In the 2x2 method, you might end up with the PBL-case where one layer is solved and the other layer is scrambled (opposite corners need to be swapped). But this case requires 11 turns, and that equals the diameter of the 2x2 cube group. It is also called an antipode, as it is furthest away from the solved state. This seems inefficient.

To summarize the steps of a version of Guimond I use, with an estimated average of steps:

1. solve a face in opposite colors: 2 htm (max. 3 htm)

2. solve the opposite face in opposite colors with a (Guimond) OLL: 5 htm (max. 6 htm)

3. separate the colors: 4 htm (max. 5 htm)

4. PBL: 8 htm (max. 11 htm)

5. AUL (aka AUF): 1 (max. 1 htm)

So, I was thinking this: Can't we prevent the aforementioned PBL case that requires 11 htm moves?

Not only do we need more turns than the average sum of the previous steps, we need the theoretical maximum number of steps of God's algorithm for 2x2. It seems awfully inefficient.

The analogy would be that in order to go down a mountain (solving the cube), you will sometimes have to move to the top (reaching an antipode) before you can go down.

The only rational explanation for this phenomenon is this: the PBL case in question can be recognized, but we did not know that that's what we were going to end up with. As we are struggling to solve the 2x2, we sort all the corners and finally "see" that we are furthest away.

In the analogy of the mountain, it means that we have avoided local hills and chose a locally optimal route that nevertheless led us slightly up. Once on top, we understood that we've been on a somewhat rough terrain that wasn't too steep and it was difficult to see whether we went up or down on average. But once on top, we had a clear view of where "down" is (we saw a river).

So, my questions are:

**1. How can we recognize at an early time (at inspection time?) that our solution involves an "opposite corner swap"?**

2. Can we reverse our fortune by performing this swap on our way to the PBL phase (perhaps by using special algs), so we can eliminate that possibility once we have reached the PBL phase?

2. Can we reverse our fortune by performing this swap on our way to the PBL phase (perhaps by using special algs), so we can eliminate that possibility once we have reached the PBL phase?

If we can, then perhaps this can improve the Varasano/Ortega/Guimond methods with a couple of moves.

Any ideas are welcome.