Mehta, APB, and other EO + RB systems: An analysis on speedsolving use and classification

[Athefre]

In the past year there have been a few ideas posted to the community which start by building a 2x2x3, perform EO shortly after along with a piece or two, do a step or two more, and end with ZBLL or a couple of other possible steps. The two most well-known are Mehta and APB. This post will cover the systems that have been proposed, their speedsolving qualities, and classification.

The systems and their steps​

Mehta:
Step 1: Solve a 2x2x3 on the left + the BR edge. Typically done as 1x2x3 on the D layer then adding any three E layer edges with the FR edge slot unsolved.
Step 2: Orienting the remaining edges while solving the FR edge.
Step 3: Solve the DR line of pieces.
Step 4: ZBLL.

APB:
Step 1: Solve a 2x2x3 on the left.
Step 2: Create a pair then insert the pair while orienting all edges.
Step 3: Solve the remaining three F2L pieces.
Step 4: ZBLL.

VDBR (voidrx's DBR):
Step 1: Solve a 2x2x3 + the BR edge.
Step 2: Orient the remaining edges while solving the DBR corner.
Step 3: Solve the remaining three F2L pieces.
Step 4: ZBLL.

B1CEZ (from V Achyuthan):
Step 1: 2x2x3.
Step 2: FR edge.
Step 3: D layer corners.
Step 4: EO + DR + BR edges.
Step 5: ZBLL.

V Achyuthan's idea #2:
Step 1: 2x2x3.
Step 2: EO + DR + BR edges.
Step 3: DFR Corner.
Step 4: FR edge + DBR corner.
Step 5: ZBLL.
Note: Can combine Steps 3 and 4 to make it an L3P step.

RPSI (Random Petrus System Idea just for this post):
Step 1: 2x2x3 on the left.
Step 2: Solve the DR edge + DBR pair.
Step 3: Solve DFR while orienting all remaining edges.
Step 4: FR + BR edges.
Step 5: ZBLL.

Petrus:
Step 1: 2x2x3.
Step 2: EO.
Step 3: RB.
Step 4: ZBLL.

Speedsolving qualities

Best in each category (with the same number of steps):
Movecount: APB
MCC: APB
Algorithm count: Petrus

All of the numbers were generated using a solver created by trangium. The total averages are calculated by summing the numbers from all algorithm sets into one, not by taking the average of the averages. So which should you choose? Overall it looks like if you want the shortest movecount and best ergonomics, APB produces the best results. If you want the fewest algorithms, you should simply use intuitive EO then intuitive RB as in standard Petrus.

• APB has the lowest movecount and the best MCC (which is a combination of ergonomics and movecount).
• Mehta and B1CEZ have the worst MCC.
• Mehta requires by far the most algorithms to memorize.
• RPSI contains a low number of algorithms and has good ergonomics. VDBR has good ergonomics as well. The disadvantage of these two is that they have a lot of algorithms in the EO step. The EO steps of all systems is where it is most likely to have several cases which are difficult to find good algorithms due to the movegroup.
• B1CEZ has the a low number of algorithms and good ergonomics. But it is a 5 step system that doesn't take care of many pieces at once. This is why it wasn't included in the best in each category above. It also has a much higher movecount than the rest.
• The differences among the systems seems to come from the fact that some have freedom of movement restrictions. These reduce algorithm ergonomics and slightly increase the potential movecount. Such as the FR and BR edges being placed first in Mehta or the D layer corners in B1CEZ.
• V Achyuthan's idea #2 contains 719 algorithms for the EO + two edges step. So that wasn't generated.
• In the algorithm sheets, TDR uses a weighted average for the movecount. That number wasn't used in the comparison since it wouldn't have been fair to everything else because they don't have weighted averages applied.
• Some potentially work better with pseudo than others. That has yet to be tested. Although pseudo in this case doesn't provide a major advantage due to often needing to use a move to correct the pseudo at the end of the solve.
• The movecount for almost all is close to the same. The biggest difference comes in ergonomics and the number of algorithms that a system requires the user to learn.
• Recognition will overall be a tie among all. The systems other than standard Petrus have the same number of edges to be recognized in their EO steps and don't require finding many specific pieces in each step. Petrus EO recognition isn't really bad, but it is a common complaint against the method. Intuitive RB doesn't really require recognition in the way that an algorithm based step does, so it evens out.

