Understanding Parity.

[miniGOINGS]

I think the OLL parity is caused by an odd number of twists in the inner layers like Rw..

Can somebody tell me, is this true?

Yes.

 

[Stefan]

I think the OLL parity is caused by an odd number of twists in the inner layers like Rw..

Can somebody tell me, is this true?

Inner layer quarter turns, yes. And you're joking, right? I'd expect any good big cube blindsolver to know this...

 

[Chuck]

Yes, I'm totally joking.

Spoiler

No I'm not I rarely solve big cubes with open eyes, thus I never thought why I get orientation parity. Stupid me.

Thank you

 

[qqwref]

Parity means two different things in math and cubing. In math, parity refers to whether a cycle is even or odd - whether you can construct it out of 3-cycles or not. This is sometimes a useful concept in cubing, for instance in BLD solving, but most of the time it isn't (since a parity constraint can only exist if there are some single turns which change parity, but then fixing it is trivial so it isn't helpful to talk about it).

In cubing, the idea of parity comes about when you think you have reduced a puzzle into a given group, so that the puzzle can hopefully now be solved using a more restrictive set of moves. Basically, there is parity when the puzzle can't be solved with only those moves (but *looks* like it can) and so you have to do a special algorithm to move back into the solvable group. Generally each set of positions with a given type of parity is the same size as the set of solvable positions. The pure-math analogue of this idea is the coset.

This sounds a bit contrived, maybe, but it's easy to see why this definition is useful if you think about common cubing scenarios where people would say there can be a parity problem. In 4x4, once you reduce to 3x3, the assumption is that the puzzle can be solved with outer-layer turns only; if it can't, you have a parity problem. In fact, in this case, there are three parity conditions (OLL parity, PLL parity, double parity). In void cube, we can define the set of allowed moves to be "algorithms that, on a 3x3, would keep F2L intact", so if we are on the last layer and see two edges swapped we have encountered a parity case. In Square-1, the idea is that once you get to cubeshape you'll never have to leave (and in fact every non-parity cubeshape position can be solved without ever leaving square/square), but if you get parity you have to go outside of your restricted set of moves. All of these cases fit into the coset-like definition of parity we use in cubing, but they don't all fit into the mathematical definition. (In the mathematical definition, a pure swap of two corners doesn't change parity, but a (1,0) does - not very useful!)

 

[Innocence]

I have a question:
Do you use K4?

As you still seem to be oblivious to the hilarity, let me spell it out for you.

Spoiler

Spoiler

Kirjava invented it. :O

 

[masterofthebass]

I know I'm a little late on this response:

Fisher Cube doesn't have parity, it only has rotated centers like a picture cube would have.

So... center orientation parity?

I don't think this should be still considered parity as in all actuality at least 2 centers need to be rotated. One of the centers just has to be trivially rotated along with the piece that needs solving.

I personally still don't get how sq-1 parity works

Not even the setup-to-turn-six-corners solution?

I have never looked into the inner workings of sq-1 parity, so my original statement stands. I seem to understand it a little now, based off of chris's explanation of the corners just being bandaged edges.

 

[Sakarie]

I have a question:
Do you use K4?

As you still seem to be oblivious to the hilarity, let me spell it out for you.

Spoiler

Spoiler

Kirjava invented it. :O

That doesn't mean somebody still necessary use it. Pochmann doesn't use Classic Pochmann, in spite of that it's his "classic" solvingmethod.

 

[Stefan]

I don't think this should be still considered parity as in all actuality at least 2 centers need to be rotated. One of the centers just has to be trivially rotated along with the piece that needs solving.

And how do you know that? That you can't just rotate one center alone (by 90 degrees)?

 

[qqwref]

I don't think this should be still considered parity as in all actuality at least 2 centers need to be rotated. One of the centers just has to be trivially rotated along with the piece that needs solving.

And how do you know that? That you can't just rotate one center alone (by 90 degrees)?

I guess what he was saying was that you don't need to know whether that's possible. Even if the 3x3 did have a center orientation parity issue, where one could be 90 degrees off, you wouldn't have to be able to solve that for a Fisher Cube because the two-center-twist commutator is intuitive/straightforward and would fix that problem. So it's not really right to call it a parity, just a tricky situation that is part of the puzzle, when the solution is just a matter of realizing you can do a commutator on a piece you want solved and a piece that doesn't show orientation.

 

[Deleted member 4527]

Square-1 parity happens i think because of an odd permutation occuring outside of cube shape, so when you get it back to cube shape you get parity.

