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 Post subject: Puzzling ParitiesPosted: Wed Sep 28, 2011 8:21 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Hi All,

I have recently become aware that, even though I know and use the term `parity`, I am not absolutely sure of its definition.

As I understand it, a parity occurs when `2 pieces that are the same (or unrecognisable- perhaps hidden) are exchanged` to create a seemingly unsolvable situation (where 3cycles are not possible). Is this definition a good one? I’ll give some examples to start it off..

I think `parity` is one of the most interesting things about puzzles. And I would not mind if this thread turned into a photo thread of interesting Parity situations (like the `patterns` thread) along with or after the discussion.

Examples:

On a standard 4x4x4 there are 2 types of parity:
The 1st one that is caused by the `hidden` centre pieces, similar to the Void Cube parity (where 2 of the `3x3 edge groups` become exchanged). This occurs because we have been unable to `see` which faces to build the cube on and we have accidentally offset it by 1 face. (This does not occur on the odd large cubes because we can see the centre).

The 2nd one is caused by the incorrect placement of `indistinguishable` centre pieces (where one `3x3 edge group` is flipped). I don’t think this parity is possible on a 4x4 Supercube because the centres have now become distinguishable. If you solve the centres, you will remove the `apparent edge parity`?

Tower Cubes:
A simmilar centre piece parity occurs on Tower Cubes because the `indistinguishable` centres can be placed in the wrong layer (in my photo the green centres on the 2nd & 6th layers have been exchanged). The `evidence` of the parity occuring can be seen in the edges again, but the solution is in exchanging the centres. This parity exists on a 2x2x3 (or 2x2xN) cube also, but you have to switch the imaginary parts (is this, for example, parity?).

I think under permutation rules of parity these common 2x2x2 and 3x3x2 examples are actually a type of `parity`? The switched pieces are unseen. Are these situations traditionally seen as parity do you think?

Dayan Gem 3:
This parity occurs because 2 of the small same coloured `square-face edge triangles` are exchanged.

Vulcano:
This parity occurs because, similar to the 4x4x4 Cube, we have been unable to `see` which faces to build the cube on and we have accidentally offset it by 1 face.

Opinions?
Cheers,
Burgo.

 Attachments: parity.jpg [ 234.96 KiB | Viewed 3620 times ]

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)
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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 12:04 am

Joined: Fri Jan 07, 2011 2:37 pm
It seems like parities on one puzzle are related to an impossible situation on another one. With the two 4x4x4 parities, for example, it seems like those positions shouldn't be able to exist because they aren't possible on a 3x3x3. It is the same with some of the other examples shown. On a 3x3x3, two edges can't be switched like they are on the 2x3x3, and two corners can't be switched like on the 2x2x2. So, when you mentioned "a seemingly unsolvable situation," I think that statement should be something like "a situation that is unsolvable on a similar puzzle."

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 1:20 am

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Hi Zeotor,

It may be that our perception of another puzzle `tricks us` to think a situation is a parity when it is not. Especially if we are reducing it to `be` that other puzzle, like in the 4x4x4 and the Vulcano. I often hear terms like `seeming parity`, and wonder , so I thought I'd ask. But I just love parities (if I can figure out what that is ) so why not make a thread.

I am throwing curly ideas about, for sure. But in the cases I have shown, I think they might be parities because of `hidden pieces` that need to be exchanged. Like, on the 3x3x2 and 2x2x3, for example, if we imagine a 3x3x3 we would switch 4 edges to solve it (2 of them are not physical edges on those puzzles). Of course I have used them because they are easy examples of trickier situations on larger puzzles). Don't worry, I have been confusing myself for some time over this .

Cheers,
Burgo.

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 1:26 am

Joined: Mon Oct 18, 2010 10:48 am
My understood definition of parity, though this may not be the technical definition, is where two pieces are switched that would otherwise be dependent in orientation in comparison to a hidden piece.

For example, take a 4x4 edge parity where one edge is flipped. By revealing the 5x5 piece beneath, one discovers that both edges are incorrect, and the 5x5 edge is correct. Thus, the 4x4 edge flip parity would not exist on a 5x5, as the parity is dependent upon the 5x5 center edge piece not being there.

Which brings us back to the definition of parity. There is parity on a vulcano since, if one were to reveal the edge beneath the pyraminx edges (the master pminx edge), you would discover that both the vulcano edges are flipped, and the mpminx edge is correct.

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I don't know half of you half as well as I should like and I like less than half of you half as well as you deserve.

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 1:55 am

Joined: Sat Jun 11, 2011 2:34 pm
If I recall correctly, the mathematical definition of parity of a permutation is that an even number of 2 cycles is even, and an odd number is odd (pretty easy to remember )
When this is applied to puzzles, things get a bit more complicated.
the parity situation when an edge pair is flipped in place, is a parity. In the sense that it is a pure 2 cycle. However, the situation when 2 edge pairs are swapped with each other, is not a mathematical parity, because it is 2 2cycles. The situation combining these two is a 4cycle so again, no parity.

