# Parity on higher order puzzles.



## mrCage (Aug 10, 2011)

The higher order cubes and even the mosaic cube has parity. So is this always the case for "higher order" puzzles. Proof or counterexample anyone???

Per


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## vcuber13 (Aug 10, 2011)

the mosaic cube parity is a centre twist right?


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## Tao Yu (Aug 10, 2011)

Gigaminx has no parity http://www.speedsolving.com/forum/showthread.php?15861-Gigaminx-Parity

Not fully sure if teraminx has parity. Searching for "teraminx Parity" didn't return any threads specifically on that subject


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## StachuK1992 (Aug 10, 2011)

Sq1? Void cube?
A list of these would be nice here.


I'm thinking my interpretation of this thread may be off?


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## Kirjava (Aug 10, 2011)

Not sure how you're defining "higher order puzzle" and "parity" here.

Anyway, the 1x2x17 doesn't have parity.


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## Escher (Aug 10, 2011)

Kirjava said:


> Not sure how you're defining "higher order puzzle" and "parity" here.


 
Ya the question is a little ambiguous...


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## mrCage (Aug 10, 2011)

By higher order i mean a puzzle with more layers than the base puzzle. The 4x4x4 is a higher order 3x3x3 (or 2x2x2) and the gigaminx is higher order megaminx. A 1x2x17 is not really higher order with my interporetation. It has to be extended in all 3 dimensions. A 2x2x17 might be considered a higher order 1x1x16. Hard to define precisely. I think the gigaminx is good couterexample already...

Let us compile a list of higher order puzzles with and without parity.

Per


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## Kirjava (Aug 10, 2011)

Define "base puzzle".


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## mrCage (Aug 10, 2011)

Kirjava said:


> Define "base puzzle".



I think my post gave a "definition by example" 

Parity is kind of a nightmare to define though. Apparent swap would do for now ...

Per


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## Kirjava (Aug 10, 2011)

I know, but I don't know why you gave 3x3x3 as an example of a base puzzle when 2x2x2 has a lower order than it.


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## mrCage (Aug 10, 2011)

Kirjava said:


> I know, but I don't know why you gave 3x3x3 as an example of a base puzzle when 2x2x2 has a lower order than it.


 
Maybe because it's a more common puzzle?? Ok, then 2x2x2 is a lower order 3x3x3 

Per

And i believe the 3x3x3 was invented before 2x2x2 right?? Or at least was released before it


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## JonnyWhoopes (Aug 10, 2011)

Not trying to be facetious, but then how do we determine what the base puzzle is? Is the megaminx just a lower order gigaminx?


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## mrCage (Aug 10, 2011)

JonnyWhoopes said:


> Not trying to be facetious, but then how do we determine what the base puzzle is? Is the megaminx just a lower order gigaminx?


 
Correct! Teraminx is higher order gigaminx (or megaminx). Petaminx is higher order teraminx (or etc ...)

Per

Then what is flowerminx?? ...


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## Kirjava (Aug 10, 2011)

mrCage said:


> Maybe because it's a more common puzzle??





mrCage said:


> And i believe the 3x3x3 was invented before 2x2x2 right?? Or at least was released before it


 
These are both very silly reasons.

Your definition of "base puzzle" and "higher order puzzle" is arbitrary. Nothing wrong with this, it just makes it somewhat meaningless.


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## JonnyWhoopes (Aug 10, 2011)

mrCage said:


> Correct! Teraminx is higher order gigaminx (or megaminx). Petaminx is higher order teraminx (or etc ...)
> 
> Per
> 
> Then what is flowerminx?? ...


 
So, you're pretty much just asking "what puzzles have parity" right? If we have no definition of a base puzzle, or what higher order means, then this is what it comes down to.


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## Kirjava (Aug 11, 2011)

oddlespuddle said:


> 2x2x2 and 3x3x3 are small cubes or "lower order cubes".



