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