However, switching invisible or hidden parts could only be considered parity if it is a pure 2 cycle of those hidden parts, otherwise it is an example of void-cube style parity. Which is not a pure 2-cycle but could be a 2+2cycle, on in the visible pieces, and one in the invisible.

Pretty sure this is all correct

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 3:26 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
MaeLSTRoM wrote:
If I recall correctly, the mathematical definition of parity of a permutation is that an even number of 2 cycles is even, and an odd number is odd (pretty easy to remember )
When this is applied to puzzles, things get a bit more complicated.
the parity situation when an edge pair is flipped in place, is a parity. In the sense that it is a pure 2 cycle. However, the situation when 2 edge pairs are swapped with each other, is not a mathematical parity, because it is 2 2cycles. The situation combining these two is a 4cycle so again, no parity.

However, switching invisible or hidden parts could only be considered parity if it is a pure 2 cycle of those hidden parts, otherwise it is an example of void-cube style parity. Which is not a pure 2-cycle but could be a 2+2cycle, on in the visible pieces, and one in the invisible.

Pretty sure this is all correct

I agree with all of this.
TomZ wrote:
In mathematics, parity can be either even or odd. The way cubers use the word is just plain wrong.
Wikipedia wrote:
In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective mappings from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation σ of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements x,y of X such that x < y and σ(x) > σ(y).
Seems you are right. ...

I think, I understand the mathematical definition - at least vaguely - my question is, is there consensus how cubers use the term and how they should use it?
To me a 4x4x4 has only one parity case: A single 2 cycle of two edges (one 3x3x3 edge flipped). The other one that is called a Parity situation - two pairs of edges need swapping (two 3x3x3 edges swapped) - is none in my view (as Maelstrom has said).

Do cubers talk about parity situations on a 3x3x3? Usually not. In a mathematical sense you can get an odd parity of corners, if there is an odd parity of edges, as well.

I have used the term "parity" (in quotation marks) for situations that seem impossible to solve at the first glance.
I will refrain from doing this in the future.
Also, if some identically looking pieces hide another 2-cycle of pieces, I would not use the term 'parity', but talk about a 'seeming parity' (which is unmathematically, completely )

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 3:30 am

Joined: Sun Oct 08, 2006 1:47 pm
Location: Houston/San Antonio, Texas
The meaning of parity has devolved by quite a bit in the context of twisty puzzles.

Now, I'm not saying we must strictly enforce this definition of parity, but there is a proper meaning to this that most people have either completely forgotten, or never knew, because most people are using the term parity incorrectly. I know I tend to make dreadfully long posts, and I really try to avoid it whenever possible, but I really think it is worth it here

If anyone is interested in knowing exactly what it means, continue reading otherwise ignore this post and carry on your blissful way

Parity in the context of twisty puzzles is referring to the parity of a permutation. If the mathematical language doesn't scare you away, read that and skip to "NOW", below. Otherwise, here is a very basic way of explaining it:

Let's say you have three letters: A, B, and C. These letters begin in order: ABC. Now let's say you want to rearrange the order of these letters, but you only allow yourself to manipulate them in a specific way: you may swap any two letters, while the third must stay in the same spot. For example, in one swap you can reach any of the following states: BAC CBA ACB

If you keep swapping letters around for awhile you will find there are 6 total arrangements: ABC, ACB, BAC, BCA, CAB, and CBA. Note this is exactly the total number of permutations of 3 objects ( = 3! = 3x2x1 = 6). In fact, you can reach any permutation of any number of objects just by starting with the objects in order and then swapping only two at a time. Where it gets interesting is when you start counting how MANY swaps it takes to reach a specific permutation.

For example, let's say you want to reach the state ACB. Remember, you must start with ABC. You could do:

ABC -(swap B & C)-> ACB
Done! It took 1 swap. Let's try a different way...

ABC -(swap A & B)-> BAC -(swap A & C)-> BCA -(swap A & B)-> ACB
Done! It took 3 swaps. Not to beat a dead horse, but let's try one last one...

ABC -(swap A & C)-> CBA -(swap B & C)-> BCA -(swap A & C)-> BAC -(swap B & C)-> CAB -(swap A & C)-> ACB
Done! It took 5 swaps.

Notice that we can do it 1, 3, or 5 swaps, but what about 2? or 4? It turns out this is impossible. If it takes an ODD number of swaps to reach ACB in one way, it will take an ODD number of swaps in every way and it is IMPOSSIBLE to do it in an EVEN number of swaps. Likewise with permutations that require an even number of swaps. Furthermore, since you can break down any permutation into sequences of swaps, doing a complicated rearrangement all at once (like cycling the numbers ABC -> BCA) is equivalent to doing a sequence of swaps that has the same affect one at a time. This is called the parity of the permutation. Since it takes 0 moves to get from in-order to in-order, we say the in-order (or solved ) state has EVEN parity. permutations that take an odd number of swaps to reach from solved are said to have ODD parity and permutations that take an even number of swaps to reach from solved are said to have EVEN parity. Here are the 6 permutations of ABC again, this time organized by their parities:

even: ABC, BCA, CAB
odd: ACB, BAC, CBA

Everytime you swap two letters, you must switch to some state that has the opposite parity (i.e. ODD parities must go to EVEN parities in one swap and vice-versa). It will also always be true that exactly half of all permutations of n objects will be EVEN while the other half will be ODD.

