Parity?

[CubeThing]

I know this is a really noobish question but what is parity?

 

[byu]

Parity is when you have an odd number of swaps on the cube. It occurs on 4x4 and 2x2, Square-1, and other puzzles.

 

[Yes We Can!]

I HATE PARITIY!!
I just solved the 4x4x4 four times and got 4 parity cases!! -.-

 

[cpt.Justice]

I HATE PARITIY!!
I just solved the 4x4x4 four times and got 4 parity cases!! -.-

That's completely average, if you think of both OLL parity and PLL "parity" as parity cases. Both have a 50% probability of occuring, so you will on average get one of them per solve.

 

[Yes We Can!]

I HATE PARITIY!!
I just solved the 4x4x4 four times and got 4 parity cases!! -.-

That's completely average, if you think of both OLL parity and PLL "parity" as parity cases. Both have a 50% probability of occuring, so you will on average get one of them per solve.

I know, but I only counted OLL parity, PLL parity was 2 or 3 times...

 

[moogra]

Parity is when you have an odd number of swaps on the cube. It occurs on 4x4 and 2x2, Square-1, and other puzzles.

2x2? I don't see how that happens. Maybe it's just that I don't know any method at all for 2x2.

 

[mcciff2112]

Parity is something that can happen on a larger (or smaller (2x2 is possible)) cube, but is impossible to happen on a regular 3x3.

 

[Ellis]

but is impossible to happen on a regular 3x3.

I get parity on 3x3 all the time

 

[royzabeast]

After reading all of these, i still dont get what it means :/

Can somebody explain it more thoroughly?
I'm not the dude that posted this btw, but i was going to pose the same question

 

[mystera]

there's parity on 5x5 too...like the center edge is flipped

 

[Johannes91]

After reading all of these, i still dont get what it means :/

Don't worry, neither do the people who posted those "definitions".

http://www.geocities.com/jaapsch/puzzles/theory.htm
http://www.ryanheise.com/cube/parity.html
http://en.wikipedia.org/wiki/Parity_of_a_permutation

 

[blgentry]

Even the web page definitions above (the first one doesn't work) don't really say how parity presents itself. I'm sure the definitions are technically correct, but I'd bet most people reading them rolled their eyes and closed the page.

I'm not sure about the exact definition or why that particular word is used to describe it, but I can tell you how parity appears. This gonna take a couple of paragraphs, but stick with me here.

In solving a 4x4x4 or a 5x5x5 (or bigger) one way of doing it is called "reduction". Basically, you solve the centers, then pair up the edge pieces into groups. By doing this, you reduce the problem back down to solving a 3x3x3: The centers are the same as the center caps on a 3x3x3, the edge groups are the same as the edge pieces on a 3x3x3, and the corners are just the corners. It rather elegant and brilliant actually.

Parity occurs (randomly) in several of these steps and always shows up as a condition that would be impossible on a 3x3x3 (without taking it apart or physically altering it). The first place it can appear is when you are grouping the edge pieces. Sometimes, you'll come to a scenario where you can't (intuitively) group the last remaining edge pieces. That would be the equivalent of having half of an edge piece swapped with half of another edge piece on a 3x3x3. An impossible condition. There are algorithms to solve those parity conditions.

Next, it can appear during OLL. You can end up with three edge groups flipped properly and one *not* flipped properly. That condition is impossible on a 3x3x3 also. There are algs to solve this type of parity too.

Finally, there is PLL parity which also manifests as a condition that would be impossible on a 3x3x3. ...and there are algs to solve that too.

Perhaps a picture is worth a thousand words here. This page shows both OLL and PLL parity on a 4x4x4 and the algs to solve them:

http://www.bigcubes.com/4x4x4/finalsolve.html

Hope that helps.

Brian.

 

[Johannes91]

Jaap's site worked when I posted that, and will probably be back up sooner or later. For now, just use archive.org.

