Metrics for Contemplating, Comparing, and Categorizing Methods

[kubnintadni]

What 3 things make a method feel different from any other? Well, probably alg-count, ergonomics, recognition, reliance on intuition, and many other factors are important, but I'd like to focus on the answers to 3 questions one might ask to characterize a method they have just been told about:

1. Does it rely more on direct solving or reduction?
2. Is it a corners first method?
3. How many cubies are affected per step?

I have not been able to find records of the following metrics which I came up with today, but they are sufficiently obvious, I think, that I would not be surprised if they were hidden behind search terms I failed to think of. That said, here they are:

1. DSR (Direct Solving Ratio): `20/(20 + total cubies reduced but not solved over all steps)`
2. CFA (Corners First Adherance): `[8 - ((2/3) * edges solved when corners are solved)]/[8 + edges solved when corners are solved]`
3. MCAPS (Mean Cubies Affected Per Step): `avg(cubies oriented, permuted, or both during step n) as n goes from 1 to the last step of the method)`

Here are some example values from popular methods (more examples at end):

* CFOP (Cross, 4x F2L Pairs, OLL, PLL)
- DSR: ~0.71 (`20/(20 + {0 + 0 + 0 + 0 + 0 + 8 + 0})`)
- CFA: 0.00 (`[8 - ((2/3) * 12)]/[8 + 12]`)
- MCAPS: 4.00 (`(4 + 2 + 2 + 2 + 2 + 8 + 8)/7`)
* Roux (FB, SB, CMLL, EOLR, L4EP)
- DSR: ~0.83 (`20/(20 + {0 + 0 + 0 + 4 + 0})`)
- CFA: ~0.29 (`[8 - ((2/3) * 6)]/[8 + 6]`)
- MCAPS: 4.80 (`(5 + 5 + 4 + 6 + 4)/5`)
* ZZ (EO-Cross, 4x ZZF2L Pairs, ZBLL)
- DSR: ~0.71 (`20/(20 + {8 + 0 + 0 + 0 + 0 + 0})`)
- CFA: 0.00 (`[8 - ((2/3) * 12)]/[8 + 12]`)
- MCAPS: ~4.67 (`(12 + 2 + 2 + 2 + 2 + 8)/6`)

1. DSR: The number of cubies eventually solved divided by the number of cubies oriented, permuted, or finally solved over all steps. This will always be in the interval (0, 1] with 1 being a method such as CFCE which has only direct solving.
2. CFA: Plot a point at (0, 0) in the Cartesian Plane. Let c = e = 0. For each step in the method, if and only if 1 or more cubies are solved (not just reduced) in that step, increment c by 1 for each corner solved thusly, increment e by 1 for each edge solved thusly, and plot (c + e, c - ((2/3)e)). Then find the slope of the line segment from (0, 0) to the plotted point at which the last corner was solved. This will always be in the interval [0, 1], with 1 being a pure CF method.
3. MCAPS: The average number of cubies which are partially or fully solved each step.

None of these metrics are meant to be measures of how "good" a method is, but rather neutral distinctions that may well play a insignificant or even non-existent part in how good a given method is. They may of course, as with any other metrics, be used to argue for a method being good or bad, but the onus will be upon the apologist to establish the relevance of the metric.

You may find that you use slightly different steps for a method. For example, maybe you can consistently plan XCross in inspection for CFOP. In that case you can count that as one step and calculate the correct values. If you can do XCross half the time, then you could calculate the values asuming both XCross and Cross respectively, and then do a weighted average of the two. The same principle applies for option select.

Methods with a higher MCAPS might tend towards more algorithms, more reliance on intuition, or both.

More Examples:

* Mehta-6CP (FB+1, 2 Belt Edges, EOLE, 6CO, 6CP, L5EP)
- DSR: ~0.65 (`20/(20 + {0 + 0 + 5 + 6 + 0 + 0})`)
- CFA: ~0.22 (`[8 - ((2/3) * 7)]/[8 + 7]`)
- MCAPS: ~5.17 (`(6 + 2 + 6 + 6 + 6 + 5)/6`)
* Mehta-CDRLL (FB+1, 2 Belt Edges, EOLE, DCAL, CDRLL, L5EP)
- DSR: 0.80 (`20/(20 + {0 + 0 + 5 + 0 + 0 + 0})`)
- CFA: ~0.22 (`[8 - ((2/3) * 7)]/[8 + 7]`)
- MCAPS: ~4.17 (`(6 + 2 + 6 + 2 + 4 + 5)/6`)
* Mehta-TDR (FB+1, 2 Belt Edges, EOLE, TDR, ZBLL)
- DSR: 0.80 (`20/(20 + {0 + 0 + 5 + 0 + 0})`)
- CFA: 0.00 (`[8 - ((2/3) * 12)]/[8 + 12]`)
- MCAPS: 5.00 (`(6 + 2 + 6 + 3 + 8)/5`)
* Beginner's (that I used) (4x Cross Edges, 3x Belt Edges, 4x D Corners, Last F2L Edge, EOLL, Permute LL Edges, L4CP, 4x Orient Corner LL)
- DSR: ~0.71 (`20/(20 + {0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 4 + 0 + 4 + 0})`)
- CFA: 0.00 (`[8 - ((2/3) * 12)]/[8 + 12]`)
- MCAPS: ~1.47 (`(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 4 + 4 + 4 + 1 + 1 + 1 + 1)/19`)

 

[Silky]

I think this is a really neat idea for categorizing methods. Generally I usually group them into 4 categories (1) LBL, (2) Blockbuilding, (3) Corners First, (4) Orient first. Methods are usually comprised of a combination of these categories:

CFOP: LBL
ZZ: Orient first + LBL
Roux: Blockbuilding + corners first
Petrus: Blockbuilding + LBL
LMCF: Corners First
SSC: Orient first + Corners first

etc..