NOW,

What does this have to do with twisty puzzles? Whenever you make a single move on a puzzle, you are applying some permutation to the pieces. If you really wanna break it down, you can say you are simultaneously applying some permutations to different groups of pieces: one group for each orbit of pieces.

Let's look at an example: The megaminx!

The megaminx has 3 types of pieces: centers, edges, & corners. For now, ignore orientation; we're just focused on the location of the pieces. Also, since the centers don't move relative to the core, let's ignore them as well. So we're down to looking at the locations of the edges and the corners. It is also important to note that every edge can get to every edge location and every corner can get to every corner location (this is not true for puzzles like a non-jumbled Helicopter Cube!) Thus we have two ORBITS of pieces: the corner orbit and the edge orbit.

Make any single move on the megaminx (just 72 degrees) Note that you just moved 5 corners and 5 edges. Let's break those down seperately: if we label the edges ABCDE, the move resulted in the permutation BCDEA. Note this takes ABCDE -> BACDE -> BCADE -> BCDAE -> BCDEA 4 swaps, so this permutation is EVEN. If we make another move, it will also be EVEN because it looks just like this move once we relabel the pieces correctly. So if we consider the permutation of all 30 edges, we started in the solved position (which is defined to be EVEN) and after every move, we are applying an EVEN permutation. Just as EVEN + EVEN = EVEN, an EVEN permutation composed with an EVEN permutation will result in an overall EVEN permutation. So, no matter what move you make on a megaminx, the permutation of the edges will travel from one EVEN permutation to another EVEN permutation. It is impossible to ever make it to an ODD permutation. We say that it is impossible to swap two edges on a megaminx correct? That's because a single swap clearly takes only one swap which makes it an ODD permutation. It's also true that you can never reach a state where 2 edges are swapped and another 3 edges are all cycled because that is also and ODD permutation, but that's a little harder to remember Anyway, a single move puts the corners in an EVEN permutation as well, so the corners can only ever reach EVEN permutations. Thus the permutation of both the edges and corners of a megaminx are permanently lock in EVEN permutations.

Has anyone zoned out yet? I never said you had to read this, so let me have my moment

Now consider a Rubik's cube. Again, looking at the locations of only the edges and corners, a single move (90 degrees!) cycles 4 edges and 4 corners. A four cycle: ABCD -> BACD -> BCAD -> BCDA takes three swaps so is an ODD parity! That means every move on a Rubik's cube toggles the parity of the edges! This, by the way, guarantees that the number of 90 degree moves it takes to scramble a Rubik's Cube + the number of 90 degree moves it takes to solve that scrambled cube is always guaranteed to be even! But wait, there's more! The corners ALSO get cycled by four, which is also an ODD parity. We must consider the parities of both the edges and corners simultaneously! The puzzle starts with both orbits in EVEN parity, but after one move, both orbits have ODD parity, and they will continue to toggle back and forth after every move, but the important fact is they always match. That's why when you solve a Rubik's Cube, REGARDLESS of your method, at some point you must be considering either the corners or the edges to have either parity, but at a point later (or perhaps simultaneously) in the solve, you assume the other piece type must lie in EVEN parity (because with the other piece type corrected to EVEN parity, it has to). This also explains why it is impossible to solve a cube with just two swapped edges or just two swapped corners, but entirely possible to solve a cube with BOTH two swapped edges and two swapped corners (and I can guarantee it will take an ODD number of 90-degree moves to solve )

Expanding this idea to other puzzles becomes very fun (to me anyway...). You find crazy things like 1)On a 2x3x3, a 90 degree turn of a square face toggles the parity of both the edges and the corners, but a 180 degree turn of a rectangle face toggles the parity of the edges ONLY, thus making ALL permutations of the whole puzzle possible to solve. 2)On a void cube, you are missing the centers so have no idea if the puzzle as a whole has been rotated from the solved position. Although you are imagining a certain orientation of the puzzle, it is possible that your idea is 90 degrees off of where the puzzle "really is" (note that if you consider the rotation of the entire puzzle 90 degrees excluding centers you have moved 12 edges in three 4-cycles [an overall ODD permutation] while moving 8 edges in two 4-cycles [an overall EVEN permutation] and thus toggled the permutation of edge pieces ONLY [something that is impossible with the traditional assumptions that the centers don't move] Of course, if you keep track of which "center" is which as you scramble and put all the colors back where they started, you will avoid this error.) 3) The 24 centers on a helicopter cube fall in 4 distinct orbits, assuming you disallow jumbling moves; therefore you must consider a total of 5 orbits of pieces simultaneously. Every move on the helicopter cube toggles the parity of the corners and 2 of the 4 center orbits, though not always the same 2. It is even possible to find three moves on a helicopter cube that together toggle 3 of the 4 center orbits twice (keeping each EVEN) but toggling the corners 3 times (switching it to ODD) and so it is possible to have just two corners switched on a helicopter cube.

