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# Solving five edges and two corners

## The two-pairs approach

### The "two pairs" approach

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 Difficulty Medium Efficiency Produces average to good solutions. Applications Useful in speed cubing and fewest-moves solving.

#### Building 2 pairs

The first step in this approach is to build two corner/edge pairs while preserving the orientation of the edges. A result from group theory tells us that we can preserve the orientation of the edges by using only a restricted set of moves, which depends on the frame of reference for edge orientation. The two possibilities are described below:

 In this case, using only the red and white faces of the cube is guaranteed to preserve the orientation of the edges. In this case, using only the green and white faces of the cube is guaranteed to preserve the orientation of the edges.

After you have built both pairs, you may notice that each pair can face in one of two directions. Together, there are three possible results:

In the first two cases, assuming a frame of reference that allows turns of the red and white sides, it is possible for one of the pairs to be temporarily stored into the free slot in the first two layers which makes manipulation of the other pieces easier. In the third case, this is not directly possible since both pairs point in the opposite direction to the direction of the free slot. The third case is therefore the hardest to manage.

#### Inserting 3 edges

Once you have two corner/edge pairs, you will need to fit the three remaining edges between them. Which strategy you should apply depends on which of the three pair cases above you have, and also on what edge permutation you have.

• ##### Strategy #1: pair 3-cycles.

With this kind of pair configuration, pair 3-cycles are the most useful tool. In fact, all permutations with this pair permutation are solvable using either one or two applications of a pair 3-cycle.

Some examples are below:

This should be the default strategy, but there are sometimes more efficient ways using other techniques.

• ##### Strategy #2: simple right/top moves.

While this strategy is rarely applicable in this pair configuration, sometimes you will be able to reorder the pairs and edges using only right/top moves:

• ##### Strategy #3: move it out of the way.
Incorrect Correct

Ignoring the corners, we could solve the edges in three moves. But clearly, this will break up one of our corner/edge pairs, as shown in the incorrect solution.

However, in the correct solution, we anticipate that our planned three moves will destroy that corner/edge pair, so we move it out of the way. Once the three moves are done, we bring that pair back.

Incorrect Correct

Another example.

This time we move the pair out of the way using a slightly longer maneuver. This is actually the same number of moves as a pair 3-cycle, but depending on the previous and next moves, choosing this strategy over a pair 3-cycle may allow you to cancel some moves.

• ##### Strategy #4: F2 conjugation.

Some difficult edge permutations can be solved by an F2 conjugation, i.e. a 180 degree rotation of the front (green) side, which also guarantees that edge orientation will be preserved. The F2 move changes the effective pair configuration from two normal pairs to two opposite-facing pairs or two inverted pairs, depending on how you use it.

• ##### Strategy #5: Down-under.

Some cases that can't be solved easily on the top of the cube can be solved with the help of the bottom face using the "down-under" (B2 D') congugate. This positions the top/rear pieces onto the bottom/right side. There are three variations of the conjugated move:

 R Variation R2 Variation R' Variation

The whole sequence will affect the front/right and the back/left slots of the first two layers. Since the back/left slot is not free, we must use a preparation move - often, inserting a free, normal-facing pair into the back/left slot (if the R or R' variations are used). Here are some examples:

• ##### Strategy #6 - shuffle

If the cube is not in a state where one of the other strategies can be applied, then it is typically possible to get the cube into such a state relatively quickly using right/top/right/top moves. We call this a shuffle.

Examples:

When solving for few moves, a shuffle should be used only if there is no shorter, more direct solution using one of the other strategies, although the shuffle can sometimes be the strategy that uses the fewest moves.

• ##### Strategy #1 - simple right/top moves.

In this pair configuration, many cases can be solved using right/top moves.

• ##### Strategy #2 - R conjugation

When you have an inverted pair, the R conjugation allows this pair to be temporarily hidden away in the back/right/bottom position while the rest of the top layer is manipulated:

• ##### Strategy #3 - edge 3-cycles

When the two pairs are positioned correctly relatively to each other, it is possible to move the edges between them using edge 3-cycles.

• ##### Strategy #4 - shuffle

As with the first pair configuration, it is also possible to use a shuffle in this pair configuration to move the cube into a position that is directly solvable using one of the other strategies.

##### Inverted pairs

When both pairs are pointing in the reverse direction, the R conjugation can be used to make the pairs appear to be facing in the normal direction. For example, the following cube is solved using a pair 3-cycle conjugated by R: