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Relationship of OLL Parity + PLL Parity + L2E

Logiqx

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Introduction

This post gives a simple overview of the different parities that can occur during a reduction solve on 4x4x4 upwards.

Even layered cubes (e.g. 4x4x4 and 6x6x6) have the concept of OLL parity and PLL parity whereas odd layered cubes (e.g. 5x5x5 and 7x7x7) only have the concept of Last Two Edge (L2E) cases.

It may not be immediately apparent how these cases are related so I thought I would try to summarise in simple terms.

OLL Parity

Pure algorithm: Rw U2 x Rw U2 Rw R' U2 x' Lw' L U2 Lw L' U2 Rw' R U2 Rw R' U2 Rw' U2 Rw'

upload_2017-4-5_13-22-21.png

The image above shows OLL parity on 4x4x4 and its equivalent on 5x5x5. It is clearly evident during L2E on 5x5x5 but it isn’t easily spotted on 4x4x4 until the OLL stage of a reduction + CFOP solve.

It occurs at the moment you complete your centres during reduction and once the centres have been completed the OLL parity is fixed, until such time as you execute an algorithm which affects OLL parity.

Mathematically speaking it occurs whenever the total number of inner slice turns (QTM) applied whilst scrambling and solving the centres is an odd number. If you were to track all of the t-centre pieces whilst executing a parity algorithm on 5x5x5 you would see that the number of pair swaps is an odd number.

Note: You won’t encounter this parity during a reduction solve on an odd layered super-cube (e.g. 5x5x5 picture cube) because you will solve all of the t-centres correctly, prior to edge pairing.

Intuitive solution for OLL parity...

I mentioned above that OLL parity is caused by an odd number of quarter slice turns.

The intuitive solution is simple. Just do a quarter slice turn (e.g. Rw) and re-solve the centres:

e.g. Rw U2 Rw U2 Rw U2 Rw U2 Rw

Notice how the example above uses five Rw turns and therefore switches the OLL parity.

Whilst this will fix the OLL parity on 4x4x4, you'll have to resolve the L4E and F2L.

PLL Parity

Pure algorithm: Rw2 F2 U2 Rw2 R2 U2 F2 Rw2

upload_2017-4-5_13-22-39.png

The image above shows PLL parity on 4x4x4 and its equivalent on 5x5x5. It is clearly evident during L2E on 5x5x5 but it isn’t easily spotted on 4x4x4 until the PLL stage of a reduction + CFOP solve.

It occurs at the moment you complete your edge pairing during reduction on a 4x4x4 and once the edges have been paired the PLL parity is fixed, until such time as you execute an algorithm which affects PLL parity.

The reason that PLL parity doesn’t appear to exist on 5x5x5 is because it presents itself as just another L2E case. It is therefore solved during edge pairing, prior to the 3x3x3 stage.

PLL parity is somewhat simpler than OLL parity and some people do not even consider it to be a parity. It essentially boils down to whether you think of paired wings as a "composite edge" or remaining as individual wings. The arguments are as follows:
  • If you restrict yourself to outer layer turns (with solved centres and paired edges) then "PLL parity" is a violation of the regular permutation parity, just like "OLL parity" is a violation of edge orientation parity.
  • You can solve "PLL parity" using an even number of inner slice turns, without moving centres. Unlike "OLL parity", "PLL parity" is not a parity when you look at the permutation of the 24 wings.
The first argument seems most appropriate for a reduction solve imho.

Intuitive solution for PLL parity...

The images above clearly show which pieces need to be exchanged and this can be achieved using the intuitive slice-flip-slice approach.

e.g. Rw' (U' R U R') (F R' F') Rw + Lw (U' R U R') (F R' F') Lw' which affects UF and UB

Whilst this will fix the PLL parity on 4x4x4, you'll have to resolve the F2L and OLL.

L2E Cases

There are 16 distinct cases in total and for illustrative purposes, I have derived them from the following three cases:

upload_2017-4-5_13-25-8.png

I'll refer to the images above as “solved”, “diagonal wing swap” and “adjacent wing swap” but the adjacent swap is essentially the diagonal swap after a tredge flip.

The OLL and PLL parity algorithms were repeatedly applied to these three cases and the resultant cases to produce the full set of L2E cases; 16 in total, including solved.

upload_2017-4-5_13-29-51.png

I have created a web page listing all 16 cases and decent algorithms - http://cubing.mikeg.me.uk/algs/l2e.html

You may find it insightful to execute the algorithms for A2-A6 on a 4x4x4 and compare them to a 5x5x5.

