# Hierarchy of Last Layer Sub-Steps, Subsets of OLLCP and ZBLL



## Logiqx (Jun 17, 2015)

I've created a diagram which shows the hierarchy of last layer sub-steps with OLLCP and ZBLL at the top.

The diagram should be pretty self-explanatory. Please let me know if I've made any mistakes or major omissions.

Here's the link - http://cubing.mikeg.me.uk/LLSS.pdf

What I mean by a "pure" subset is where the exact cases can be found in the parent set; recognition and execution. The typical OCLL (orienting corners, retaining edge orientation) consists of 7 cases from OLL and is what I call a "pure" subset. If you arrange the 57 OLL cases in a grid with corner orientation on one axis and edge orientation on another axis the 7 OCLL cases are a straight copy of an entire row / column.

What I mean by a "loose" subset is where you pick one alg from a number of cases to achieve the desired effect. An example of this might be for the basic EOLL as the first step of a 4LLL. For 2 handed solving you might pick something like F (U R U' R') F' whereas for one handed solving you might pick something like r U2 R' U' R U' r'. These are both selected from the full set of OLL algorithms but they are for different OLL cases and aren't being used for the original purpose. Any OLL case which flips edges can be used for basic EOLL as the effect on corners is unimportant. Going back to the grid (corner orientation on one axis and edge orientation on another axis) this type of subset is not made up from entire rows / columns of the parent set.

Updates:

1.0.7 - 2016-12-18 - ZZLL actually has 169 algorithms
1.0.6 - 2016-01-03 - OLLCP is actually a "loose" subset of 1LLL
1.0.5 - 2015-12-29 - Minor cosmetics
1.0.4 - 2015-12-29 - Added OCELL
1.0.3 - 2015-06-18 - Mentioned Pi and H subsets for ZZ-Blah
1.0.2 - 2015-06-18 - Added number of unique 1LLL algorithms (3,915)
1.0.1 - 2015-06-18 - Cosmetic changes
1.0.0 - 2015-06-17 - Initial release


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## 4Chan (Jun 17, 2015)

This is really nice!

However on my monitor, light blue and light grey are really similar.
Otherwise, it's great!


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## Logiqx (Jun 17, 2015)

4Chan said:


> This is really nice!
> 
> However on my monitor, light blue and light grey are really similar.
> Otherwise, it's great!



Cool. Nice to hear that given your ZB expertise. 

I'll tweak the colors tomorrow.


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## cashis (Jun 18, 2015)

1LLL instead of last layer?


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## Logiqx (Jun 18, 2015)

cashis said:


> 1LLL instead of last layer?



Funnily enough, I started off with 1LLL but I'll put it back as I think it will look better.

I'd also like to know the number of algs for 1LLL without treating mirrors and inverses the same.

1) LL cases = 62,208
2) ... treating rotations as the same = 15,552
3) ... treating rotational symmetries as the same = ?,???
4) ... treating reflections and inverses the same = 1,212 (Bernard Helmstetter)

What I really want is the correct figure for line 3. I can calculate an upper bound in various ways:

Unique PLL cases * Distinct OLL cases
22 * 216 = 4,752

Unique OLL cases * Distinct PLL cases
58 * 72 = 4,176

Unique OLLCP cases * Distinct EP cases
332 * 12 = 3,984

Unique ZBLL cases * Distinct EO cases
494 * 8 = 3,952

The upper bound is therefore is 3,952 unique cases / 3,951 algs but the actual number will be lower.

I might try to figure it out later...


Edit: I make it 3,916 cases / 3,915 algs - see message 8


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## pdilla (Jun 18, 2015)

Useful for when trying to explain substeps to friends.


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## Logiqx (Jun 18, 2015)

Logiqx said:


> The upper bound is therefore is 3,952 unique cases / 3,951 algs but the actual number will be lower.
> 
> I might try to figure it out later...



I calculate 3,916 unique 1LLL cases (treating mirrors and inverses as different) and therefore 3,915 algorithms.

I just considered how OLL cases can be overlaid on PLL cases - (216 * 16) + (114 * 2) + (58 * 4) = 3456 + 228 + 232 = 3916

I've updated the diagram accordingly.


