Can Varasano/Ortega/Guimond be easily improved?

[Cuc]

I noticed something this week that I've never seen pointed out.

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?

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

Any ideas are welcome.

 

[xyzzy]

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

It's not too difficult to determine the corner permutation during inspection, and then all you need to know is just how your OLL alg affects corner permutation.

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 you learn two different CLL algs for each orientation case, at least one of them won't be diag-solved. (Unless the algs are related by an R2 F2 R2, so pick actually different algs to learn.)

But anyway the serious solvers all use EG, where this isn't so much of a problem because diag-solved is pretty rare with EG.

 

[Cuc]

@xyzzy I was surprised by your answers and I had to think a while whether your answers are actually answering my question.
--
Your answer to question 1 does not actually answer the how. I am confused how this could be done quickly. Yes, you could actually inspect all the corners and determine their permutation. But I could use some help how to do this quickly. I am also at a loss how to predict how the "OLL alg affects corner permutation."
--
Your answer to question 2 proposes a different solution method, so it is also not exactly answering the question.

In CLL you solve one layer first. I don't do that with the method described. In fact, it is in the PBL phase that we actually solve both layers at once. Your answer suggests devising pairs of OLL algs usable with my method, such that the algs in each pair are not related by R2 F2 R2. It required some mind warping to understand what you meant, but I got it. I can see how this idea could optimize some cases, given the algs (which I don't have yet). But I need some help to see how I can determine which one of each pair of algs to use. Perhaps that will answer itself when I have the algs?

Note. If we are going to do this correctly, there shouldn't be a case with 11 (or more) moves at all.

Looking at http://www.cyotheking.com/cll2-2/, I noticed that
- one Sune case requires 11 moves (the 5th),
- a Pi case has two 11 move algs (the 3rd),
etc.

I realize that the algs mentioned here are "good" algs; most of them are at most 3-gen and use only U, R, and F moves, but some are 4-gen and use L moves as well. That's OK. I haven't checked the optimality of these algs. But again, I don't see how this will help me right away. I believe I'll have to create those algs I need, don't I . . .

Thanks for the inspiration. Any comments and ideas are welcome.

 

[xyzzy]

In CLL you solve one layer first. I don't do that with the method described.

I meant "OLL"! Silly me. (I was thinking of "CLL" as in algs that do something to the corners, not the method.)

I can see how this idea could optimize some cases, given the algs (which I don't have yet). But I need some help to see how I can determine which one of each pair of algs to use.

You'll have to do some kind of CLL recognition. Here's one possible system. For each of the seven non-skip OLL cases, learn two algs where one of the algs preserves the D layer and one of the algs does an adj swap on the D layer. (Check a list of EG algs to find something good. I don't actually know the algs needed to do this.)

Learn how the first alg permutes the corners, and also learn to recognise the case that's "related by a diag swap". (E.g. for R U R' U' R' F R F' to solve the T case, the "related" CLL case looks like this. This is basically the case where, if you apply the usual alg, you'll end up with a diag swap. To set it up, do a Y perm then the inverse of the OLL alg.)

If the D layer is a diag swap and you get exactly the same CLL as what the first alg solves, applying the first OLL alg will give you a solved-diag PBL, which you don't want, so apply the alternative OLL alg instead to get a nicer PBL.

If the D layer is solved and you get the "diag related" CLL, again, applying the first OLL alg will give you a diag-solved PBL, so apply the alternative OLL alg instead.

In every other situation, just apply the first OLL alg, and you'll definitely not get a diag-solved or solved-diag PBL.