# Understanding Parity.

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#### andyt1992

##### Guest
I often hear people say if you can understand why parity happens you can work out how to solve it, I am not just talking about 4x4's here but about any cube or non cube puzzle that can get parity.
The puzzles I am most interested in understanding why parity happens are:

SQ-1
Fisher Cube
4x4 and upwards
Void Cube - Shifted Centers?

The reason is I dont want to have to remember a load of algorithms and solve cubes in robot mode, I wish to have a greater understanding of the logic of each puzzle and be able to solve each intuitively.

##### Member
I'd like to know that too.

#### cmhardw

First hint: Parity is the same concept on all puzzles. It refers to the parity of the permutation of the pieces. If the overall piece permutation parity is odd, then you have "parity error." If the overall piece permutation is even you do not have "parity error." Also, "parity error" is a very poor choice of wording, as the real concept behind what we term parity is really referring to odd parity permutations.

Search for parity on this forum for some other topics with much more detailed explanations. Consider this only a nudge in the right direction.

Chris

#### masterofthebass

Fisher Cube doesn't have parity, it only has rotated centers like a picture cube would have.

The case of a void cube parity is quite simple. If you were to take a 3x3, and solve your f2l with the Eslice centers cycled clockwise (red cross piece on the green center, green piece on the orange center, etc.) you would get the same scenario as you do on a void cube. This happens on the void cube quite easily, obviously, because you don't have the centers there to reference off of.

I personally still don't get how sq-1 parity works, and I'll let you search around for the bigcube edge parity.

#### Kirjava

##### Colourful
First hint: Parity is the same concept on all puzzles. It refers to the parity of the permutation of the pieces. If the overall piece permutation parity is odd, then you have "parity error." If the overall piece permutation is even you do not have "parity error." Also, "parity error" is a very poor choice of wording, as the real concept behind what we term parity is really referring to odd parity permutations.

Search for parity on this forum for some other topics with much more detailed explanations. Consider this only a nudge in the right direction.

Chris

Either cubers don't use this exact definition or "PLL Parity" is really badly named.

#### vcuber13

##### Member
PLL parity basically means a 2 cycle, so you switch only 2 corners or 2 edges (sq-1, even cubes 4x4+, etc.). On big cubes I think the OLL parity is caused by an odd number of twists in the inner layers like Rw, and the parity alg does another odd with makes the total even. I don't know but I think sq-1 parity happens while you get it into a cube.

#### Kirjava

##### Colourful
PLL parity basically means a 2 cycle, so you switch only 2 corners or 2 edges

PLL parity is not a single 2 cycle, rather two sets of 2 cycles. In Chris' definition, this is not parity.

#### vcuber13

##### Member
PLL parity basically means a 2 cycle, so you switch only 2 corners or 2 edges

PLL parity is not a single 2 cycle, rather two sets of 2 cycles. In Chris' definition, this is not parity.
Well, I meant i can be thought of like a 2 cycle. (or 4 cycle (O-Perm) and so fourth)

And, two 2 cycles isn't parity, a Z and H Perm arn't

#### Kirjava

##### Colourful
PLL parity basically means a 2 cycle, so you switch only 2 corners or 2 edges

PLL parity is not a single 2 cycle, rather two sets of 2 cycles. In Chris' definition, this is not parity.
Well, I meant i can be thought of like a 2 cycle. (or 4 cycle (O-Perm) and so fourth)

I'm specifically addressing Chris' definition here; you're forgetting that an edge pair is two pieces.

However, PLL parity is defintely not a 4 cycle.

And, two 2 cycles isn't parity, a Z and H Perm arn't

Nice edit. You do know that I stated exactly this in the post you replied to, right?

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#### Stefan

##### Member
Fisher Cube doesn't have parity, it only has rotated centers like a picture cube would have.
So... center orientation parity?

The case of a void cube parity is quite simple. If you were to take a 3x3, and solve your f2l with the Eslice centers cycled clockwise
More direct perspective, not talking about something that isn't there:
Turning an inner slice is a 4-cycle of edges, that's an odd permutation right there.