These results are also confirmed by comparing the movecount and ergonomics of the available alg sheets for some of the various systems. The solver results that were used to make the comparison can be seen below.

Spoiler: APB

Spoiler: Mehta

Spoiler: Petrus

Spoiler: RPSI

This one could be really good if the pair is kept on the U layer instead of being completely solved in step 2. Then the algorithms would use ARFs instead of AUFs. u moves as an option in the orientation step may further reduce the movecount.

Spoiler: VDBR

Spoiler: B1CEZ

Spoiler: V Achyuthan's idea #2

This system would likely have a good movecount and good ergonomics. Then combining steps 3 and 4 to make L3P would be 116 algorithms with around the same ergonomics as APB's L3P. However, this system contains a lot of algorithms (over 800).

Classification​

With all of these methods following the steps of 2x2x3 > EO + RB in some way > LL and having the same or similar possible variant endings such as L5EP and PLL, it seems to make sense to classify them as Petrus systems. Just as APB was initially. Soon after APB was proposed it was switched to advertising as a separate method for more widespread attention. Though it is of course still completing EO + RB after 2x2x3 as in Petrus. It is pretty easy to think of various orders of solving EO along with RB. Each varying in speedsolving quality. But would it really be necessary to call them new methods?

Reasons that they could be called Petrus systems or a single new group:

• All start with a 2x2x3.
• Can use FB + E layer edges, 2x2x2 > 2x2x3, or any 2x2x3 building strategy in each.
  • FB + E layer edges can be used with APB, Petrus, or any others and they are still the same method. 2x2x2 > 2x2x3 > BR edge can be used with Mehta and it is still the same method.
• Perform EO soon after the 2x2x3 is built.
• Have a primary or optional variant that completes the right side block then ends with ZBLL.
• Have the same or similar variants such as ending with L5EP.

If there are any suggestions or corrections, please provide them.

 

[Reirto-RRNF]

Well great job for making this

Well seeing this make me debating to myself should i keep use Mehta or not, but yeah i more comfortable with Mehta so i gonna keep using Mehta

 

[Jack Reiman]

classifying them as petrus systems is something that i've also thought of doing, and this makes a lot of sense

 

[S1neWav_]

Well great job for making this

Well seeing this make me debating to myself should i keep use Mehta or not, but yeah i more comfortable with Mehta so i gonna keep using Mehta

use whatever is fun for you

 

[tsmosher]

Classification​

With all of these methods following the steps of 2x2x3 > EO + RB in some way > LL and having the same or similar possible variant endings such as L5EP and PLL, it seems to make sense to classify them as Petrus systems. Just as APB was initially. Soon after APB was proposed it was switched to advertising as a separate method for more widespread attention. Though it is of course still completing EO + RB after 2x2x3 as in Petrus. It is pretty easy to think of various orders of solving EO along with RB. Each varying in speedsolving quality. But would it really be necessary to call them new methods?

Reasons that they could be called Petrus systems or a single new group:

• All start with a 2x2x3.
• Can use FB + E layer edges, 2x2x2 > 2x2x3, or any 2x2x3 building strategy in each.
  • FB + E layer edges can be used with APB, Petrus, or any others and they are still the same method. 2x2x2 > 2x2x3 > BR edge can be used with Mehta and it is still the same method.
• Perform EO soon after the 2x2x3 is built.
• Have a primary or optional variant that completes the right side block then ends with ZBLL.
• Have the same or similar variants such as ending with L5EP.

If there are any suggestions or corrections, please provide them.

Would the various methods just be Petrus-VDBR, Petrus-M (or Petrus-Mehta), Petrus-APB, etc. then?

EDIT: You missed Petrus-W. Great post as always.

 

[Reirto-RRNF]

Would the various methods just be Petrus-VDBR, Petrus-M (or Petrus-Mehta), Petrus-APB, etc.