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## Daniel Lin (Dec 28, 2015)

OCELL is a subset of ZBLL. It orients corners and permutes edges, while preserving edge orientation and possibly ruining corner permutation. Anything that preserves EO is a ZBLL algorithm. 
And BLL is not a specific set of algs because it can be done in different ways depending on the number of looks. For the 2look version it is the same thing as LLEF/L4C. 

On the OLLCP side, you can add SuneOLL, which Kirjava posted about here https://www.speedsolving.com/forum/showthread.php?23222-SuneOLL 

Also, I don't get what the difference is between a pure subset and a loose subset. Can you please explain it?


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## Kudz (Dec 28, 2015)

I might be wrong, but I think You can add things like Tripod, Line, Reverse Line etc
If I am not wrong those are subsets of 1LLL like ZBLL so they should be included, but as I said I might be wrong.


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## Logiqx (Dec 29, 2015)

Daniel Lin said:


> OCELL is a subset of ZBLL. It orients corners and permutes edges, while preserving edge orientation and possibly ruining corner permutation. Anything that preserves EO is a ZBLL algorithm.
> And BLL is not a specific set of algs because it can be done in different ways depending on the number of looks. For the 2look version it is the same thing as LLEF/L4C.
> 
> On the OLLCP side, you can add SuneOLL, which Kirjava posted about here https://www.speedsolving.com/forum/showthread.php?23222-SuneOLL
> ...



Thanks. OCELL was an oversight and has now been added. I've also added a comment about SuneOLL which essentially relates to execution as opposed to actual cases.

What I mean by a "pure" subset is where the exact cases can be found in the parent set; recognition and execution. The typical OCLL (orienting corners, retaining edge orientation) consists of 7 cases from OLL and is what I call a "pure" subset. If you arrange the 57 OLL cases in a grid with corner orientation on one axis and edge orientation on another axis the 7 OCLL cases are a straight copy of an entire row / column.

What I mean by a "loose" subset is where you pick one alg from a number of cases to achieve the desired effect. An example of this might be for the basic EOLL as the first step of a 4LLL. For 2 handed solving you might pick something like F (U R U' R') F' whereas for one handed solving you might pick something like r U2 R' U' R U' r'. These are both selected from the full set of OLL algorithms but they are for different OLL cases and aren't being used for the original purpose. Any OLL case which flips edges can be used for basic EOLL as the effect on corners is unimportant. Going back to the grid (corner orientation on one axis and edge orientation on another axis) this type of subset is not made up from entire rows / columns of the parent set.



Kudz said:


> I might be wrong, but I think You can add things like Tripod, Line, Reverse Line etc
> If I am not wrong those are subsets of 1LLL like ZBLL so they should be included, but as I said I might be wrong.



The diagram only shows subsets of OLLCP and ZBLL since there are so many of them that inter-relate. There are obviously a number of unrelated LL subsets which aren't shown since this is just about OLLCP and ZBLL.


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## Daniel Lin (Jan 2, 2016)

According to your definition of pure and loose subsets, i think OLLCP should be considered a loose subset of 1LLL rather than a pure one because it randomly affects EP. Only about 1/12 of the algs preserve EP.


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## Logiqx (Jan 3, 2016)

Daniel Lin said:


> According to your definition of pure and loose subsets, i think OLLCP should be considered a loose subset of 1LLL rather than a pure one because it randomly affects EP. Only about 1/12 of the algs preserve EP.



Thanks, fixed.


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## OLLiver (Jan 3, 2016)

Daniel Lin said:


> According to your definition of pure and loose subsets, i think OLLCP should be considered a loose subset of 1LLL rather than a pure one because it randomly affects EP. Only about 1/12 of the algs preserve EP.




wait.......you do realise that OLLCP doesn't randomly preserve EP. OLLCP is to 1LLL as COLL is to ZBLL. OLLCP+EPLL recog =1LLL. 
OLLCP doesn't solve all the edge cycles, but it definitely doesn't randomise it. There are only 12(?) Edge cycles so you learn one of those algs as your OLLCP, if you get good you will recognise it as a 1LLL case. i.e. when it will completely solve the cube (EPLL skip). OLLCP is not 1LLL by any standards.


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