SO WHAT IS A PARITY ERROR?!?!?!?!?!

This term first appeared when the reduction method of solving the 4x4x4 was developed, and it does NOT mean that some orbits of pieces are currently in their wrong orbit (or at least it didn't then - I actually believe it SHOULD now ) The parity error came about because of an ASSUMPTION. When you solve a 4x4x4 by reduction you group the centers and edges and then ASSUME it can be solved as a 3x3x3. There were two situations that were called parity errors on a 4x4x4 - 1) when a pair of edges was swapped with another pair and 2) when two edges side by side were swapped with eachother, thus looking like a single flipped 3x3x3 edge in the reduction step.

Case 1) is an "error" only because the assumption that the puzzle can be solved as a 3x3x3 is wrong. If you consider the edges as 2 conjoined edges, you have two 3x3x3 "edges" swapped while the corners are correct. This is an INVALID PARITY for the 3x3x3 - hence, a PARITY ERROR!!! (it's glorious!!!! ). To correct the error, you must break your assumption and use some 4x4x4 moves to solve it. Interestingly, if you take the puzzle for what it really is (a 4x4x4 and NOT a 3x3x3), you have two pairs of swapped pieces - an EVEN and therefore totally acceptable and common permutation. It is only under the assumption of a 3x3x3 that the issue arises.

Case 2) came about for the same reason - a single flipped edge for a 3x3x3 is impossible, so it was also called a PARITY ERROR, but when you remove the assumption and look at the puzzle as a 4x4x4 again, the edges really are in an ODD parity (a 90 degree slice move toggles the parity of both the edges and the centers, but since centers come in 6 groups of 4 indistinguishable pieces, a truly ODD permutation can be disguised as solved by swapping two identical centers).

So there you have it, a PARITY ERROR was originally an assumed reduction of a puzzle into an impossible combination of orbit parities of a subgroup of the puzzle. (i.e. a 4x4x4 being treated as a 3x3x3 when the assumed 3x3x3 was impossible)

So a PARITY ERROR was not:
1) a tower cube with the edges in an ODD parity
2) any situation on a Dayan Gem 3 where two 'square-face edge trianles' are exchanged
3) any situation where the parity of an orbit is ODD due to an invisible piece being incorrect

Burgo, your situation about the vulcano is EXACTLY the same type of problem that the first solvers of a 4x4x4 encountered. You assumed the puzzle was reduced to another, correct? Also, Zeotor was on the right track when he talked about reducing a puzzle to another

HOWEVER, I BELIEVE IT IS MORE USEFUL TO TALK ABOUT A SITUATION ON A PUZZLE WHERE ONE OR MORE ORBITS ARE CURRENTLY IN ODD PARITY. BASIC ALGORITHMS LIKE COMMUTATORS OR CONJUGACY OF AN EVEN PERMUTATION WILL ALL FAIL TO SOLVE A PUZZLE THAT HAS AT LEAST ONE ORBIT IN ODD PARITY. YOU MUST RECOGNIZE THE PERMUTATION AS HAVING ODD PARITY AND TRACK DOWN THE CAUSE. THUS, ATTEMPTING TO APPLY COMMUTATORS OR CONJUGACIES OF EVEN PERMUTATIONS IS ERRONEOUS IN STATES WHERE ONE OR MORE ORBIT IS IN AN ODD PERMUTATION. YOU COULD EVEN SAY THAT YOU ARE IN AN ERRONEOUS PARITY, OR BETTER YET... YOU HAVE A PARITY ERROR (In which case, flip all my answers - tower cube, Dayan Gem 3, and most situations with hidden stuff all occur because an orbit is in an odd permutation, but the vulcano situation is actually 2 pairs of swapped edges, which is even and totally not problematic)

So there's what a parity error ACTUALLY meant when it was first used, and what I believe we should (and most people already do) take it to mean now, which in some cases contradicts the original meaning

If you perservered and read through that entire post, you deserve a cube -> and I hope you learned something (like hopefully exactly how to identify the cause of a parity situation on a puzzle you've never seen before)

Peace,
Matt Galla

PS It also appears to be true that any move on any puzzle that causes an ODD permutation to occur also affects some "hidden" piece. While this is clearly true for a puzzle like the Rubik's cube, and a little less obvious, but nevertheless applicable to a puzzle like the void cube, it can be quite a stretch for a puzzle like the tower cube... However, if you're loose enough with the definition of hidden piece, I have never found a puzzle where a puzzle has a move that alters parity without negatively affecting a piece that could somehow be made visible by altering the puzzle in some way. I do believe such a puzzle could theoretically exist however....