Half of the L2E cases do not have any rotational symmetry and have a probability of 1/12 whereas the other half have rotational symmetry and thus a probability of 1/24.


I hope this summary is useful / interesting to some people.
 
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mark49152

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Nice guide, thanks Mike.

I'm not sure I understand the difference between adjacent and diagonal wing swaps though. Aren't they effectively the same thing, which is that a single pair of wings needs to be swapped? One can be reduced to the other with a tredge flip.

I think all 16 L2E cases reduce to two simple cases, depending on whether an odd or even number of wing swaps needs to happen. Simple case meaning 1 or 0 swaps respectively.
 
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Logiqx

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I'm not sure I understand the difference between adjacent and diagonal wing swaps though. Aren't they effectively the same thing, which is that a single pair of wings needs to be swapped? One can be reduced to the other with a tredge flip.
You're quite correct. They are essentially the same thing after a tredge flip.

The reason I separated them out is because they create 3 distinct groups of cases when you only apply OLL / PLL parity.

I've documented the 3 groups and how their cases cycle from one to another on my web page:

http://cubing.mikeg.me.uk/algs/l2e.html

Edit: If you change the view from "groups" to "parity" you'll see the two groups that you refer to in your post - even / odd parity.
 
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mark49152

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The reason I separated them out is because they create 3 distinct groups of cases when you only apply OLL / PLL parity.
I guess it depends how you look at it. If you imagine a solved cube with any two wings swapped, it's clear that you could easily set those up to be diagonal, adjacent or opposite. For example, OLL parity A3 can be set up to a B2 diagonal swap with a simple Rw.

On the other hand, you'll never solve that swap without a parity alg, by definition:).
 

Logiqx

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I guess it depends how you look at it. If you imagine a solved cube with any two wings swapped, it's clear that you could easily set those up to be diagonal, adjacent or opposite. For example, OLL parity A3 can be set up to a B2 diagonal swap with a simple Rw.

On the other hand, you'll never solve that swap without a parity alg, by definition:).
Yes, there is more than one way to skin a cat. It's also possible to derive all 16 cases using basic edge flips and the slice-flip-slice approach (midges or wings).

I simply chose to use the OLL + PLL parity algorithms because the group bindings are relatively interesting. ;)
 

mark49152

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It's also possible to derive all 16 cases using basic edge flips and the slice-flip-slice approach (midges or wings).
You can convert between any parity cases using flipping/slicing, and between any non-parity cases. But you can't convert parity to non-parity or vice versa because flipping/slicing does only even numbers of wing swaps. Which is why I think of them as separate groups.

I use AvG edge pairing, so flipping/slicing usually reduces parity to a single case, your C1, or occasionally I'm left with A3. Which is nice because only two algs :).

Thinking about 5BLD, it's interesting that wing parity is equivalent to OLL parity (single swap) and midge parity effectively creates PLL parity in the wings (double swap). I never thought of it that way before.
 

Logiqx

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I use AvG edge pairing, so flipping/slicing usually reduces parity to a single case, your C1, or occasionally I'm left with A3. Which is nice because only two algs :).
That's nice! I don't use AvG edge pairing but I thought that might be the case.

Thinking about 5BLD, it's interesting that wing parity is equivalent to OLL parity (single swap) and midge parity effectively creates PLL parity in the wings (double swap). I never thought of it that way before.
Cool. I wondered whether it would complement your existing knowledge of 4BLD + 5BLD. :)
 

Lucas Garron

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Nice description!
I have two comments for things to clean up:

Mathematically speaking it occurs whenever the total number of quarter slice turns applied whilst scrambling and solving the centres is an odd number.
Nit: this only applies to inner slices.

PLL parity is far simpler than OLL parity and some people would argue that it isn’t really “parity” in the same sense as OLL parity.
I would just state the perspectives directly:
  • If you restrict yourself to outer block turns (with solved centers and paired edges), this is a violation of the regular permutation parity, just like "OLL parity" is a violation of edge orientation parity.
  • You can solve PLL parity using an even number of inner slice moves, without moving centers. That is, unlike OLL parity, "PLL parity" is not a parity when you look at the permutation of the 24 wings.
 
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