# Conjugate

A conjugate is a sequence of moves that consists in doing a sequence A, then a sequence B, and finally the inverse of the sequence A. In other words, a conjugate is an algorithm like A B A', where A and B can be any sequence of moves. In intuitive terms, it means: do something to set up another task which does something useful, and undo the setup.

## Mathematical definition

Given a group, a conjugate is an element of the form ghg', where g and h are elements of the group with inverses g' and h'.

Cube notation is very close : A B A'

It is sometimes written as [A: B] = A B A'

## Effect

In A B A', the sequence A is the setup move or sequence and A' puts the cube in its original state. In between, there is an embedded sequence B. Alone, this sequence would affect some set of locations K on the cube. Because of the conjugation, the affected pieces will be KA', which means that the locations will be transformed by the inverse of A.

Indeed the sequence B will be applied on a cube which space has been modified by A. The sequence B is supposed to change some pieces in the locations K and A consists in bringing the piece you want to change to the location K. Once you've done the sequence B, you bring those pieces back with A'.

### Trivial case

If the set of locations K is not modified by A, then the conjugation does not change anything, and A B A' = B.

### General case

Pieces can be affected by the following subsequence of ABA':

• by B only: this part is outside of the conjugation
• by AB: this is the first semi-conjugated part
• by BA': this is the second semi-conjugated part
• by ABA': this is the conjugated part

Let's call J the set of pieces modified by A, and N the intersection of J and K.

The initial location of the outside part is: O = ((K \ N)B \ N)B'

The initial location of the first semi-conjugated part is: H1 = (NB \ N)B'A'

The initial location of the second semi-conjugated part is: H2 = NB' \ N

The initial location of the conjugated part is: C = (NB inter N)B'A'

The slot location is: NA'

### Usage

The conjugation is often used to provide a setup for some transformation. The conjugation ABA' is very similar to B, except that B is applied on other pieces as it would be without conjugation. For example, if a sequence exchange some pieces, or rotate some pieces, you can choose to exchange other pieces or rotate other pieces :

[F': [M2D2M2, U]]

where [M2D2M2, U] exchange the four top edges, so applying F' exchange the front-right edge instead of the front-top edge.

The intersection is N = front-top edge

The slot is NA' = NF = front-right edge

The conjugated part is:

C = (NB inter N)B'A' = empty

Surprisingly, in this example, the conjugated part is empty. It means that no individual piece is actually following the whole conjugation. F' and [M2D2M2, U] are only connected so that some pieces are transfered through the intersection N. The transfer occur between both semi-conjugated parts.

In this simple case, we do not really distinguish the semi-conjugated parts. But it is possible to take this difference into account, and also to use a conjugated part.

#### Semi-conjugated parts

Semi-conjugated parts can be used to move pieces inside or outside a slot. This is widely used to insert or extract a pair of pieces in F2L or to insert or extract corners or edges individually.

H1 is extracted from the slot and H2 is inserted into the slot.

Example: RU'R'

N = top-right edge, top-right-back and top-right-front corners

Slot location : NA' = NR' = top-right-front corner, front-right edge and front-right-bottom corner

H1 = (NU' \ N)UR' = top-right-front corner and front-right edge

These pieces are extracted from the slot and replace the top-back edge and top-back-left corner.

H2 = NU \ N = top-front-right corner and top-front edge

These pieces are inserted into the slot, replacing front-right edge, and front-right-bottom corner.

Example: RUR'

N = top-right edge, top-right-back and top-right-front corners

Slot location : NA' = NR' = top-right-front corner, front-right edge and front-right-bottom corner

H1 = (NU \ N)U'R' = front-right edge and front-right-bottom corner

These pieces are extracted from the slot and replace the top-front edge and top-front-left corner.

H2 = NU' \ N = top-back-left corner and top-back edge

These pieces are inserted into the slot, replacing front-right edge, and front-right-top corner.

#### Conjugated part

The conjugated part is used to move pieces inside the slot (the pieces do not enter nor leave the slot):

RUR'

where the intersection is N = top-right edge, top-right-front corner and top-right-back corner

and the slot is NA' = NR' = front-right edge, front-right-top corner and front-right-bottom corner

The conjugated part is :

C = (NB inter N)B'A' = top-right-front corner

This piece is moved to the front-right-bottom corner and rotated so that the right facelet is moved to the bottom facelet.

The outside part O is only affected by B', so it can be put back into place by adding B' to the sequence, giving ABA'B', which is the commutator of A and B.

The first semi-conjugated part H1, which was affected by AB, is then only affected by A, becoming the container of pieces PA.

The second semi-conjugated part H2 is split into two parts:

• one part which is affected by B', so affected by BA'B', and is then a conjugated part of the commutator
• one part which is not affected by B', so affected only by BA', becoming the container of pieces PB.

The conjugated part is split into two parts:

• one part which is affected by B', becoming the quirk part
• one part which is not, becoming a conjugated part of the commutator

Pieces that were not affected by the conjugate, but that are affected by the last B' move, are the PN pieces of the corresponding commutator.

## Differences among conjugation, transformation, and pseudo/non-matching blocks

As of August 2020, it seems that the terminology has been used interchangeably recently.

### Conjugation

Conjugation is very broad. This is the general term for A B A'. You perform a setup, do some moves, then undo the setup. You have performed a setup so that you can easily perform the moves in "B". The A setup moves cause the puzzle to be in an offset state, changing the appearance. Then you undo the offset later. However, traditionally there is no further intent within this setup move. It wasn't until around 10 years ago that the technique of offsetting a layer was applied with the intent to reduce large algorithm sets and the overall move count of a method. That is where the term Transformation comes in. It refers to this specific technique.

### Transformation

 Main Article : Transformation

Transformation is the application of a conjugate to change the state of the cube to gain a large advantage in the future. This is a relatively new term in the community. Transformation technically is conjugation, but there is a big difference in the intent. The reason for the application is completely different from the traditional use of A B A' setup, moves, setup undo. In transformation you are intentionally trying to change a case into another. You are taking advantage of the state of the pieces to improve the rest of the solve. Transformation is used to reduce the number of moves and the number of cases in a method. Conjugation doesn't specifically refer to this intent; it is a general term. Transformation is a different technique under the conjugation umbrella.

Setup = L' U R U' L U R' U'. This is one of the Sune orientation cases. But if you use the URF+UR pair and do an R' turn, you get the below state.

You have now transformed a Sune case into an L case. This is the L case solved by r U R U' L' U R' U'. So if you do r U R U' L' U R' U' then R (which undoes the R' turn in the last example), it will be solved.

Example methods/applications: NMLL, A2, CTLS, 42, and other case reduction applications.

### Pseudo/Non-Matching

 Main Article : Pseudoblocks

This is when pieces are put together in such a way that the colors don't match. This is referring to the building process. We are only talking about the current state of the pieces. It is all about what everything looks like right now - not about your future intent.

Example methods/applications: Roux, Heise, ZZ, FMC, A2, A3, and it can be used in pretty much everything else.

### Differences among conjugation, transformation, and pseudo/non-matching blocks

In summary, the difference is in intent. In language, we have many separate words to clarify intention and reason. It is important to understand the differences for clarity. When someone says "conjugated Roux" or "conjugated ZBLL", it isn't clear what is meant and it requires them to provide an explanation. Do they mean Roux/ZZ with non-matching blocks? For Roux, do they mean 42? For ZZ, do they mean CTLS? Or do they mean something else? Because pseudo/non-matching and transformation fall under conjugation, it is important to use the correct term for the specific technique that is being used.