# Odd layer cubes parity



## Gafimp (Jun 11, 2010)

Hi, I am new to this forum so I don't know if this is in the right place.

My question is, why don't you get parity errors on big cubes with odd layers, like 5x5x5, 7x7x7?
Surely, because the centre pieces are indistinguishable from each other, two centre pieces could be swapped, and you could force a parity error when the cube is into 3x3x3 form, like you get on even layer cubes.


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## amostay2004 (Jun 11, 2010)

Centres are fixed on odd layer cubes.


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## Kirjava (Jun 11, 2010)

You /can/ get parity errors on the 5x5x5 and 7x7x7, but I don't see how centres would affect parity on odd /or/ even cubes.


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## zachtastic (Jun 11, 2010)

Gafimp said:


> ..two centre pieces could be swapped, and you could force a parity error when the cube is into 3x3x3 form, like you get on even layer cubes.



Not sure what you mean by that. The center pieces on a 5x5 are fixed to the core. This thread discusses parity. Apparently the way most people refer to parity is a misnomer, and it actually has to do with even vs. odd number of swaps? But, I still don't have a clue what I'm trying to say, so here is yet another thread.


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## Gafimp (Jun 11, 2010)

Thanks for replies so far.

To my knowledge, a cube must have an even number of "pairs of pieces" swapped. So if a pair of pieces are swapped, another pair of pieces somewhere else on the cube must be swapped too to retain parity. So I guess a 3-cycle is just two swaps.

It is easier to explain with pictures.







On this picture, you can see a 4x4x4 and a 5x5x5.

You get a parity error on the 4x4x4 when A and C are swapped, and B and D are swapped. So the cube as a whole is parity free, but if you are solving it as if it's in 3x3x3 form, you cannot do it.

My question is, why can't pieces E and F be swapped, then the group of pieces G swap with the group of pieces H? Is it just that two centre pieces can't be swapped? 8 swaps in total (even number), so it should work.


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## Kirjava (Jun 11, 2010)

Permutation isn't shared between centres and edges like it is between corners and midges.

EDIT: On a sidenote, this is what I meant when I was referring to parity on a 5x5x5;


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## vcuber13 (Jun 11, 2010)

For your pictures, you can't switch H and G because the middle edges and corners have the same limits as a 3x3, yoiu cant swithc only 2 edges on a 3x3 so you can't on a 5x5 (the middles ones). Also 2 centres can be switched without affecting the rest of the cube.


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## Kirjava (Jun 11, 2010)

vcuber13 said:


> Also 2 centres can be switched without affecting the rest of the cube.




Technically, no.


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## miniGOINGS (Jun 11, 2010)

Kirjava said:


> Technically, no.



Technically?


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## Kirjava (Jun 11, 2010)

Technically.


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## vcuber13 (Jun 11, 2010)

miniGOINGS said:


> Kirjava said:
> 
> 
> > Technically, no.
> ...



Yes its true I just tried it on a super cube in gabbasoft, and it did a 3 cycle.


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## Kirjava (Jun 11, 2010)

vcuber13 said:


> Yes its true I just tried it on a super cube in gabbasoft, and it did a 3 cycle.




I'm impressed. Well done.


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## cmhardw (Jun 11, 2010)

Gafimp said:


> Thanks for replies so far.
> 
> ...
> 
> ...



To answer your question I would like you to consider a scenario. Assuming you did swap pieces E and F, why *would* that have any affect on the pieces from groups G or H? (Hint #1: it could affect *some* of those pieces, but not all).
Hint #2: Switching pieces E and F would also affect other pieces that you have not marked on the diagram



vcuber13 said:


> Also 2 centres can be switched without affecting the rest of the cube.



This is definitely false, no matter what size cube you consider, and no matter which center piece type you consider. I'm not trying to be harsh or bash you, and in the interest of good discussion, do you realize why your statement is false? To be very clear, I am considering your statement to have been made as if you consider switching two centers on a supercube.