Don't really like calling those like that. Just APB, VDBR and Mehta is just more simple

 

[OreKehStrah]

Don't really like calling those like that. Just APB, VDBR and Mehta is just more simple

It matches the way we name ZZ variants. ZZ-A, ZZ-B, ZZ-C, ZZ-CT, etc. We usually just say we use ZZ and then add on the extra part to be more specific when needed. I would imagine most people would get used to saying they use Petrus, and specify what kind if needed as well.

 

[DuckubingCuber347]

I'm up to referring to 223 variants as Petrus-[x] but I think you'll have a hard time convincing Mehta users on this.

 

[Reirto-RRNF]

I'm up to referring to 223 variants as Petrus-[x] but I think you'll have a hard time convincing Mehta users on this.

Well yeah is this the problem because i'm Mehta users

 

[V Achyuthan]

I'm up to referring to 223 variants as Petrus-[x] but I think you'll have a hard time convincing Mehta users on this.

I agree

 

[V Achyuthan]

Well yeah is this the problem because i'm Mehta users

lol

 

[Jack Reiman]

It matches the way we name ZZ variants. ZZ-A, ZZ-B, ZZ-C, ZZ-CT, etc. We usually just say we use ZZ and then add on the extra part to be more specific when needed. I would imagine most people would get used to saying they use Petrus, and specify what kind if needed as well.

I can't say I really like the sound of putting petrus as a prefix, the names apb and mehta sound really good on their own. I can get behind the classification change, but the naming scheme would definitely need work.

 

[GenTheSnail]

Would LEOR fit into this system?
As a method that achieves eo223 by way of fb > dfdbeo, that seems to fall in line with these.

 

[Athefre]

Would the various methods just be Petrus-VDBR, Petrus-M (or Petrus-Mehta), Petrus-APB, etc. then?

EDIT: You missed Petrus-W. Great post as always.

Petrus-W would need a variant which ends in ZBLL. I checked the wiki and it looks like the F2L variant is just EO then 1x2x3 (Petrus on the front). Is there some other difference?

Would LEOR fit into this system?
As a method that achieves eo223 by way of fb > dfdbeo, that seems to fall in line with these.

Interesting question. It's right on the line. The edges are oriented while the 2x2x3 is being completed versus immediately after. Though the intended group of this post contains systems which orient all edges while solving the right side block. LEOR could just use any of those but without their EO steps. I don't think I would put it in this group.

 

[DuckubingCuber347]

I've already adopted referring to 223 on the left followed by EO methods as Petrus-[x] as can be seen in my signature and PP thread but I do have a few problems with it:

Do we keep the names themselves and just add the Petrus prefix or use a naming scheme nearly identical to ZZ, i.e. ZZ-a, ZZ-d. I am assuming that we would go with the first option, referring them not as Mehta but Petrus-Mehta. My problem with this is that it does not exactly roll off your tongue as well as something like ZZ-CT and it just seems to complicate matters when we could simply teach others that x-method is a variant of y-type (in this case y=dl followed by EO).
If we go with the latter option this still raises problems as it not only partially discredits the creators, but it would still have a tricky naming scheme. One would have to assume the base Petrus should be called Petrus-a but what would we name APB? We cannot have two identical P-a's as one would often be clueless as to what the writer is truly referring to. We could call base Petrus simply, Petrus, but that would still complicate matters since it is simply to vague and one would wonder if another were referring to a variant or Petrus itself.

My other, related, problem has to do with variants, and how they would fit on with the titles. Petrus-Mehta-TDR sounds bad (or if using the alternative naming Mehta would be shortened to "M" or "YM") so how can we distinguish variants within the variants?

Recap: I strongly support the idea but there are some problems to be addressed with the fluidity of the words.

Edit: I forgot to add, what would we do about APB with 223 at db? It doesn't fit into the dl>EO system but it's still essentially the same method as APB.

 

[tsmosher]

Petrus-W would need a variant which ends in ZBLL. I checked the wiki and it looks like the F2L variant is just EO then 1x2x3 (Petrus on the front). Is there some other difference?

It is simply: db > EO using the free F layer > L2P > CODFLL > L5EP-DF. There is no variant that ends with ZBLL.

