Lars Petrus in his Petrus method.

R U R' U R U2 R'

Anti-Sune

As referenced by the name, Anti-Sune is the opposite of Sune. It is still an OCLL, but the algorithm is mirrored. It was also coined by Petrus in the method of the same name.

R U2 R' U' R U' R'


Sune and Anti-Sune

Sledgehammer

This is a trigger that is used in a lot of algorithms, and in F2L. It also has a much lesser-known reverse, hedgeslammer. If repeated 6 times, it will bring the cube back to its previous state.

R' F R F'

Sexy Move

This is another trigger that is heavily used in almost everything. You can find it in F2L, OLL, and PLL and, if repeated 6 times on a cube, will bring it back to the same state it was in before.

R U R' U'

Reverse Sexy

This is, as the name implies, the reverse of sexy. It is less used, but is still quite prominent in F2L, where the triple sexy is frequently replaced with triple reverse sexy as it is said to be quicker. Either way, if repeated 6 times it will bring the cube back to its original state, as with most 6 move triggers.

U R U' R'

U Perms

This is a PLL (Position Last Layer) algorithm. There are 2 variants, the Ua and Ub perms. They are used when all the corners are permuted and there are 3 edges to permute in a triangular fashion. Doing either one 3 times will bring the cube back to its original state and executing either one once will make the case that the other one solves.

Ua: M2 U M U2 M' U M2

Ub: M2 U' M U2 M' U' M2

T Perm

The T perm is perhaps the most well-known PLL algorithm, with its only competition being the U perms (above) and the J perms (below). It is used to permute 2 opposite edges and two adjacent corners, and the shape of those pieces to permute when viewed from above makes a T, hence the name.

R U R' U R' F R2 U' R' U' R U R' F'

J Perms

These are 2 PLL algorithms that permute 2 adjacent edges and 2 adjacent corners. It is recognisable by the sheer number of blocks it has. There is one solved line, and 2 unsolved blocks. The Jb tends to be the faster one, as it is an RUF algorithm, but the Ja – being either an RUL or LUF algorithm – can also be very fast with practice.

Ja: R' U L' U2 R U' R' U2 R L or L' U' L F L' U' L U L F' L2' U L U

Jb: R U R' F' R U R' U' R' F R2 U' R' U'

H Perm

The H perm is a PLL algorithm that swaps 2 sets of opposite edges. All the corners are already solved. The directions of the U turns can be switched.

M2' U' M2' U2' M2' U' M2'

Key (OLL 33)

This OLL looks like a T when viewed from the top. It is not the only OLL to look like this but it is recognizable from the two opposite facing blocks that the unoriented pieces make.

R U R' U' R' F R F'

Bottlecap (OLL 51)

This OLL is a very famous one, and a good example of an OLL that contains the ‘sexy' trigger. It is a Line case, and it has 2 opposing edge and corner blocks, along with two adjacent corners that make headlights.

f ( R U R' U' ) ( R U R' U' ) f' or F U R U' R' U R U' R' F'

T (OLL 45)

This OLL is probably one of the most famous out of all the full algorithms. It is 6 moves long, and the other T shaped algorithm in addition to Key, above. The main body of the algorithm is the sexy trigger.

F R U R' U' F'

Checkerboard

This is a pattern but has a famous algorithm to make it. This is quite an intuitive algorithm, and one that many beginner cubers will be taught or figure out or see done.

M2 E2 S2 or R2 L2 U2 D2 F2 B2

Z Perm

This permutation is a PLL algorithm. It switches two sets of adjacent edges. All the corners are already solved when this algorithm is used. It is used in every method that forces an EPLL (a PLL of only the edges, in other words, all corners are pre-solved) including Petrus.

M' U' M2' U' M2' U' M' U2 M2' U

There will undoubtedly be more, as with any list, but these are the most famous and well known.

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:

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