Define "Parity"

[JLarsen]

It's not possible, on a 4x4, to have a permuted correctly but not oriented correctly, right? So isn't the OLL parity 2 edges swapped?

Assuming that you mean an edge piece, no, it is physically impossible (as in, the piece won't fit upside down)

Yeah one edge piece is a no. Even if it would physically fit in there upside down.

 

[beingforitself]

Therefore it's actually not "parity" but "pairity" which is derived from "pair" and "pity".

meh, i lol'd

 

[Lord Voldemort]

It's not possible, on a 4x4, to have a permuted correctly but not oriented correctly, right? So isn't the OLL parity 2 edges swapped?

Assuming that you mean an edge piece, no, it is physically impossible (as in, the piece won't fit upside down)

That's what I'm saying though.
They're swapped and flipped, but that still means only 2 edges are switched.

 

[fanwuq]

It's not possible, on a 4x4, to have a permuted correctly but not oriented correctly, right? So isn't the OLL parity 2 edges swapped?

Assuming that you mean an edge piece, no, it is physically impossible (as in, the piece won't fit upside down)

That's what I'm saying though.
They're swapped and flipped, but that still means only 2 edges are switched.

Really? http://speedcubing.com/chris/4speedsolve3.html

 

[brunson]

So the consensus seems to be, there is a rigorous definition of parity, but is mostly tossed around incorrectly on the forums.

 

[lowonthefoodchain]

Parity: Bombs your standard deviation for big cubes.

 

[JLarsen]

Another reason even numbered cubes are less pleasant than odd numbered ones.

 

[qqwref]

There is indeed a mathematical definition for parity, but in cubing we actually use "parity" as a shorthand for "parity problem", that is, something which is the wrong parity and thus cannot be solved using normal moves but which must instead be dealt with by a special algorithm. Hence we don't talk about even parity or odd parity, just parity or no parity (and I prefer it this way; it makes much more sense for speedsolving). Of course the standard for "normal moves" changes from puzzle to puzzle... for 3BLD a normal move is a commutator, for the last step of reduction a normal move is an outer layer move, for Square-1 a normal move is any short sequence which stays in cubeshape (or very close to it). So for all of these things, what we call parity is actually slightly different, but the concept is really the same.

The only type of parity problem I can think of that this doesn't work for is corner orientation parity, which nobody seems to have brought up (but if you did and I missed you I apologize). Instead of traditional parity where everything has to add up to 0 mod 2, in corner orientation parity everything has to add up to 0 mod 3. The concept is the same, but there's no nice way to refer to things that have to add up mod 3. Fortunately it is still fine to talk about parity problems (i.e. anything that isn't 0 mod 3), and the cubing definition of "parity" still makes sense

 

[Stefan]

Fortunately it is still fine to talk about parity problems (i.e. anything that isn't 0 mod 3)

It can certainly be considered a "problem", but I have doubts about "parity problem". Cause parity really seems to be a two-cases thing. If I had to use a word for corner orientation, I might use invariant. But you can also just say "corner orientation is violated" or "false corner orientation" or something like that.

 

[AvGalen]

I think the simplest way to describe parity is by doing a J-Perm on a 2x2x2. You should now have 2 corners swapped.

On 3x3x3 this would be an impossible position
On 2x2x2 you can just do a U or U' and now you have a 3-cycle of corners left.

Try to think about this example for a while.



I like Stefans pair-pity
And Michaels "everything that is not considered normal" is the best description for the way most people use it.

 

[blah]

I think the simplest way to describe parity is by doing a J-Perm on a 2x2x2. You should now have 2 corners swapped.

On 3x3x3 this would be an impossible position
On 2x2x2 you can just do a U or U' and now you have a 3-cycle of corners left.

Try to think about this example for a while.

Sorry, but I really don't get the message you're trying to convey Can you make it a little clearer?

 

[mrCage]

I think the simplest way to describe parity is by doing a J-Perm on a 2x2x2. You should now have 2 corners swapped.

On 3x3x3 this would be an impossible position
On 2x2x2 you can just do a U or U' and now you have a 3-cycle of corners left.

Try to think about this example for a while.



I like Stefans pair-pity
And Michaels "everything that is not considered normal" is the best description for the way most people use it.

What would not be normal about 4x4x4 parity (parities). They happen all the time.
More interesting is to analyse how the parities can arise. First of all i do not see the 4x4x4 PLL parity as a parity at all, It is commonly called a parity because you are supposed to be in the "3x3x3 stage" when you spot it, when using reduction at least. This parity is indeed solvable by doing 2 3-cycles on edges. This is not the case with the OLL parity. This parity can occour because a 4-cycle on edges is possible by an inner layer turn. In fact the effect on centers by this inner layer turn is completely fixable also on a 4x4x4 supercube. On bigger cubes the OLL parity (or parities) are fixable (of course), but it (they) cannot occur without a centers side effect. The inner layer quarter turn centers side effects are not completely fixable.

Hope this makes sense to most of you!

Per

PS! You see more parities on larger cubes when you do direct solving than with reduction solving. The reason why should be quite obvious by now ...

 

[qqwref]

Fortunately it is still fine to talk about parity problems (i.e. anything that isn't 0 mod 3)

It can certainly be considered a "problem", but I have doubts about "parity problem". Cause parity really seems to be a two-cases thing. If I had to use a word for corner orientation, I might use invariant. But you can also just say "corner orientation is violated" or "false corner orientation" or something like that.

