Parity on higher order puzzles.

[Kirjava]

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.

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.

 

[mrCage]

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

 

[Kirjava]

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.

 

[mrCage]

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

 

[Owen]

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...

 

[cmhardw]

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)

 

[mrCage]

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

 

[Christopher Mowla]

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.

 

[qqwref]

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?

 

[riffz]

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

 

[JonnyWhoopes]

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.

 

[StachuK1992]

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.

 

[BC1997]

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.

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

 

[Kirjava]

'regular' version

Define 'regular version'.

 

[MaeLSTRoM]

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.

 

[BC1997]

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.

 

[Kirjava]

So 3x3x3 is a higher order puzzle?

 

[BC1997]

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.

 

[qqwref]

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.

 

[mrCage]

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