Define "Parity"

[AvGalen]

@blah: You are overthinking because I used the word "describe". I should have used the word "explain".

You know exactly how odd/even permutation parity works. You just didn't realise this from my example so I will explain:

On 3x3x3 a J-Perm swaps 2 corners and 2 edges. Solving that with 3 cycles (A-Perm, U-Perm) is impossible so you have parity that you need to fix with a quarterturn (U or U')

because this might already be to complicated it is also possible to explain this on a 2x2x2 where there are no edges.

On 2x2x2 a J-Perm swaps 2 corners and 2 (imaginary) edges. Solving that with 3 cycles (A-Perm) is impossible so you have parity that you need to fix with a quarterturn (U or U')

 

[Weston]

parity is the relationship between pieces

not necessarily whether its possible on a 3x3 cube.
but you have even and odd parity

 

[Muesli]

Bit of Self Advertising here.

http://www.speedsolving.com/wiki/index.php/Parity

 

[qqwref]

Bit of Self Advertising here.

http://www.speedsolving.com/wiki/index.php/Parity

If you wrote a good part of the article I don't mean to offend you, but... that article is terrible. The definition of parity makes no sense, the entire thing seems to be about NxNxN cubes (does the writer not even understand what parity is? or what it means on Square-1, for instance?) and uses weird notation, and the "switched corner parity" shouldn't be there at all. I'd fix it but I've long since abandoned the wiki for being impossible to maintain quality-wise.

 

[CuBeOrDiE]

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.

a nuisance

 

[mrCage]

Come on. This is very easy to define. If after reduction to a "3x3x3" cube you vannot solve the cube with solely outer layer turns you have some kind of parity issue. You can have a "flipped edge" on even size cube. Or 2 "swapped edges" also on even sized cube. On odd sized cube you cannot have any kind of parity after full reduction.

You may however have to do some "special algorithm" to complete the full reduction.

Per

 

[Johannes91]

Come on. This is very easy to define. If after reduction to a "3x3x3" cube you vannot solve the cube with solely outer layer turns you have some kind of parity issue.

Uh... Parity is a much more general concept than that. It applies to other puzzles than big cubes and other methods than reduction. You should know that.

 

[mrCage]

Come on. This is very easy to define. If after reduction to a "3x3x3" cube you vannot solve the cube with solely outer layer turns you have some kind of parity issue.

Uh... Parity is a much more general concept than that. It applies to other puzzles than big cubes and other methods than reduction. You should know that.

Yes i'm aware of that. But the first post was about cubes, not other puzzles. I'm also aware that parity may also be on corners as well as edges. And yes of course on bigger cubes also appear for other solving methods than reduction. But about 95% use reduction ...

Sometimes a short answer is better than a fuller more detailed answer

Per

 

[Am1n-]

http://www.ryanheise.com/cube/parity.html

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.

When considering the permutation of all edges and corners together, the overall parity must be even, as dictated by laws of the cube. However, when considering only edges or corners alone, it is possible for their parity to be either even or odd. To obey the laws of the cube, if the edge parity is even then the corner parity must also be even, and if the edge parity is odd then the corner parity must also be odd.

An interesting fact is that a commutator always represents an even permutation on both edges and corners. Given a commutator X.Y.X-1.Y-1, regardless of whether X.Y involves an even or odd number of swaps, X-1.Y-1 will involve the exact same number of swaps, and 2 times any number gives us an even number overall. What this means is that commutators cannot directly solve positions where the edges and corners have odd parity.

While parity tends not to be an issue at the beginning of a solve, it may become an issue in the endgame. If the edges and corners have odd parity, the easiest way to correct them is to apply a single 90 degree turn of any face. However, in the endgame, we typically have a need to employ sequences that affect only a select few pieces while preserving everything else, and a single 90 degree turn does quite the opposite, dislodging 8 pieces. Commutators allow us to affect a small number of pieces while preserving the rest, however these do not directly apply when the parity of the edges and corners are odd.