Why?



oddlespuddle said:


> A flowerminx is a 2x2x2 dodecahedron. The creator decided not to use the mathematical term I guess. The real name for it would be "whatever comes before mega" and minx.



Kilominx is something else.


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## uberCuber (Aug 11, 2011)

oddlespuddle said:


> OLL edge parity: Only even numbered cubes
> PLL parity: Only even numbered cubes.
> OLL corner parity: Uh... only pops?
> Edge paring parity: All big cubes that involve edge pairing. (This means "cubes" not other twistypuzzles like dodecahedrons).


 
These are assuming reduction method, though, which not everybody uses. Because PLL parity isn't parity at all, and what you are referring to as "OLL edge pairty" and "edge pairing parity" become the same thing.


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## qqwref (Aug 11, 2011)

A pretty good rule of thumb is that any turn of 1/(2n+1) of a full rotation can't cause parity (whereas turns of 1/(2n) can). So, any higher-order Pyraminx, Megaminx, Dino Cube, Skewb, and so on can't have parity, in the sense of being able to do a pure 2-cycle without affecting any other pieces.


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## TMOY (Aug 11, 2011)

I think you mean any other pieces of the same orbit, or else the corners+centers parity of the 4^3 for example wouldn't qualify as a parity.


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## mrCage (Aug 11, 2011)

Kirjava said:


> These are both very silly reasons.
> 
> Your definition of "base puzzle" and "higher order puzzle" is arbitrary. Nothing wrong with this, it just makes it somewhat meaningless.


 
Then how about this: 3x3x3 is the base cube puzzle because it's the lowest order of it's kind with all the various types of pieces. 2x2x2 is not the base because it has no center pieces. Same goes with flowerminx versus megaminx. We have always used higher order term without a proper definition so it's about time we define it now. This forum is the right place to discuss this 

Per


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## Kirjava (Aug 11, 2011)

mrCage said:


> Then how about this: 3x3x3 is the base cube puzzle because it's the lowest order of it's kind with all the various types of pieces. 2x2x2 is not the base because it has no center pieces. Same goes with flowerminx versus megaminx.



3x3x3 doesn't have wings or obliques.



mrCage said:


> We have always used higher order term without a proper definition so it's about time we define it now. This forum is the right place to discuss this


 
Higher order doesn't need any other definition because it's meaningless. It's just used like 'bigcubes' is.


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## mrCage (Aug 11, 2011)

Kirjava said:


> 3x3x3 doesn't have wings or obliques.
> 
> 
> 
> Higher order doesn't need any other definition because it's meaningless. It's just used like 'bigcubes' is.


 
You begin to sound like Stefan .... So would 'bigminx' be OK for you then?? Nah, there is pyraminx and megaminx ...

Per


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## Kirjava (Aug 11, 2011)

I don't understand the point in trying to do what you're doing.

You could just admit that your definition of 'base puzzle' is arbitrary.


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## mrCage (Aug 11, 2011)

Kirjava said:


> I don't understand the point in trying to do what you're doing.


 
It's a free world. You think another forum is more suitable for this thread?? (branch of thread).
You think defining higher and lower order is meaningless also?

Per


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## Owen (Aug 11, 2011)

There used to be an active user on this forum called "parity". He hasn't posted in a while, I wonder what happened to him...


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## cmhardw (Aug 11, 2011)

I don't understand the contention about what qualifies as a base puzzle. It _is_ an arbitrary definition. 3x3x3 *is* the base puzzle, because Per defined it so. Megaminx *is* the base puzzle, because Per defined it so.

[/rant]

As to the rest of the discussion, what exactly are we meaning when we say "parity"? I greatly dislike defining what people call PLL parity (2 double swaps of wings) as a "parity" as this is only a parity when viewing the cube through the eyes of a reduction solving method. Having an odd permutation in wings would qualify as "parity" for the wing orbit. Having an odd permutation of the corners would qualify as "parity" for the corner orbit.