PPS If you think all this stuff is childish and want a REAL challenge, try analyzing a Rubik's cube where you are restricted to 180 degree turns only and then explain why it does NOT have (4! * 4! * 4!/2) * (4! * 4!/2) positions even though it does have 5 orbits of four pieces each: 3 orbits of 4 edges each and 2 orbits of 4 corners each where the parity of the 3rd edge orbit is dependent on the first 2 and the parity of one corner orbit is dependent on the other. (Note all orientations are uniquely determined by position, so they don't even factor into the calculation)

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 6:25 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Thank you Allagem for your long article.
I agree with most of it.
One thing, though:
Allagem wrote:
...Case 2) came about for the same reason - a single flipped edge for a 3x3x3 is impossible, so it was also called a PARITY ERROR, but when you remove the assumption and look at the puzzle as a 4x4x4 again, the edges really are in an ODD parity (a 90 degree slice move toggles the parity of both the edges and the centers, but since centers come in 6 groups of 4 indistinguishable pieces, a truly ODD permutation can be disguised as solved by swapping two identical centers)....
As you can see on my Supercube 4x4x4, this a real parity situation. All centres are correct (I assure you, I have them correct on the other three faces, as well )

So, all corners are correct = even number of permutations
all centres are correct = even number of permutations
edges need a single 2-cycle = odd parity

The inner, invisible 2x2x2 is in a odd parity situation, but I would not count this.
Too me, this case is an odd parity situation (I would not call it an error, though. Why should it be an error? Just a very special legal situation.)

When you solve a Crazy 4x4x4 you will never see this parity situation, when you have solved th inner 2x2x2.

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 8:04 am

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Thanks Allagem,
What an incredible response, I learned A LOT, now let’s see how much went in..

I’m not a mathematician (and saying that in this company I probably should use that little embarrassed guy, but I don’t like him very much because he just keeps on going forever. And I never know whether he is going to flash an odd or an even amount of times, but at least now I know that if he reverses, and provided he has kept count, that the total will be even).

I always knew intuitively that, `whichever puzzle`, had an end game, and during this end game there is a `fixed amount of permutations` possible to solve it (even I can only try so many times to get 5 pieces in place with 3cycles before I start frowning). Truly, every twisty puzzle is solvable in the sense that, it went away, there is a way back.

To have an `actual parity error` is, of course, impossible, unless you have something like the original version of the 15 puzzle. So surely it is when there `appears to be a parity error`, that I think, it has become synonymous with saying `I have parity`.

With the case of the Gem 3, it appears as though `parity has been broken` because of the exchange of identical pieces. So, I guess: Do we always say `apparent parity` or do we say `parity`? Which is easier to say? I think once something becomes imbedded in colloquial language, it is there.

The thing that confuses the issue is the 3 types:
1) An error in reduction
2) An offset face
3) Identical pieces are exchanged
Is this what we should say instead of saying `parity`?

Either way, we know we have to seek 2 identical pieces to swap, or perhaps some other fundamental problem. Like in the case of the Master Skewb: where we need to have the correct armature for building the puzzle around (sorry, couldn’t resist this beauty, food for thought- a 4th type).

Cheers,
Burgo.

 Attachments: master-skewb.gif [ 186.98 KiB | Viewed 3473 times ]

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)
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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 8:33 am

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
Hi,

I think parity is an odd number of single permutations.

Thanks for the picture with the super 4x4x4.
A sliceturn causes an odd number of edgepermutation.
A outer turn causes an odd number of cornerpermutation, too.
So a outer turn (-1)*(-1)=+1 causes an even number of permutations.
On my picture the 4x4x4 the square-1 and the helicoptercube shows a paritiy situation.

Burgo:
Why do you think the Vulcano has a parity situation ?

There are 4 pieces and this can solved with two 3 cycles. I solve this flipping 2 paired edges turn the babyfaces and reflipp 2 paired edges.

On a megaminx each turn causes an even cornerpermutation and even edgepermutation. No parity on megaminxes.
If 2 inneredges( crazy series) are exchanged and 2 inneredges with same color that is no parity, I think.

In my helicopter-picture 2 centers in the same orbit are exchanged. This is parity, I think.

~1982 I got a 4x4x4. There was no solution , no internet. I solved the parity makeing a quartersliceturn and rearrange the centers ( with (U2r)*5 U2) and made two 3 cylces for permuting the edges.
Attachment:

parity.JPG [ 95.43 KiB | Viewed 3466 times ]

The Rex Cube shows 2 exchanged centers. This is not parity I think.
( look at the color-scheme the scheme is mirrored)

Cheers,
Andrea

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 8:40 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Burgo wrote:
The thing that confuses the issue is the 3 types:
1) An error in reduction

I usually call an error in reduction a "parity in the groups". For example, on a 4x4x4 when you are forming reduced 3x3x3 edges you can put the reduced edges into an odd permutation but the 4x4x4 wing-edges are still in an even permutation. This isn't a parity in the 4x4x4 wing-edges but it is a parity in the 3x3x3 edge groups.

I'll borrow an image from the Gelatinbrain solving thread made by Stefan Schwalbe to demonstrate:
Attachment:

444wedge_fake_parity.png [ 6.71 KiB | Viewed 3462 times ]

As you can see two pieces need to be swapped and two swaps is even. In can be done with two 3-cycles.

