Mirek and dual Mirek?

[Stefan]

This is a suggestion/request for studying two methods for the 3x3x3 last layer.

Mirek Goljan covered this LL step a while back, finishing the LL after having solved an edge-corner-edge block of it:
http://www.ws.binghamton.edu/fridrich/L1/ece.htm

I don't know whether his algs are optimal, he doesn't provide an average, and also doesn't cover how to get the ECE block. I'd like to know the distribution and average for optimal algs for both steps (getting the ECE block and finishing the LL), both for QTM and HTM. And for building the ECE block there are four possible blocks you could choose from, so fixed-color vs 4-fold color-neutral should be analyzed.

Furthermore, I'd also like to know all this about the "dual version" of it, where corners and edges are switched, so the first step doesn't build an ECE block but a CEC block. I see these potential advantages:

- CEC block might be easier to build. Consider ECE first. If the corner is solved (or just properly oriented), it takes three moves to add one of the edges. For CEC, it takes only one move to add one of the corners. This is overly simplified, but I hope you get the point.

- CEC block can be built 2-gen more often than ECE block. For ECE, you need two adjacent edges to be correctly oriented. For CEC, one correctly oriented edge suffices.

- Fewer cases: The ECE block leaves you in one of 108 cases, the CEC block only in one of 72. Means fewer algs to learn and probably easier/faster recognition.

- After the CEC block, you can't have cases where you need to swap diagonally opposite corners, which I think often require more moves and are uglier than cases swapping adjacent corners (think of the PLLs E, N, V, Y).

- CEC block might be easier to recognize. A built CEC block can be recognized by just looking at one side of the cube (in addition to the LL side which you see anyway), the ECE block requires to see two sides of the cube.

- Compared to PLL, both ECE and CEC might offer quicker recognition of the last step. Move/solve the ECE/CEC block to the back and you can see pretty much all there's left to solved in one glance. Here ECE wins as seeing two sides (plus the LL side) is enough, CEC needs three.

 

[Johannes91]

- CEC block can be built 2-gen more often than ECE block. For ECE, you need two adjacent edges to be correctly oriented. For CEC, one correctly oriented edge suffices.

But there's the CP restriction. The probability is 1/4 for ECE and 1/6 for CEC, I think. (I also think that UR-algs are over-estimated.)

- Fewer cases: The ECE block leaves you in one of 108 cases, the CEC block only in one of 72. Means fewer algs to learn and probably easier/faster recognition.

Doesn't building the CEC block have more cases than building ECE? It also seems harder to me to do intuitively (although that could be because I haven't really practised it).

 

[Stefan]

But there's the CP restriction. The probability is 1/4 for ECE and 1/6 for CEC, I think.

Ah yes, good point. Have to think about the probabilities.

Doesn't building the CEC block have more cases than building ECE?

Yes.

Hmm, I thought the first step (building the block) has far fewer cases than the second step (finishing the last layer) because it only solves three pieces instead of five, so that the number of cases for the first step doesn't matter. But now I checked it and it's wrong. Consider the middle piece of the block as reference:

ECE: 144 cases (C=3, E1= 8, E2=6)
CEC: 216 cases (E=2, C1=12, C2=9)

Although usually you don't have only flipped edges (or you can prevent that during F2L) so the Factor E=2 would be 1, so "only" 108 cases. But still, this would probably be more reasonable by using intuition or two short.

 

[mrCage]

Hi

I remember Mirek mentioning his method before. As far as i can recall it .. there was slightly more cases in total than for the (today) standard OLL/PLL approach. However, on average the sequences were a bit shorter. I can only imagine that recognition would be quite terrible compared to OLL/PLL recognition ...

Per

 

[Robert-Y]

Erm what about this?: (Sorry if it's a bad idea...)

1. Build cross + 3 F2L pairs

2. Build an ECE block on the U face

3. Final F2L pair (for colour neutral cubers, sometimes you don't have to solve the F2L pair which corresponds to the cross colour you began with, you could fix a CE pair to link up with the ECE block on the U face, then do a whole cube rotation, giving you a new U face.)

4. Use Mirek's LL step to finish the solve.

The trouble is I don't know what the quality of the cases are like for the last two steps but I'm hoping it's good

Btw, I noticed that recognition of Mirek's LL step is actually kinda easy you can just look at the 3 CEC pieces in front of you.