 

[Lucas Garron]

Square-1 parity happens i think because of an odd permutation occuring outside of cube shape, so when you get it back to cube shape you get parity.

That is very vague and almost false.
But you can say that move sequences from one cube shape state to another can have odd net permutation parity.
It's just not that "an odd permutation occurs" except as an abstraction (or using some precise definition, which I don't think you really had in mind). Square-1 is just confusing.

 

[Deleted member 4527]

All I know is that something that is uncertain gets changed (bad way to phrase it)
like void cube, parity happens because the centres can be changed
4x4 because centres can be changed as well (at least this causes permutation parity i think)
and square-1 it's because of the cube shape.

and yeah square-1 is very confusing

 

[Stefan]

void cube, parity happens because the centres can be changed

Void cube doesn't have centres.

4x4 because centres can be changed as well (at least this causes permutation parity i think)

No.

 

[cmhardw]

Here is how I view Square-1 Parity.

Perhaps I view parity the way I do because Square-1 is the only puzzle I truly did learn to solve on my own, with no help at all - even parity. I was angry at myself for always using the internet to either learn parity for the big cubes, or to learn the entire 3x3x3 solution.

Ok, so to me and in my own personal view the Square-1 is a reduction style puzzle. It is made up of twenty four 30 degree edges pieces, and the 2 pieces of the middle layer. What has happened is that the puzzle has been solved, reduction style, so that 8 bandaged corners have been created each out of two 30 degree edges. Every state of the Square-1 is the first stage of reduction of this 24 "edge" puzzle. Now, we choose to reduce it further (to cubeshape). By reducing to cube shape it is possible that the corner parity does not match the 30 degree edge parity.

Now keep in mind that, regardless of the parity of the corners, by viewing them as bandaged edges the "bandaged piece" parity is always even. Whether the corners are in an even or odd permutation, viewing them as bandaged edges they could always be solved, if they could be solved singly, with an even number of two swaps. Now the remaining 30 degree edges could be in either an even or odd permutation. This accounts for the fact that the Square-1 has the potential for odd permutations, and (1,0) from the solved state is a perfect example of an odd permutation when viewing the puzzle as 24 "edge" pieces all of 30 degrees.

So, the corners could have either even or odd parity, and the 30 degree edges could have either even or odd parity. This all implies after reducing to cube shape.

Ok, so to fix corner-parity-does-not-match-30-degree-edge-parity I undo cubeshape in such a way as to have 6 corners on one face, and I do a 60 degree turn. This 6 cycles the corners, toggling their parity to match the edges (as long as I redo cubeshape exactly in reverse as how I undid it, or at least in such as way as to not again toggle either corner or edge parity).

You could also fix parity by somehow changing the parity of the permutation of edges by undoing, and then redoing cubeshape. I don't know enough about Square-1 to know how to do this.

I hope this at least clears up the idea of Square-1 parity. There are, in fact, two parities. Michael described them very well in his post, so there is no need to redo it. Basically what we call "Square-1 parity" is actually "bandaged-edges/corner permutation parity does not match 30 degree edge parity" in my view.

Chris

 

[c1829]

Fisher Cube doesn't have parity, it only has rotated centers like a picture cube would have.

Fisher cube does have parity. You get a edge orientation parity when trying to solve the top cross if one of the edges with no obvious orientation is flipped.

 

[Deleted member 4527]

Michael described them very well in his post, so there is no need to redo it.
Chris

Can you please direct me to his post? I'm really curious about square-1 parity.

 

[cmhardw]

Michael described them very well in his post, so there is no need to redo it.
Chris

Can you please direct me to his post? I'm really curious about square-1 parity.

Post #44 in this thread.

 

[Deleted member 4527]

Michael described them very well in his post, so there is no need to redo it.
Chris

Can you please direct me to his post? I'm really curious about square-1 parity.

Post #44 in this thread.

ohh I searched this thread thinking that you were talking about Michael Hughey lol

 

[Ranzha]

Fisher cube OLL "parity" occurs only when one middle edge is flipped.
Obviously, this is unnoticeable considering that they look oblivious to orientation, as a centre on a normal 3x3x3.
When I solve my Fisher, on the last slot I insert the edge first, check the orientation, and then insert the corner.
This way, parity is avoided.
Simple.

For centre parities, I made some algorithms that are in a thread here somewhere....
Too lazy to find it, but I like my idea of making Fisher Cube PLLs.

 

[Moose Owl]

Fisher cube 4x4

I appreciate the R L F2 B2... algorithm, but it doesn't solve my problem, which is that I can't figure out how to change the top (yellow) edges around without messing everything else up.