I personally still don't get how sq-1 parity works
Not even the setup-to-turn-six-corners solution?

Search for parity on this forum for some other topics with much more detailed explanations.
These are also good:
http://www.jaapsch.net/puzzles/theory.htm
http://en.wikipedia.org/wiki/Parity_of_a_permutation

PLL parity is not a single 2 cycle
It *is* from the perspective of solving as a 3x3x3, where it's a 2-cycle of dedges.

#### Kirjava

##### Colourful
PLL parity is not a single 2 cycle
It *is* from the perspective of solving as a 3x3x3, where it's a 2-cycle of dedges.

I know it is if you reduce the puzzle into another group, but Chris' definition doesn't allow that abstraction, referring specifically to pieces.

#### Kirjava

##### Colourful
Nice edit. You do know that I stated exactly this in the post you replied to, right?
No you said it
PLL parity is not a single 2 cycle, rather two sets of 2 cycles. In Chris' definition, this is not parity.

I don't know what this is supposed to mean, but here's what I meant that you didn't understand;

PLL parity is not a single 2 cycle, rather two sets of 2 cycles. In Chris' definition, this is not parity.

I'm saying PLL Parity is really two sets of 2 cycles. Which is not parity. (I thought this was obvious)

And, two 2 cycles isn't parity, a Z and H Perm arn't

And then you seem to tell me this as if I haven't already stated it.

Now, tell me how you think PLL Parity is a 4 cycle please.

#### vcuber13

##### Member
Nice edit. You do know that I stated exactly this in the post you replied to, right?
No you said it
PLL parity is not a single 2 cycle, rather two sets of 2 cycles. In Chris' definition, this is not parity.

I don't know what this is supposed to mean, but here's what I meant that you didn't understand;

PLL parity is not a single 2 cycle, rather two sets of 2 cycles. In Chris' definition, this is not parity.

I'm saying PLL Parity is really two sets of 2 cycles. Which is not parity. (I thought this was obvious)

And, two 2 cycles isn't parity, a Z and H Perm arn't

And then you seem to tell me this as if I haven't already stated it.

Now, tell me how you think PLL Parity is a 4 cycle please.
if your saying pll parity is really two 2 cycles, then what do you call it when 2 adjacent / opposite edges are switched on a 4x4?

#### Kirjava

##### Colourful
Chris' definition doesn't allow that abstraction, referring specifically to pieces.
I have no problem considering a dedge to be a piece.

I do, my problem is that it's two pieces.

if your saying pll parity is really two 2 cycles, then what do you call it when 2 adjacent / opposite edges are switched on a 4x4?

two 2 cycles (assuming you meant dedges and not edges, if you did mean edges (which I seriously doubt) I would simply call it a 2 cycle)

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#### Kirjava

##### Colourful
Chris' definition doesn't allow that abstraction, referring specifically to pieces.
I have no problem considering a dedge to be a piece.
I do, my problem is that it's two pieces.
You're wrong, it's at least *four* pieces, don't forget the stickers!

Come on, you know what I mean. You do however make a valid point. It seems silly to me to think of it as a single piece when they weren't always a single one in the solve, and can return to being two.

It just irks me that it's universally referred to as "PLL Parity" when it's a matter of perspective. For example, vcuber doesn't even understand what I'm talking about.

#### Stefan

##### Member
it's universally referred to as "PLL Parity"
Is it? Do *you* refer to it as that? Or is it just universally referred to that in the pseudo-3x3x3 context? There I think it's alright.

You do however make a valid point.
Yeah, I was hoping to . What one considers a piece depends on the purpose and usage.

It seems silly to me to think of it as a single piece when they weren't always a single one in the solve, and can return to being two.
Good point, though if you insist that it shouldn't be called piece, you're just as stubborn as people who insist it should be. Like you said, matter of perspective. And neither is wrong except for calling the other wrong.