Define "Parity"

[brunson]

With respect to cubes.

It's thrown around a lot, I know what it means in computer science and mathematics, but, in the words of Inigo Montoya, "You keep using that word. I do not think it means what you think it means."

I'd just like to see a rigorous definition of what people that use the term think it means.

 

[StachuK1992]

http://en.wikipedia.org/wiki/Parity

Apart from (mostly false) wiki definitions, I would say that, in a cubing sense, that there are different definitions for different situations.

For example, while speedsolving a 3x3, one may encounter a turned corner, a flipped edge, or an 'impossible' permutation case, while in blindfolded cubing, the cube may be completely solvable, yet still have 'parity.'

Therefore, we can make no and infinite definitions.

 

[Johannes91]

With respect to cubes.

It's a mystical magic synonym for "trouble".

 

[soccerking813]

Parity means: Bad

 

[JBCM627]

1. What Tim enjoys doing to Dan's cubes. (Will I ever stop referencing that?)
2. http://www.jaapsch.net/puzzles/theory.htm#permpar

 

[blah]

The parity of a permutation refers to whether that permutation is even or odd. An even permutation is one that can be represented by an even number of swaps while an odd permutation is one that can be represented by an odd number of swaps.

 

[jcuber]

IMO, parity is when there is an "impossible" case (as in impossible on a 3x3). Also synonym for bad, horrible, suckish, etc.

As well as what Tim likes doing to Dan's cubes.

 

[Lord Voldemort]

Does parity only apply to permutation or to orientation as well?
From what I know, it's the number of 2-Cycles done of the cube (because 3-cycles can be broken down as 2 two cycles). The reason you can have parity in blindsolving is because you're doing edges and corners separately, but an odd number of swaps + an odd number of swaps (edges and corners) = a total even number of swaps for the whole cube.
@ jcuber - Only odd parity is an impossible case...

 

[Tim Reynolds]

1. What Tim enjoys doing to Dan's cubes. (Will I ever stop referencing that?)
2. http://www.jaapsch.net/puzzles/theory.htm#permpar

1. Apparently not? Oh, and I wasn't the only one as I recall...I just did the obnoxious ones like v-cubes and megaminx.

As for parity...the problem is that it is colloquially used in so many different ways. My attempt at a definition for the way that it is most frequently used:
A position of a puzzle which, due to an odd permutation of some set of pieces, requires additional steps to solve the cube beyond the standard solving method.

Or basically, these

It's a mystical magic synonym for "trouble".

Parity means: Bad

work.

 

[Ton]

With respect to cubes.

It's thrown around a lot, I know what it means in computer science and mathematics, but, in the words of Inigo Montoya, "You keep using that word. I do not think it means what you think it means."

I'd just like to see a rigorous definition of what people that use the term think it means.

The mathematics term would be odd and even permutation

In plain English

If you swap two element it is odd - one swap of two pairs-
If you swap three elements it is even -this will need two swaps of two pairs
if you swap four elements it is odd -this will need three swaps of two pairs
etc

Examples :

if you need to swap ( 1 2 3 ) to e.g. (3 1 2)

You can swap position 1 with 2 and position 1 with 3 (even)

so (1 2 3 4 ) swap to (4 1 2 3)
Swap position 1 with 2 and position 1 with 3 and position 1 with 4
So this is odd permutation (parity)

so will be (1 2 3 4) to (3 4 1 2) even as this needs two swaps of pairs


odd parity is an impossible case for a normal 3x3 , but possible on a void cube

 

[Johannes91]

blah and others: Yes, we are aware of that definition, but it's not how a lot of speedcubers use it. One difference is that it's usually "parity" or "no parity" instead of even or odd, and often it's used for something completely unrelated because many just don't know the real meaning. See for example posts #4 and #7 in this very thread.

 

[Lord Voldemort]

How does a 4x4 orientation parity happen?
Isn't that 2 pieces being swapped?

 

[blah]

blah and others: Yes, we are aware of that definition, but it's not how a lot of speedcubers use it. One difference is that it's usually "parity" or "no parity" instead of even or odd, and often it's used for something completely unrelated because many just don't know the real meaning. See for example posts #4 and #7 in this very thread.

Oh, so this was supposed to be a sarcastic thread for people (who don't know what parity means) to look silly

Edit:

I'd just like to see a rigorous definition of what people that use the term think it means.

Kinda missed that earlier on

 

[blah]

How does a 4x4 orientation parity happen?
Isn't that 2 pieces being swapped?

It's an odd permutation parity of two edge pieces that appears to be an "orientation parity" (which doesn't exist) at first sight.

That's why it's called OLL parity, not orientation parity.

 

[Stefan]

It comes from solving 4x4 with the reduction method and describes the two problems you can run into after building the edge pairs. Therefore it's actually not "parity" but "pairity" which is derived from "pair" and "pity".

 

[JLarsen]

The way that I think of parity on say...a 4x4, is looking at the laws of the cube, and how something like OLL/PLL parity does not actually break the laws of the cube. The thing that people don't realize, is that even though during the final solve of a 4x4 we treat paired dedges as a single edge, they are not. OLL parity is not one flipped edge, but two, thus it does not violate the laws of the cube. The same goes for PLL parity. That's just my way of understanding it best.

 

[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?

 

[rjohnson_8ball]

This is what I believe. I am a long time computer programmer, so maybe I am biased.

The term "parity" was originally derived from data transfer for computers. Occasionally, when data got sent and received, a single bit was accidentally interpreted the wrong way (as "on" when it should be "off" or the other way around). If it was toggled the wrong way it was called a "parity error". This was sometimes a problem with magnetic tapes -- a very common way of storing and transferring large amounts of data back in the 60's through 80's. Someone devised a solution -- append one extra bit to each piece of data and assign it "on" or "off" so that the sum of "on" bits would be even. Then as the data got read it would do parity checking, and re-read a piece of data if it looked odd. The chances of two bits being wrong in a piece of data (causing a parity error to go unnoticed) would be very slim.

Today we use checksums in executables for a similar purpose.

EDIT: Okay, I suppose the word existed before the computer data transfer problems. Generally I think "parity error" signifies a missing pair, when things should be paired.

 

[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?

I believe it is possible to have 2 edges permuted correctly but not oriented. I'll try it. Having 2 sets of edge swapped is PLL parity.

Edit:Yes you can flip just 2 edges.

 

[Tim Reynolds]

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)