Burgo wrote:
2) An offset face

Can you elaborate? I can't figure out what you mean by this.

Burgo wrote:
3) Identical pieces are exchanged
I call this an "apparent parity due to indistinguishable pieces". The trouble with indistinguishable pieces is that if you swap two in the process of doing an even permutation, one of the swaps isn't visible so it looks like you did an odd permutation. The end result is that you can solve the whole puzzle into a situation that looks like it requires an odd permutation. You just have to find the two identical pieces and make sure they get swapped while you go about fixing the other two swapped pieces.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 8:49 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Andrea wrote:
In my helicopter-picture 2 centers in the same orbit are exchanged. This is parity, I think.

This is an apparent parity due to two identical pieces being swapped. It is true the parity of that orbit is odd but with jumbling moves you can put a duplicate piece in the orbit so you have two identical pieces, swap them, and then move the duplicate back out. It will change the apparent parity of the orbit.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 9:02 am

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
Hi bmenrigh,

bmenrigh wrote:
This is an apparent parity due to two identical pieces being swapped.

Cheers,
Andrea

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 9:52 am

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Hi Brandon, hi Andrea,

By `offset face` I meant: Offset by 90 deg (Like the 4x4x4 / void cube parity where you choose the wrong face (50/50) to begin the solve). I think that the Master Skewb parity is a very interesting variant on this, that could represent another type?

Andrea, the Vulcano parity depends on how you are solving it, I reduced the Vulcano to a Pyraminx (like Matt said), the situation is a parity error on a Pyraminx in the same sense as the `offset face`is a parity error on a reduced 4x4x4. I overcame it by switching one face and resolving the Pyraminx. Like your original way on the 4x4x4.

Cheers,
Burgo.

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 10:12 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
bmenrigh wrote:
Burgo wrote:
3) Identical pieces are exchanged
I call this an "apparent parity due to indistinguishable pieces". ...
I have called this a 'seeming parity'. As a non native speaker I'm not sure if apparent and seeming have a simillar or identical meaning.
Brandon, would you agree that my 4x4x4 Supercube picture above shows a 'parity' without any adjective?
@Burgo
Burgo wrote:
...By `offset face` I meant: Offset by 90 deg (Like the 4x4x4 / void cube parity where you choose the wrong face (50/50) to begin the solve). ....
I do not view this as parity on a 4x4x4. You mean the swapped 3x3x3 edges as shown in Brandon's /Stefan's diagram above, right? I agree with Brandon that this a double swap of 4x4x4 edges, no parity.

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 Post subject: Re: Puzzling ParitiesPosted: Thu Sep 29, 2011 11:02 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Brandon, would you agree that my 4x4x4 Supercube picture above shows a 'parity' without any adjective?
Indeed. You have a permutation parity in the edge-wing pieces. That's possible on a 4x4x4 because a slice turn to put the parity in the the edge wings also does two 4-cycles in the X center pieces and two 4-cycles is ODD + ODD = EVEN. You can fix the centers and still have a parity in the edge-wings.

It isn't possible to have a pure parity in the edge-wing pieces on a 5x5x5 though because in addition to the two 4-cycles in the X centers, it also does one 4-cycle in the + centers so you will end up with two + centers swapped in addition to the edge-wing pieces.

@Burgo
Burgo wrote:
...By `offset face` I meant: Offset by 90 deg (Like the 4x4x4 / void cube parity where you choose the wrong face (50/50) to begin the solve). ....
I do not view this as parity on a 4x4x4. You mean the swapped 3x3x3 edges as shown in Brandon's /Stefan's diagram above, right? I agree with Brandon that this a double swap of 4x4x4 edges, no parity.
The 4x4x4 and void cube are poor examples here. I think a much better example is TomZ's Curvy Copter II:
Attachment:

curvy_copter_ii_parity.png [ 205.9 KiB | Viewed 3399 times ]

This puzzle can have what appears to be two center pieces (the small squares) swapped and everything else solved. This arises by a similar reason to the void parity where if the outer pieces that define the color scheme are rotated relative to the core of the puzzle by an odd number of 90 degree rotations then the centers will be wrong.

The reason for this is that you can move all of the outer pieces around an axis by 90 degrees except the centers. To do so in the centers is a 4-cycle and a 4-cycle is ODD. You can do so in the corners because it is two 4-cycles and you can do so in the in the other pieces too.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Puzzling ParitiesPosted: Sun Oct 02, 2011 3:41 pm

Joined: Sun Aug 29, 2010 1:56 pm
I find there is everything said about 'puzzling parities' in this topic, so I have nothing to add here.
I only wish to add a picture of my 'Tower of Babylon', wich shows an odd parity - two marbles have to swap.

 Attachments: tower.png [ 176.77 KiB | Viewed 3284 times ]
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 Post subject: Re: Puzzling ParitiesPosted: Sun Oct 02, 2011 3:44 pm

Joined: Fri May 06, 2005 10:13 am
Location: Norway
You could make a "super" babylon tower by marking the white rings. Now parity is no longer possible!!

