A corner 3-cycle is a cycle operation that affects three corner pieces and nothing else. This is particularly useful for solving the final corners in the Heise method. Corner 3-cycles turn out to be a simple application of commutators and conjugates.
First, we will learn how to apply a basic commutator to cycle three corners in cases where this is directly supported, and then we will learn how to conjugate these commutators to support all other cases.
Let's demonstrate the method, move by move, with some examples.
The first step is to pick any side that has exactly two unsolved corners on it, where one of those corners has correct orientation and simply needs to be moved to the other corner spot on that side. Here, the yellow side is the only choice (notice that the yellow/orange/green corner has correct orientation but just needs to be moved within that side).
We will construct the commutator out of two sequences X and Y which, for the most part, try to affect completely different parts of the cube, but which will also intersect with each other on a few very specific pieces: the 3 corner pieces. Sequence Y will affect only the bottom layer, and sequence X will affect -- for the most part -- the upper two layers, and a very small part of the bottom layer. After applying the commutator X.Y.X-1.Y-1, only the pieces caught in the intersection will be affected.
What we want to do is pull the yellow/orange/green corner out onto the top (let's call that operation X). Then we are going to insert it back into the bottom layer, but in its correct place. Since it already started with correct orientation, we can just do X-1 to insert it back in, as long as the bottom layer is first aligned ready to receive that corner in the correct position. Let's call aligning the bottom layer Y. So, X.Y.X-1 will pull the yellow corner out, shift the bottom, then insert the corner back in. Now we have to undo the shift so everything returns back to normal. That's X.Y.X-1.Y-1, which is a commutator.
But don't forget we have to solve the other two corners simultaneously. Well, if we solve one of the other corners, the third one must fall into place automatically by the laws of the cube, so we only really have to solve two corners simultaneously. There are 6 different ways to pull the yellow corner out, but only one of those ways is useful to us. We need to pull it out in such a way that the corner on top moves in and takes its place and is solved.
Can you see how to do it? (press play)
Using the principle of the commutator, the rest falls out quite easily:
The whole solution is 8 moves which is quite a bit more efficient than, for example, shifting all pieces into one layer and applying a memorised sequence.
This is a familiar example that perhaps every experienced cubist knows how to solve. But can you "figure out" how to solve it? Maybe now you can! (You can try to solve this cube with your mouse)
Hint: I have oriented the cube the same way as in the previous example, so that you will find two corners on the bottom (one of which is oriented but needs to be moved within the layer), and one corner on top that can be solved in 3 moves. Those are the two corners you should look for before you attempt to solve any position.
Can you see how to insert the top corner in 3 moves so that it replaces the oriented corner on bottom?
Sometimes, the 3 corners will not be in a position where we can directly apply a commutator, so we have to position them first, then apply the commutator, then undo the positioning move. This is really what a conjugate is.
We can pick white/red/green as the oriented corner, but the corner to replace it needs to be on the opposite half of the cube. If we rotate the right side 180 degrees, then we are in a position to apply a commutator (press play). Can you see which commutator to apply?
Here is the full solution. First, move the right two corners down to the bottom (call that X), then apply a commutator (Y), then undo the initial positioning move (X-1). That's X.Y.X-1, which is a conjugate. Notice that one move cancels out, so the whole solution is really only 9 moves.
Another familiar example...
It looks like we could pick any of the three corners as the oriented corner since on the blue side they are all oriented. However, the corner to replace it cannot be inserted in its place in 3 moves. In general, this is easier when the replacement corner has a different orientation from the oriented corner. A better option in this particular case is to consider orientation relative to the yellow/white sides. Then the yellow/orange/blue corner has correct orientation (yellow) and can be easily replaced by the blue/white/red corner.
We can position for the commutator by rotating the right side 180 degrees.