Thus the 4x4x4 cube has 4 possible permutation parity states:
1) even parity for both corners and wings
2) even parity for corners, odd parity for wings
3) odd parity for corners, even parity for wings
4) odd parity for corners, odd parity for wings

Are we defining cases 2, 3, and 4 all to be "parity"? Are we trying to find situations where a base puzzle has no situation where any piece orbit may have an odd parity permutation, and at the same time a higher order version of this puzzle which can have an odd parity permutation in a certain piece orbit?

I see the following argument applicable to standard cube puzzles:

Base puzzle (3x3x3): Has parity (corners may be in an odd permutation, edges may be in an odd permutation)
Higher order cube puzzles: Have parity (see my example for the 4x4x4 cube, and my parity state matrix for the n x n x n cube)


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## mrCage (Aug 11, 2011)

Jorghi said:


> Use mathematical logic and cubing terminology to make your definitions.


 
I think i have defined all terms by now. Not in same post though. I can make my base puzzles axioms, unattackable ... hehe
What still worries me slightly is a solid definition of parity. I dont like to include centers or corners, solely the edges.

Per


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## Christopher Mowla (Aug 11, 2011)

mrCage said:


> I think i have defined all terms by now. Not in same post though. I can make my base puzzles axioms, unattackable ... hehe
> What still worries me slightly is a solid definition of parity. I dont like to include *centers* or corners, solely the edges.
> 
> Per


Well, you kind of have to include the centers if you are talking about supercubes (excluding the 2x2x2 and 4x4x4) and void cubes.


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## qqwref (Aug 11, 2011)

cmhardw said:


> I greatly dislike defining what people call PLL parity (2 double swaps of wings) as a "parity" as this is only a parity when viewing the cube through the eyes of a reduction solving method.


Indeed, it IS only a parity in terms of a reduction method. I have no problem with the idea that the method can affect the types of parities involved, because to me parity is actually affected by your method. When you reduce the puzzle to the point where you expect it can be solved within a certain class of moves, anything that can bring the puzzle out of that class while still visually appearing to be in it counts as parity. So this definition includes PLL parity, but also the parity that appears on Square-1, and a few others as well. This is quite different from the BLD/mathematical sense of parity, which would (for instance) define a parity on the 3x3x3 even though no speedsolver would have to deal with parity during their solves.


PS: I think I solved the problem here - any comments?


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## riffz (Aug 12, 2011)

Very well put, qq. That is by far the most practical way to define parity IMO.


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## JonnyWhoopes (Aug 19, 2011)

qqwref said:


> Indeed, it IS only a parity in terms of a reduction method. I have no problem with the idea that the method can affect the types of parities involved, because to me parity is actually affected by your method. When you reduce the puzzle to the point where you expect it can be solved within a certain class of moves, anything that can bring the puzzle out of that class while still visually appearing to be in it counts as parity. So this definition includes PLL parity, but also the parity that appears on Square-1, and a few others as well. This is quite different from the BLD/mathematical sense of parity, which would (for instance) define a parity on the 3x3x3 even though no speedsolver would have to deal with parity during their solves.
> 
> 
> PS: I think I solved the problem here - any comments?


 
Well said. I do like the idea of the term parity conforming to the context in which it is presented.


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## StachuK1992 (Aug 19, 2011)

Stop the bickering.
Each person arguing here, present a concise definition of what you think the following terms should be:
'parity'
'higher-order'

Otherwise, ignore the fact that this is only concerning 'higher-order' puzzles and just respond with an appropriate list of what does and/or doesn't have parity.

Thanks, qq, for providing a working definition of parity.


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## BC1997 (Aug 19, 2011)

StachuK1992 said:


> Stop the bickering.
> Each person arguing here, present a concise definition of what you think the following terms should be:
> 'parity'
> 'higher-order'
> ...