Per

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 Post subject: Re: Puzzling ParitiesPosted: Sun Oct 02, 2011 9:47 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Hi Stefan,
Thanks for sharing that.

I have had the same situation on my Babylon Tower, although mine occured at the bottom. I don't remember now how I overcame it but I know it involved using `both recesses`. What is the cause of it? I know I didn't exchange entire columns so it must be a single turn of the bottom layer and re-solve? Is the parity caused because there is an even number of columns and that makes an odd permutation (non-mathematician here)?

This is what I'd hoped to see in this thread.. all those interesting `situations`, (technically a parity or not), but with a bit of a discussion of what the cause of the situation is.

It would be good if there was a consensis of what we can name each type of `situation`, because I don't want to make it hard for people to post a picture here with any interesting case..
1) An error in reduction "parity in the groups".
2) A misaligned face (Void cube parity) "?"
3) Identical pieces are exchanged "apparent parity due to indistinguishable pieces".
4) Incorrect armature for building the puzzle around (as in the Master Skewb example above- which I think is a special case of type 2?)
5) An actual `parity` (with brief a descriptor of what that is).
But someone better do this `who is not me` (perhaps a mathematician ).

To tell you the truth I am still struggling to understand `completely` what a parity is in cubing. Allagem was making a case for it being a mathematical term that in layman's terms is something like the symetry of a puzzle's solve. That, to me would mean that there was no such thing as a parity error `at all` in a twisty puzzle? The puzzle is just either odd or even. But, if a parity error is just a disguise or a mask, and it is `removed` by the puzzle being `super`, then why is Konrad's Super 4x4x4 situation a parity? (Because of No2? If it is a case of No2, then why does the same case occur on the 5x5x5)? And why is it that the tower cube `2 identical centres switched` not a parity? Is it because even if `two identical pieces` are switched that the parity of the puzzle is still either odd or even (am I doing OK for a non-mathematician)?

Cheers,
Burgo.

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: Puzzling ParitiesPosted: Sun Oct 02, 2011 11:31 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Hi Burgo, I'm really struggling to understand your post but I'll try to address a few points.

The Tower of Babylon parity does indeed arise from an even number of columns. A twist of a ring is a 6-cycle and thats an ODD permutation. To solve the parity that Stefan showed you would twist the upper ring 1/6th to solve one of the swapped pieces. There 5 pieces would be broken and a 5-cycle is EVEN so you can solve it via 3-cycles. Swapping two corners on a 2x2x2 works the exact same way.

I see no reason why you couldn't exchange two columns on the Tower of Babylon. Swapping 6 pairs is an EVEN number of swaps.

I think the "error" in "parity error" is a bit misleading. There is no error, it's just yet another situation on a twisty puzzle. The word "parity" just refers to the the permutation required to get to a particular situation. If the permutation has an even number of swaps the parity is EVEN and if the permutation is an odd number of swaps the parity is ODD.

Just by definition all of the pieces in a solved puzzle are in an EVEN permutation. That is, the parity of the permutation is EVEN. If you do a move that changes the parity of the permutation then the pieces are now in an ODD permutation.

On a scrambled puzzles with pieces all of the place you can't usually tell how many swaps are needed to get the pieces back to solved so you don't know if the parity is EVEN or ODD. It isn't until you get down to just a few pieces remaining to be solved that you can see if it requires an EVEN or ODD number of swaps to finish solving them.

First a refresher. On a normal 3x3x3 when you do a quarter turn of a face you change the parity of the corners and the edges. If the edges were in an EVEN permutation, a quarter turn makes them ODD. If the corners were in had ODD parity the quarter turn makes them EVEN. Also, if you do a middle slice move on the 3x3x3 you toggle the parity of the permutation for the edges and also for the centers (relative to the corners). So when you're solving a 3x3x3 you don't really run into any problems because you are solving relative to the centers you don't have to worry about if they are in an EVEN or ODD permutation. Then, when you get to the last layer, assuming you first solve either the corners or edges first then you have some sequence to swap either two corners or two edges. It turns out though that the parity of the corners and edges are linked as long as you are solving relative to the fixed centers. If you have two corners to swap then you also have two edges to swap and vice-versa. So when you fix either the corners or edges you don't have to worry about running into an ODD permutation parity in the other type of pieces because you already resolved it.

Now lets talk a bit about the Void Cube. You would think that it would be like a 3x3x3 where the parity of the corners and edges are linked but it is not. That is because a middle-slice move does a 4-cycle of the edges (which is an ODD permutation) and because there aren't any centers to solve relative to that "unlinks" the parity of the corners and edges. Then, when you get to the end you could have two edges swapped in an ODD permutation parity. You can think about this issue a few ways. The first way is that you got the whole color scheme rotated by 90 degrees (relative to the invisible centers) and you have to fix it. The second way to look at it is just that there are no centers and a slice move is an ODD permutation. These are different ways to look at the same problem.