You explained "parity" as many cubers use the word. This has been discussed here before (see e.g. this thread). If somebody wants to understand the mathematical meaning, just search, there have been some good explanations IIRC. Although I think the sites I linked to are very readable, too.

 

[fanwuq]

From what I understand, everything has parity at any state. The parity is either odd or even. The proper name of the algs are "parity fix." The parity fix is just a fancy conjugate of a quarter turn.
http://speedcubing.com/chris/4speedsolve3.html

I have questions about how to avoid fixing the parity at the end of the solve for 444 FMC. I think I've read somewhere on this forum that the number of qtm moves used to scramble the cube must = the number of qtm moves used to solve centers. Is that true? If so, what is defined at 1 move?
Is "r" 1 move or "Rw" 1 move? Is that only for OP?
PLL parity occurs during edge pairing, I think, but how? and how to prevent it?

 

[Johannes91]

I think I've read somewhere on this forum that the number of qtm moves used to scramble the cube must = the number of qtm moves used to solve centers. Is that true?

I'm not sure I understand what that's supposed to mean. Probably false.

Every quarter turn of an inner slice change the parity of edges on 4x4x4, because it's one 4-cycle. A quarter turn of an outer slice makes two 4-cycles, which doesn't change the parity. So only inner slice turns matter.

If the cube was scrambled with an even number of them (counting in QTM), but you use an odd number when reducing to 3x3x3, you'll get "OLL parity", i.e., odd parity on edges. If it was scrambled with odd number of turns and you use an even number, "OLL parity" again.

I don't consider "PLL parity" a parity because I see it as two 2-cycles. But yes, it can change during edge pairing, using just basic move sequences like u R U R' u'.

 

[fanwuq]

Ok, thanks! So I should count the number of inner slice quarter turns. That makes sense.

For PLL parity:
If u R U R' u' changes it, what is something that solves the same case that doesn't change it?

 

[byu]

Parity is when you have an odd number of swaps on the cube. It occurs on 4x4 and 2x2, Square-1, and other puzzles.

2x2? I don't see how that happens. Maybe it's just that I don't know any method at all for 2x2.

It happens. Do a T-Perm on a 2x2, and don't make any AUF afterwards, and you'll see what I mean.

but is impossible to happen on a regular 3x3.

I get parity on 3x3 all the time

So do I, while doing BLD.

there's parity on 5x5 too...like the center edge is flipped

The center edge is never flipped. It's the two wing edges around it that's flipped.

 

[Johannes91]

For PLL parity:
If u R U R' u' changes it

Sorry, I was unclear, I didn't mean that that alone changes it. For example, this changes it: (d R F' U R' F d') (u' R F' U R' F u). Similar idea can be done with variations of that 5-mover.

I don't know a good way to define "PLL parity" when all dedges aren't connected. It's only clear when looking at it as a 3x3x3.

there's parity on 5x5 too...like the center edge is flipped

The center edge is never flipped. It's the two wing edges around it that's flipped.

... and they're swapped, not flipped.

 

[fanwuq]

Oh. Thanks! So if you end with 2 edges that need to be swapped for the last pairing, beginning the pairing with u rather than d changes the PLL parity? And it works the same way when you end the pairing with a 3 cycle of edges?

Edit:
But that's just a pure 3 cycle, so it shouldn't happen. Unless this is what you mean by it being not a true parity? The PLL parity alg looks like it's pretty much the H perm on the 3x3. H can be solved with 2 U perms, so I guess cycling the other set of the last 3 edges can work? No, that wouldn't because it would be equivalent to just doing a U perm on the 3x3 stage. I don't know now.

 

[Tetris Cube]

I don't consider "PLL parity" a parity because I see it as two 2-cycles. But yes, it can change during edge pairing, using just basic move sequences like u R U R' u'.

PLL parity is relative to the 3x3x3 stage, isn't it? It's not an odd number of piece swaps total on the 4x4+ cube, that would be imposible to solve, I think.