Chris


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## PeterNewton (Jun 11, 2010)

@cmhardw, is Gafimp wrong because he should have said that "two centers _in the same orbit_ can be switched without affecting the rest of the cube"? Or is that also untrue? I am having similar problems..


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## Gafimp (Jun 11, 2010)

cmhardw said:


> To answer your question I would like you to consider a scenario. Assuming you did swap pieces E and F, why *would* that have any affect on the pieces from groups G or H? (Hint #1: it could affect *some* of those pieces, but not all).
> Hint #2: Switching pieces E and F would also affect other pieces that you have not marked on the diagram
> Chris



Swapping just E and F is impossible, it would result in a wrong parity on the whole cube wouldn't it? You can't swap just 2 pieces. An odd number of additional swaps would be in the cube somewhere. 3 is an odd number of additional swaps, swap the group of 3 pieces at G with the group of 3 pieces at H?

I assume this works because parity of edges on a 3x3x3 relies on parity of corners, and vice versa.
So then I assume parity of centre pieces on big cubes are dependant on the parity of the edge/corner pieces, is this wrong?


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## vcuber13 (Jun 11, 2010)

cmhardw said:


> vcuber13 said:
> 
> 
> > Also 2 centres can be switched without affecting the rest of the cube.
> ...


Yes I realized that after Kirjava said that:


vcuber13 said:


> miniGOINGS said:
> 
> 
> > Kirjava said:
> ...





Gafimp said:


> ...You can't swap just 2 pieces... is this wrong?



It would dpensd ion the peice, you can't switch only 2 centres, corners or middle edges. But you can have a 2x2 swap of corners and edges (J perm). You can however swap only 2 wing edges.


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## cmhardw (Jun 11, 2010)

vcuber13 said:


> It would dpensd ion the peice, you can't switch only 2 centres, corners or middle edges. But you can have a 2x2 swap of corners and edges (J perm). *You can however swap only 2 wing edges*.



It gets more interesting on the larger sized cubes (larger than 4x4). Yes you can swap only two wings, but on a super5x5 or larger this is not the only effect you experience (i.e. more piece types, by necessity, *have* to change parity with the wings). Maybe you already knew this, but I wanted to include it for completeness for anyone following this thread.

Chris


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## vcuber13 (Jun 12, 2010)

For the second time today I said something without checking the effect on a supercube:fp


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## Christopher Mowla (Jun 12, 2010)

cmhardw said:


> vcuber13 said:
> 
> 
> > It would dpensd ion the peice, you can't switch only 2 centres, corners or middle edges. But you can have a 2x2 swap of corners and edges (J perm). *You can however swap only 2 wing edges*.
> ...


I know, it's very interesting. For a cube size _n_=4 or greater with _r_ orbits with inner-layer odd parity, there are exactly \( 2r\left( n-2 \right)-4r^{2} \) center-edge pieces and/or obliques which cannot be solved back due to the 2_r_ wing edges out of place caused by the odd permutation. What's interesting about this is *it does not matter* which orbits have parity, whether they are spaced out or together, etc. Only the number of orbits with odd parity matters. (I have a derivation to prove this strong statement too).

For example for a 1000X1000X1000 cube with 22 random orbits with odd parity, there are exactly 41, 976 center pieces which cannot be solved back (unless the wing edges are fixed back with an odd parity algorithm). But this is nothing compared to the maximum number of center pieces which cannot be corrected on the 1000X1000X1000 for some number of orbits with odd parity: 249, 000.


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## mrCage (Jun 12, 2010)

Why is this still baffling/confusing people??

# of parities depends on the size of the cube, method and partially on parity definition...

Per


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## Johannes91 (Jun 12, 2010)

mrCage said:


> Why is this still baffling/confusing people??
> 
> # of parities depends on the size of the cube, method and partially on parity definition...



I think you just answered your own question...


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## cmhardw (Jun 13, 2010)

This is also partly a reason why parity still confuses people.


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## ~Phoenix Death~ (Jun 13, 2010)

cmhardw said:


> This is also partly a reason why parity still confuses people.


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