Parity in cubing= a case which is impossible to solve on a regular 3x3 (thats the permutation parity definition)

Higher order puzzles= puzzles that have more pieces and are harder/take longer to solve than the 'regular' version
e.g. Megmainx=Gigaminx, 3x3=4x4-7x7


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## Kirjava (Aug 19, 2011)

BC1997 said:


> 'regular' version


 
Define 'regular version'.


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## MaeLSTRoM (Aug 19, 2011)

Parity in terms of combination puzzles: The ability to do a pure 2-cycle, or a sequence of moves that performs a pure 2-cycle.


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## BC1997 (Aug 19, 2011)

Kirjava said:


> Define 'regular version'.


It was in the example, for example the puzzle with the least amount of pieces. e.g. 2x2, Megaminx. It can also be defiend as the first puzzle that came out e.g 3x3.


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## Kirjava (Aug 19, 2011)

So 3x3x3 is a higher order puzzle?


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## BC1997 (Aug 19, 2011)

It depends, if from you point of view the 2x2 is the base puzzle (first definition), if not then no. But I would say its sort of in the middle I guess.


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## qqwref (Aug 19, 2011)

StachuK1992 said:


> Each person arguing here, present a concise definition of what you think the following terms should be:
> 'higher-order'


I'm surprised nobody has said this yet: a higher-order puzzle is a puzzle with at least two different cuts on the same axis. (And I define axis in terms of rays out from the center of the puzzle, not in terms of lines through it.)

The 2x2x2 and 3x3x3 are both first-order because they only have one type of cut; the Mirror Blocks has many different types of cuts, but only one per axis, so it's still first-order. And puzzles like the 3x3x5, 5x5x5, Gigaminx, etc. have two cuts on some or all axes, so they are higher-order. It is worth noting that this definition counts the Pyraminx and Magic Crystal as higher-order (because of the trivial tips), which does actually make sense mathematically; if you want, you could disregard these puzzles by not counting any cut which only increases the number of pieces by one.


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## mrCage (Aug 19, 2011)

My idea of higher order is basically like so: more parallell layers in all directions than an otherwise equivalent puzzle.
Higher order should not introduce any new kind of external pieces, only more of them. What happens internally is not of interest to me.
The base puzzle should have all kinds of external pieces, that a lower order could possibly have. The 2x2x2 loses the 3x3x3's centers and does not qualify in my definition as the base puzzle for cubes. Flowerminx similarly loses the centers of megaminx.

It is debatable whether pyraminx or master pyraminx is the base puzzle (with my definition) for that 'family' of puzzles ...

Per


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## Christopher Mowla (Aug 19, 2011)

For me, the legitimate big cube base puzzle is the smallest cube size which is able to illustrate the bigger picture for the nxnxn regarding algorithm translation. Since many algorithms that work on the 4x4x4 can't work on larger cube sizes, it is NOT the base puzzle. The 5x5x5 is much better, as it includes the central edge pieces and its centers are affected by an odd permutation in the wing orbits. However, see this spoiler of what I believe to be the optimal opposite wing edge swap algorithm in quarter turns.


Spoiler



WCA Notation
*4x4x4* (WCA Notation)
r u Rw' Uw' r' Uw Lw Uw' r2 Fw' r' Fw r' Uw f' r' x

*5x5x5* (WCA Notation)
(M'r) u Rw' Uw' r' Uw (LwM) Uw' r2 Fw' r' Fw r' Uw f' (Mr') x

*6x6x6* (SiGN Notation)
Outer orbit
2-3r 2-3u 3r' 3u' 2R' 3u 3l 3u' 2R2 
3R
3f' 2R' 3f
3R'
2R' 3u 2-3f' 2-3r' x 

Inner orbit
2-3r 2-3u 3r' 3u' 3R' 3u 3l 3u' 3R2 
2R
3f' 3R' 3f
2R'
3R' 3u 2-3f' 2-3r' x

