# 2-Gen

**2-Gen** is an abbreviation for **2** **Gen**erator. Generally speaking, it refers to an algorithm (like Sune) or a substep (like 2GLL or LSE) which can be executed/solved using a combination of only two sides or slices of the puzzle, i.e. it can be generated by only those two moves.

## Contents

## 2-Gen in abstract algebra

### Definition

If S is a subset of a group G, we denote 〈S〉 the subgroup generated by S, which is :

- the smallest (for ⊂) subgroup of G containing every element of S
- the intersection over all subgroups containing the elements of S
- the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.

This 3 definitions are equivalent. If S contains n elements, then 〈S〉 is n-Gen.

For instance, if G is the Rubik's Cube Group (the set of all Rubik's Cube algorithm) and S = {R,U} (R and U are two algorithms), then the third definition says <S> = <R,U> = { , R, U, R2, U2, R', U', R U, U R, R U2, ..., R U R' U R U2 R' U2, ... } and <R,U> is 2-Gen. If S = {U L' F r', B d'}, then <S> = <U L' F r',B d'> = { , (U L' F r'), (B d'), (U L' F r')2, (B d')2, (U L' F r')', (B d')', (U L' F r') (B d'), (B d') (U L' F r'), (U L' F r') (B d')2, ... } is 2-Gen.

### The entire cube is 2-gen

It's been known for a long time that the 3x3 Cube can be generated by <F, B, R, L, D> as Ux = R L' F2 B2 R L' Dx L R' B2 F2 L R'. This means the entire cube is 5-gen and all algorithms can be written with <F, B, R, L, D>.

Moreover, all algorithms can be written with <U B L U L' U' B', R2 F L D' R'>. Indeed, we can write F, B, R, L and D as a finite product of (U B L U L' U' B'), (R2 F L D' R') and their inverses.

## 2-Gen algorithms

### Definition of 2-Gen algorithms (mathematically and generally speaking)

*(mathematically)*A 2-Gen algorithm is an algorithm that belongs to a 2-Gen subset such as <R,U>, <M,U> or <U L' F r',B d'>.. A 2-Gen algorithm means an algorithm that uses only two elementary algorithms.*(generally speaking)*Generally speaking, those two elementary algorithms are single slice/face moves and a 2-Gen algorithm means an algorithm that uses only two sides or slices of the puzzle. The most common sides used in 2-gen algorithm for the 3x3x3 cube, but also M and U are common.

### Examples

Sune; *R U R' U R U2 R' U2* is a classic 2-gen algorithm that uses only R and U.

### Use

2-gen algorithm that uses only R and U or M and U are the fastest for most cubers to preform, so though there might be several different ways to complete the same algorithm, the 2-Gen one, if there is one, is most often used, possibly over a shorter algorithm using awkward moves like B, f, S, etc.

The Roux and Petrus 3x3 methods have completely 2-Gen steps. In Roux, the final step is done by turning only the M slice and the U layer, with the exception of the E slice in a special case. In Petrus, the F2L is finished with only R and U turns.

There are cllections of 2-gens such as 2-Gen F2L, 2-Gen OLL, 2-Gen PLL.

There are other "Gen's" too, such as 3 and 4-Gen's There are also 2-Gens for left-handed people that use L and U instead of R and U.

## n-Gen algorithms

For examples,

Alg | R'UR'U'R'U'R'URUR2 |

a 2-Gen, and

Alg | y2F2UR'LF2RL'UF2 |

a 4-Gen.

## External links

- Speedsolving.com What are 2-gen algorithms?
- Speedsolving.com 2-GEN OLL and PLL'S]
- Speedsolving.com The entire cube is 2-gen!