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Solving five edges and two corners

The edges-first approach

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The "edges first" approach

Difficulty Easy to learn
Efficiency Sometimes produces very good solutions, other times very bad solutions
Applications Useful in fewest-moves solving where extra thinking time can be used to distinguish between good and bad situations. Not as useful for speedcubing. Learning to solve the edges on their own is a crucial first step to mastering the more advanced approaches.

Solving the edges

When solving the edges, it is important to have some idea of how close they are to being solved. When three of the top edges are in their correct positions (relative to each other), this is actually far away from being solved. When only two edges on top are correct relative to each other, this is actually close to being solved. When no edges on top are correct, this is the same distance as when three edges are correct.

To understand why this is so, we need to look at the cases.

In the cube to the left, notice that the blue/white and orange/white edges are in order, but the red/white edge is out of order. Can you see how to solve this position in three moves? (Click play to see the solution)

When only two edges are in the correct order, this is the closest position from being solved.

Similarily, here two of the top edges are in their correct positions (relative to each other). Can you see which ones, and see how to solve the the edges in three moves?

Here, all three top white edges are in the correct order. To be able to solve this position, we first need to take one of the correct edges out of the chain so that we end up in a position that is closer to being solved. Then we can use a usual 3-move solution to finish.

Here, none of the edges are correct. First, we try to get exactly two edges correct. Following that, we can use a standard 3-move solution to finish.

Solving two corners

Now that all edges are solved, we need a way to permute the corners without disturbing the edges. The way to do this is to use commutators and conjugates. In fact, this technique for moving corners around is identical to the technique used to solve the final three corners in the last step of this solution.

Let's look at a few examples.

In our first example, we solve each of the two corners individually, each time using a corner 3-cycle. In the first 3-cycle, we care only about the red/white/green corner and ignore what happens to the red/green/yellow corner and the orange/white/green corner which are also moved by the 3-cycle. Similarly, in the second 3-cycle we care only about solving the green/white/orange corner and ignore what happens to the red/blue/white corner and the yellow/green/red corner.
This time, we try to solve two corners with a single 3-cycle. Our commutator is conjugated by a rotation of the red side.
This is one of the worst positions that can occur and one that should be avoided by looking ahead and choosing a different solving approach: all corners are in their correct positions but are incorrectly oriented. Here, we use a corner twist to solve two of the corners.
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