*7x7x7* (SiGN Notation)
Outer
2-4r 2-4u 3r' 3u' 2R' 3u 4l 3u' 2R2 
3R
3f' 2R' 3f
3R'
2R' 3u 2-4f' 2-4r' x

Inner
2-4r 2-4u 3r' 3u' 3R' 3u 4l 3u' 3R2 
2R
3f' 3R' 3f
2R'
3R' 3u 2-4f' 2-4r' x


As you can see, if we apply the algorithm to individual orbits of the 6x6x6 and 7x7x7, we need interior conjugates (3R for the outer orbit and 2R for the inner orbit) to not discolor the centers. Don't believe me? Just apply the same algorithm without those conjugates. Therefore, the 5x5x5 is not good enough either.

I also agree with what Per said about that the base puzzle should have all piece types. Therefore, the 6x6x6 would not qualify as the base puzzle for all cube sizes based off of that statement by itself (it doesn't have central edges or any middle center pieces). Here is a counterexample algorithm to anyone who thinks differently.



Spoiler



*4x4x4* (WCA Notation)
x' Uw l Uw' Rw2 x Uw' l' Uw Lw' L' x Uw Rw' Uw' l' Uw (L x) Uw' x

*5x5x5* (WCA Notation)
x' Uw l Uw' M2Rw2 x Uw' l' Uw Rw'M x L' x Uw M Rw' Uw' l' Uw Lx Uw' x

*6x6x6* (SiGN Notation)
Outer
x' 3u
3L
2L 3u' 3r2 x 3u' 2L' 3u 3l' L' x
3L'
3u 3r' 3u' 2L' 3u L x 3u' x

Inner
x' 3u
2L
3L 3u' 3r2 x 3u' 3L' 3u 3l' L' x
2L'
3u 3r' 3u' 3L' 3u L x 3u' x

*7x7x7* (SiGN Notation)
Outer
x' 3u
3L
2L 3u' 4r2 x 3u' 2L' 3u 4r' x L' x
3L'
3u 4r' 3u' 2L' 3u L x 3u' x 

Inner
x' 3u
2L
3L 3u' 4r2 x 3u' 3L' 3u 4r' x L' x
2L'
3u 4r' 3u' 3L' 3u L x 3u' x


As you can see with the translations to the 5x5x5 and 7x7x7, middle center pieces are discolored. This time if you delete the interior conjugates (3L for the outer orbit and 2L for the inner orbit), it's even uglier. (Very complicated alg.)

So, from an algorithm point of view, the 6x6x6 is the base cube sizes for even cubes and the 7x7x7 is the base cube for odd cubes. However, if an algorithm works correctly on a 7x7x7, it definitely will work on a 6x6x6 (with minor adjustments should the algorithm move the just the central slice on the odd cube). So I'm in favor of the 7x7x7 being the base cube size and have stated my reasoning why I think so.


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## riffz (Aug 23, 2011)

mrCage said:


> My idea of higher order is basically like so: more parallell layers in all directions than an otherwise equivalent puzzle.
> Higher order should not introduce any new kind of external pieces, only more of them. What happens internally is not of interest to me.
> The base puzzle should have all kinds of external pieces, that a lower order could possibly have. The 2x2x2 loses the 3x3x3's centers and does not qualify in my definition as the base puzzle for cubes.


 
7x7 is a base cube then, if you count obliques as a distinct piece type.


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## mrCage (Aug 23, 2011)

Corner pieces has faces on more than 2 of the puzzle's faces. Edge pieces has faces on exactly 2 faces of the puzzle. Center pieces lie entirely on a single face. Orbitals is something else .... My DEFINITION .... 

Per


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## riffz (Aug 24, 2011)

mrCage said:


> Corner pieces has faces on more than 2 of the puzzle's faces. Edge pieces has faces on exactly 2 faces of the puzzle. Center pieces lie entirely on a single face. Orbitals is something else .... My DEFINITION ....
> 
> Per


 
That works then


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