 

[Athefre]

Edit: I forgot to add, what would we do about APB with 223 at db? It doesn't fit into the dl>EO system but it's still essentially the same method as APB.

It is part of the same categorization. Petrus does EO and the final 1x2x3 to complete F2L. The systems are the various ways in which you can do that. The 2x2x3 can be anywhere as long as EO + the block is solved.

For the naming of the systems, that is something that we are discussing in the server. Nothing perfectly complete yet, so we welcome ideas.

It is simply: db > EO using the free F layer > L2P > CODFLL > L5EP-DF. There is no variant that ends with ZBLL.

If I add a comparison table for L5EP for each system, I will include Petrus-W.

 

[Cubing Forever]

Why not divide methods into families?
There are many methods, which are fundamentally similar, but structurally dissimilar(AKA they reach the same cubestate but with different approaches), like the methods mentioned in OP.
Instead of calling these variants of each other, we can call them methods that belong to a particular family.

Here's a classification that I've thought of:

FreeFOP family: anything that solves F2L without solving EO in between.
Petrus/EO223+RBLL family: Regular Petrus, all Mehta variants, APB, LEOR etc.(anything that has EO223(+solve something) as its first steps).
Roux/reduce to LxE family: Roux, 42, WaterRoux, ZBRoux etc.(anything that has dL+dR(or a part of dR) as its first steps).
EOFirst/ZZ family: All ZZ variants, Squall, maybe EOMR(anything that does EO+something as its first step).
Corners/Columns first family: Waterman, LMCF, any columns first etc.
Francisco family: Triangular, Quadrangular and Hexagonal Francisco etc.

As for Nautilus, I'm not able to place it anywhere since some variants(like L5E, CX) look like they belong to the Roux family and some others(like ZBLL, LL) look like they belong to the FreeFOP family or the Petrus family.

ofc this isn't perfect so any improvements/suggestions/constructive criticisms are welcome.

 

[Swagrid]

Fantastic work as always from Athefre. Always a treat to see a new Athefre post, be it on speedsolving or on discord.

Cool to see that the metrics for alg count, movecount, and mcc are dominated by Petrus and APB, which is by all means Petrus with a spicy RB. Viva la Petrus method.

Also:

Mehta and B1CEZ have the worst MCC

This is hilarious to me. Mehta was sold on ergonomic alg spam. It had the worst MCC. You love to see it.

 

[Athefre]

Why not divide methods into families?
There are many methods, which are fundamentally similar, but structurally dissimilar(AKA they reach the same cubestate but with different approaches), like the methods mentioned in OP.
Instead of calling these variants of each other, we can call them methods that belong to a particular family.

Here's a classification that I've thought of:

FreeFOP family: anything that solves F2L without solving EO in between.
Petrus/EO223+RBLL family: Regular Petrus, all Mehta variants, APB, LEOR etc.(anything that has EO223(+solve something) as its first steps).
Roux/reduce to LxE family: Roux, 42, WaterRoux, ZBRoux etc.(anything that has dL+dR(or a part of dR) as its first steps).
EOFirst/ZZ family: All ZZ variants, Squall, maybe EOMR(anything that does EO+something as its first step).
Corners/Columns first family: Waterman, LMCF, any columns first etc.
Francisco family: Triangular, Quadrangular and Hexagonal Francisco etc.

As for Nautilus, I'm not able to place it anywhere since some variants(like L5E, CX) look like they belong to the Roux family and some others(like ZBLL, LL) look like they belong to the FreeFOP family or the Petrus family.

ofc this isn't perfect so any improvements/suggestions/constructive criticisms are welcome.

voidrx also talked about method families. He even has a doc on it. For Nautilus he created a Nautilus family since it isn't part of the others. If you want to talk more in-depth about classification in general, we could do that in the Method Debate and Development Server or I can make a classification channel in my development server. We can get the discussion going to potentially create something presentable to the community.

For families and Petrus specifically, I think that is essentially what is talked about in the main post. To gather the methods which do Petrus's EO > RB into a group. If we do families, there's definitely going to be a family tree like structure with various branches.