That would work, but it's kind of long and convoluted, isn't it? The concept of a parity problem works fine IMO for things that must be divisible by 3 or some other number, if you disregard the fact that parity means the state of something mod 2. Personally I prefer to refer to this kind of thing in our own internally consistent single-word way, rather than use a lot of words (or longer words) even if it might be more technically correct. As a special community, after all, we are allowed our own jargon

First of all i do not see the 4x4x4 PLL parity as a parity at all, It is commonly called a parity because you are supposed to be in the "3x3x3 stage" when you spot it, when using reduction at least. This parity is indeed solvable by doing 2 3-cycles on edges. This is not the case with the OLL parity.

Indeed, and this goes back to what I was saying about normal moves. If you reduce 4x4 to a 3x3, then both PLL and OLL parity are real parity cases, because they are both things that you can't solve with just outer layer turns. On the other hand, if you use some kind of centers-first/F3L-first direct solving method, PLL parity isn't a parity anymore because doing commutators is also counted as a normal type of move to do, whereas OLL parity still needs an algorithm. (If you use something like cage, where you are free to do an r-slice move until you are solving the very last edges, even OLL parity could be considered as not a parity case, because r is a normal move and you don't have to fix centers afterwards.)

 

[Stefan]

Personally I prefer to refer to this kind of thing in our own internally consistent single-word way, rather than use a lot of words (or longer words)

Huh? How is "corner orientation parity" shorter than "false corner orientation"?

 

[AvGalen]

I think the simplest way to describe parity is by doing a J-Perm on a 2x2x2. You should now have 2 corners swapped.

On 3x3x3 this would be an impossible position
On 2x2x2 you can just do a U or U' and now you have a 3-cycle of corners left.

Try to think about this example for a while.

Sorry, but I really don't get the message you're trying to convey Can you make it a little clearer?

Hint: Parity has to do with even cycles. What does a U or U' turn do?

 

[cmhardw]

parity => partaaaay

anyone? anyone?

Chris ;-)

P.S. on a somewhat serious note, I can't be the only one who has that word association between those words when written out. Or maybe I am?? :-s

 

[Dene]

parity => partaaaay

anyone? anyone?

Chris ;-)

P.S. on a somewhat serious note, I can't be the only one who has that word association between those words when written out. Or maybe I am?? :-s

I can assure you that you are not the only one. Lucas in particular seems to be a fan.

 

[qqwref]

I've made that mistake several times as well, and I think it's part of where the name "Martini" comes from too. (Maybe us non-Europeans should start calling it that as well, to avoid confusion.)

 

[DavidWoner]

I've made that mistake several times as well, and I think it's part of where the name "Martini" comes from too. (Maybe us non-Europeans should start calling it that as well, to avoid confusion.)

http://www.youtube.com/watch?v=b9_60vDnbLY

 

[blah]

I think the simplest way to describe parity is by doing a J-Perm on a 2x2x2. You should now have 2 corners swapped.

On 3x3x3 this would be an impossible position
On 2x2x2 you can just do a U or U' and now you have a 3-cycle of corners left.

Try to think about this example for a while.

Sorry, but I really don't get the message you're trying to convey Can you make it a little clearer?

Hint: Parity has to do with even cycles. What does a U or U' turn do?

I'm really confused now :confused: I always thought what I thought was true was true... Now I'm starting to get confused.

I really have no idea what you meant by "to describe parity". I know what you're talking about in the whole example, I've thought about it for a while, but no, I still don't see where the "description of parity" is, I can't seem to pinpoint that description in your example In fact, I don't even know what parity is. I always thought I knew. Now I realize all I know is odd parity and even parity, but I don't know what parity itself means.

Here's how I understand odd and even permutation parities:

When a quarter turn is done on any face of the cube, the corners can be solved by a 4-cycle, same goes for the edges. A 4-cycle can be seen as 3 2-cycles, hence permutation parity for corners is odd since they can be solved by an odd number of 2-cycles (3), permutation parity for edges is also odd (3). The sum of parities, however, is even (6).

When a half turn is done on any face of the cube, the corners can be solved by either (a) 2 3-cycles, or (b) 2 2-cycles, same goes for the edges. (a) 2 3-cycles can be seen as 4 2-cycles, (b) 2 2-cycles are 2 2-cycles, so either way, permutation parity for corners is even since they can be solved by an even number of 2-cycles (4 or 2), permutation parity for edges is also even (4 or 2). The sum of parities is even too.

(On a solved cube, the corners can be solved with a 0-cycle, same goes for the edges. A 0-cycle can be seen as 0 2-cycles, hence permutation parity for corners can is even (0), permutation parity for edges is also even (0). The sum of parities is even (0).)

So we can see that the sum of the permutation parities of corners and edges must be even on a legal cube. However, the respective permutation parities of corners and edges are not necessarily even. The odd-/even-ness of the permutation parity of each piece type (corners/edges) is "toggled" with every quarter turn applied to the cube, hence a cube scrambled with an even number of quarter turns can only be solved with an even number of quarter turns, same goes for odd.

Since there are no edges on a 2x2x2, the permutation parity of the corners can be odd or even depending on the odd-/even-ness of the "invisible edges", i.e. the sum of the permutation parities of the corners and the "invisible edges" must still be even.

I hope this is right because this is what I thought was right all along. But I don't see the connection between this and the example you provided