 

[kubnintadni]

I think this is a really neat idea for categorizing methods. Generally I usually group them into 4 categories (1) LBL, (2) Blockbuilding, (3) Corners First, (4) Orient first. Methods are usually comprised of a combination of these categories:

CFOP: LBL
ZZ: Orient first + LBL
Roux: Blockbuilding + corners first
Petrus: Blockbuilding + LBL
LMCF: Corners First
SSC: Orient first + Corners first

etc..

I like that. I was just thinking today about something similar to that:

1. Corners First (eg LMCF)
2. Face Edges First (sub-categories: Cross, Arrow, Line)
3. Block First (sub-categories: 1x2x3, 2x2x2)
4. Reduction First

The first step is here defined as what advanced solvers plan in inspection. If a method would fit multiple categories, favor the lower index.

This excludes plenty of methods, but I think it covers the main ones in an intuitive way and could be expanded on piecewise in order to accommodate a method.

 

[Silky]

I like that. I was just thinking today about something similar to that:

1. Corners First (eg LMCF)
2. Face Edges First (sub-categories: Cross, Arrow, Line)
3. Block First (sub-categories: 1x2x3, 2x2x2)
4. Reduction First

The first step is here defined as what advanced solvers plan in inspection. If a method would fit multiple categories, favor the lower index.

This excludes plenty of methods, but I think it covers the main ones in an intuitive way and could be expanded on piecewise in order to accommodate a method.

What do you mean favor lower index ? And what methods don't fit these categories/combination of these categories ?

 

[kubnintadni]

What do you mean favor lower index ? And what methods don't fit these categories/combination of these categories ?

ZZ solves the cross (or sometimes arrow or line) while also solving EO. It fits the requirements for both "Face Edges First (2)" and "Reduction (4)", so it is categorized as the lower index: Face Edges First.

PCMS doesn't fit this, for example. Neither does Triangular San-Francisko (though quadrangular and hexagonal would both fall under Block First: 1x2x3).

 

[Silky]

ZZ solves the cross (or sometimes arrow or line) while also solving EO. It fits the requirements for both "Face Edges First (2)" and "Reduction (4)", so it is categorized as the lower index: Face Edges First.

PCMS doesn't fit this, for example. Neither does Triangular San-Francisko (though quadrangular and hexagonal would both fall under Block First: 1x2x3).

PCMS would be corners first no doubt. I see that your point though since you're using stricter categories. And exactly how are you defining Reduction? Is this move reduction ? Is this EO ? Technically all solving methods are reduction since you are reduction the cube by ever piece you solve/orient/reduce moves ( ZZ only using LUR for F2L ).

 

[kubnintadni]

PCMS would be corners first no doubt. I see that your point though since you're using stricter categories. And exactly how are you defining Reduction? Is this move reduction ? Is this EO ? Technically all solving methods are reduction since you are reduction the cube by ever piece you solve/orient/reduce moves ( ZZ only using LUR for F2L ).

Good point; here I'm using "Reduction" to refer to the act of bringing 1 or more pieces closer to being solved without actually solving them. So EO, CP, any part of Domino Reduction, &c..

And you are correct that PCMS is not CF by this system. That said, it is CF by its CFA score.

If I was going to assign PCMS a category, I would create a new category for multiple small blocks.

 

[Silky]

Good point; here I'm using "Reduction" to refer to the act of bringing 1 or more pieces closer to being solved without actually solving them. So EO, CP, any part of Domino Reduction, &c..

And you are correct that PCMS is not CF by this system. That said, it is CF by its CFA score.

If I was going to assign PCMS a category, I would create a new category for multiple small blocks.

EWhy wouldn't it be corners first under this system ? Also I'm assuming you're referring to CP as in YruRU/Brigs.

 

[kubnintadni]

EWhy wouldn't it be corners first under this system ? Also I'm assuming you're referring to CP as in YruRU/Brigs.

It would only be Corners First if you could plan all four pairs and predict CMSLL in inspection. I doubt that's feasible.

 

[Silky]

Also why would you group it into the lower index instead of the index which is most prevalent in the method. ZZ does use cross but Reduction/EO is a far more important feature of the method with consideration that the original variant does not solve cross.

 

[kubnintadni]

Also why would you group it into the lower index instead of the index which is most prevalent in the method. ZZ does use cross but Reduction/EO is a far more important feature of the method with consideration that the original variant does not solve cross.

Because my intent when designing this was to focus on the thing which to me most makes a method look like another: how it starts. As for what makes methods similar to eachother as a whole, that's a different (and considerably less clear) question.

ZZ feels much more like CFOP with EO at the beginning than HTA with a cross at the beginning.