Difference between OLL and PLL parity on 4x4

some1rational

Member
Hey everyone, I'm new and I'm not sure where this question should go, but I'm gambling it should go in the puzzle theory section.

I've recently solved a 4x4 via the common method of reducing it to a 3x3 by making the centers and pairing up the edges. However I was unable to fix the parities for some time without referring blindly to some algorithms found online until I read the 'understanding parity' thread about even and odd numbers of inner layer quarter turns. So far I've only figured out how to fix the PLL parity where there are 2 swapped corners, but I'm still unable to do the OLL error where I have one 'dedge' flipped (without blindly following some algorithm). My question is, as far as I'm aware, are not both errors caused by having performed an odd numer of inner layer quarter turns? Because the method I have for fixing my PLL error is obviously not working for the OLL error. So then what exactly is different between the OLL and PLL parity cases? Thx!

miniGOINGS

Member
"PLL Parity" is actually two 2-cycles of edges (similar to the H or Z Perms).

"OLL Parity" is a single 2-cycle (swap) of edges. Doing a single inner layer turn cycles the 4 edges in that layer. The 2-cycle of the parity and the 4-cycle from an odd number of inner layer turns cancel out the parity.

I hope that made some sense...

vcuber13

Member
My question is, as far as I'm aware, are not both errors caused by having performed an odd numer of inner layer quarter turns? Because the method I have for fixing my PLL error is obviously not working for the OLL error. So then what exactly is different between the OLL and PLL parity cases? Thx!
OLL is that but PLL is if the centers are build on the wrong axis, similar to the void cube parity.

miniGOINGS

Member
PLL is when either 2 edge pairs, or 2 corners are swapped
OLL Is when and edge group is flipped "upside down"
They know that. Read their post.

some1rational

Member
thx miniGOINGS, I'm still a little confused but I think I just need to work on it a little more (I forgot to mention that I DID solve the degde parity once on my own, but I've forgotten exactly what I did (i.e. what that was the magical 'money' move that I performed lol)). Your comment about cycles though gives me some hope to recall exactly what I did that time haha.

trying-to-speedcube...

Member
Well, you could just do r, then you intuitively solve the centers again using only r U2 r' type moves. Then you pair up the edges again.

Cancelling that out gives r U2 r U2 r U2 r U2 r

some1rational

Member
YAY! thx alot guys, I finally worked it out! albiet I'm not really counting my cycles or moves or anything, but know I know a method that is intuitive for me (though it scrambles up my 3x3 solve but leaves edges adn centers intact...yea inefficient, but I don't mind, my goal was to be able to do it without rote memorization haha)

vcuber13 - your comment about centers being on wrong axis really did the trick

trying-to-speedcube - I may have to try what you're saying sometime, could probably work out a more efficient method that way then the one i currently have

EDIT: actually lol, I just realized what I did was exactly wat you said trying-to-speedsolve

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Kirjava

Colourful
OLL is that but PLL is if the centers are build on the wrong axis, similar to the void cube parity.

Something tells me that this isn't strictly true. (Thinking in a non-reduced context)

Kirjava

Colourful
Code:
01:25:56 <+Ethan_Rosen> Kirjava, I believe he's right
01:26:05 <+Kirjava> yeah, it sounds right
01:26:16 <+Kirjava> but it doesn't sound like a neat enough explanation for it
01:26:34 <+Kirjava> you could say that it's caused by incomplete reduction
01:26:45 <+Kirjava> improper pairing, etc
01:27:16 <+Kirjava> I want to know the actual reason, like how oll parity is caused by a quarter inner turn
01:27:22 <+Kirjava> lgarron: you following me?
01:27:41 <%lgarron> 4-cycle.
01:27:44 <%lgarron> That's all.
01:27:57 <+Kirjava> so it's unrelated to void parity?
01:28:13 <%lgarron> Yes.

Lucas Garron

Moderator
Staff member
Code:
01:28:24 <%lgarron> Except that they're both parities.
Which can both be caused by 4-cycles. But that's the nature of parities on cubes.

Member
PLL is if the centers are build on the wrong axis
Could someone explain what this means. I understand the cause of OLL parity (and there are loads of good threads on here about it). But I don't understand the root cause of PLL parity on a 4x4.

cmhardw

Could someone explain what this means. I understand the cause of OLL parity (and there are loads of good threads on here about it). But I don't understand the root cause of PLL parity on a 4x4.
PLL parity has absolutely nothing to do with how you built the centers.

If PLL parity happens, it is created when you finish pairing up the edges. After pairing up the edge groups we will use the center groups to define the "solved" orientation of the cube. Once you pair all the edge groups they will be in some permutation, and this permutation will have either even or odd permutation parity. The corners will also be in some permutation, and they will have either even or odd permutation parity.

PLL parity happens when the permutation parity of edges is even and the permutation parity of corners is odd, or when the permutation parity of edges is odd and the permutation parity corners is even.

PLL parity does not happen if the permutation parity of edges is odd and the permutation parity of corners is odd, or when the permutation parity of edges is even and the permutation parity of corners is even.

For example, let's say you are down to two edge groups to pair, and they are unsolved. You have one group at FL and one group at FR. At uFL you have yellow/red and at uFR you have the other yellow/red. At dFL you have yellow/orange and at dFR you have the other yellow/orange. Based on how you pair these last two edge groups you can either end up with a yellow/orange edge group at FL and a yellow/red edge group at FR; or you could end up with a yellow/red edge group at FL and a yellow/orange edge group at FR. This effectively swaps two edge groups, which changes the parity of the edge permutation. One of the above cases for pairing edges will create even edge group permutation parity, the other odd permutation parity.

When you complete the last edges, if you create an edge group permutation parity that is different from the permutation parity of the corners, then you have created PLL parity. If you create an edge group permutation parity that matches the permutation parity of the corners, then you will not have PLL parity.

Member
What a great explanation! As someone commented in the probability thread, you have a professor's knack for distilling and demystifying complicated puzzle theory into simple understandable chunks. Thanks for taking the time. I have a couple questions about this:

PLL parity happens when the permutation parity of edges is even and the permutation parity of corners is odd, or when the permutation parity of edges is odd and the permutation parity corners is even.
Conceptually, I get that the corners and dedges, independently, will have odd or even parity. For OLL parity, I know that the parity state is determined by whether the total number (scramble + solve) of inner-layer turns (QTM) is odd or even. (1) Is that the same measure you mean when you say "the permutation parity of edges"? (2) What is the measure of the parity state of the corners -- the total number of outer-layer turns (QTM)?

MaikeruKonare

Member
That really was a great explanation. I know 4x4 parity well And I know plenty of algorithms for it, I just wanted to comment on how vivid your description was.

elrog

Member
OLL is that but PLL is if the centers are build on the wrong axis, similar to the void cube parity.
I hadn't though about it that way. PLL parity is just 2 3-cycles of edges anyway though. So it can occur as void cube parity or the even/odd thing. Oh wait, void cube parity has to do with an even/odd number of slice moves.

vcuber13

Member
what i said earlier in the thread is wrong, for 4x4 pll parity isnt caused by how the centres are built.

PLL parity has absolutely nothing to do with how you built the centers.
But, on a 5x5, if the centre stickers (the fixed centres, not the 3x3 centres) are removed, wouldn't it be possible to have PLL parity?

Renslay

Member
But, on a 5x5, if the centre stickers (the fixed centres, not the 3x3 centres) are removed, wouldn't it be possible to have PLL parity?
Then you can have a void cube parity. (PLL parity can't, because of the middle pieces of the edges.)