In the example of the Tower of Babylon, you could use a marker to draw a line down one of the columns to mark the orientation of that ring. Then if you had all of the rings aligned with 0 twist you could not be in an ODD permutation. By keeping them all aligned during the solve you would be able to make sure you are staying in an EVEN permutation and the situation where two are swapped would no occur.

You can not always eliminate the possibility of and ODD permutation parity by using super stickers. Konrad's 4x4x4 case is a good example. As I explained earlier in this thread, that would not be able to occur on a 5x5x5 because it has the + centers which would have been 4-cycled along with the edge-wing pieces. That is, the parity of the edge-wing pieces and the 5x5x5 + centers are linked. If one of the piece types in an ODD permutation than the other one is too. This is just like the edges and corners being linked on a 3x3x3. On a 4x4x4 there aren't any + centers so the edge-wings can be in an ODD permutation all by themselves. On the 5x5x5 they can only be in an ODD permutation if the + centers are also in an ODD permutation. If you solve all of the centers on the super 5x5x5 first then you know the edge-wings are already in an EVEN permutation and you won't run into Konrad's case.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Puzzling ParitiesPosted: Mon Oct 03, 2011 12:04 am

Joined: Sun Oct 08, 2006 1:47 pm
Location: Houston/San Antonio, Texas
Thank you Allagem for your long article.
I agree with most of it.
One thing, though:
Allagem wrote:
...Case 2) came about for the same reason - a single flipped edge for a 3x3x3 is impossible, so it was also called a PARITY ERROR, but when you remove the assumption and look at the puzzle as a 4x4x4 again, the edges really are in an ODD parity (a 90 degree slice move toggles the parity of both the edges and the centers, but since centers come in 6 groups of 4 indistinguishable pieces, a truly ODD permutation can be disguised as solved by swapping two identical centers)....
As you can see on my Supercube 4x4x4, this a real parity situation. All centres are correct (I assure you, I have them correct on the other three faces, as well )

-Image-

So, all corners are correct = even number of permutations
all centres are correct = even number of permutations
edges need a single 2-cycle = odd parity

The inner, invisible 2x2x2 is in a odd parity situation, but I would not count this.
Too me, this case is an odd parity situation (I would not call it an error, though. Why should it be an error? Just a very special legal situation.)

When you solve a Crazy 4x4x4 you will never see this parity situation, when you have solved th inner 2x2x2.
*facepalm* Konrad, you are absolutely correct. And the funny part is I actually knew that, the words just came out of my mouth incorrectly (or rather the text typed out of my fingers incorrectly )

Burgo wrote:
Is this what we should say instead of saying `parity`?
Hey, you can say whatever you want I already said I didn't think the original meaning was very useful. But thinking about the parities of paticular moves (or more accurately, the positions that arise as a result of combinations of those moves) of a puzzle is tremendously helpful in solving puzzles

bmenrigh wrote:
I think the "error" in "parity error" is a bit misleading. There is no error, it's just yet another situation on a twisty puzzle. The word "parity" just refers to the the permutation required to get to a particular situation. If the permutation has an even number of swaps the parity is EVEN and if the permutation is an odd number of swaps the parity is ODD.
^This!^

Like I said in my first post, the only "error" is associated with the assumption that you are working with a properly reduced puzzle (such as 4x4x4 to 3x3x3) when that assumption has, in fact, placed you in an impossible parity. So technically speaking, the original meaning was a parity "error" but only an error by an assumption.

I guess the main point to take away from all of this is that the issues arise with odd parities, or with puzzles that have moves that result in an odd parity (relative to solved that is, i.e. the move toggles the parities of the permutations of one or more orbits of pieces) but it is by no means erroneous to be odd. That would hurt the feelings of the odd permutations!

And always remember to consider puzzles where the orientation of the "core" cannot be immediately determined by looking at the outside of the puzzle. On a standard 3x3x3 or a curvy copter, you can deduce the orientation of the core by looking at the right pieces (centers and edges respectfully). However, on a void cube or a normal helicopter cube, these pieces are not visible so in general you must consider an additional available move: a rotation of the entire puzzle WITHOUT REORIENTING THE CORE OR RELABELLING ORBITS!!! So on a helicopter cube (remember to not relabel orbits! In other words, imagine you wanted to make the puzzle look like you rotated it 90 degrees, but without actually doing it. You must get all the pieces to their new locations manually - 8 centers should not move at all...) you get 2 even cycles of corners (even parity) and a grand total of 4 4-cycles of centers (one 4 cycle in each separate orbit), so the parity of every orbit of centers actually is flipped, but this can be countered by the correct edge turns, (ex. UF UB flips the parity of all 4 center orbits without flipping the parity of the corners) So really, no problems. On a void cube, a rotation of the whole puzzle gives 2 4-cycles of corners (even parity) and 3 4-cycles of edges (odd parity!) and since there is no way to counter this with other moves, it can cause huge problems later in the solve....

Peace,
Matt Galla

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 Post subject: Re: Puzzling ParitiesPosted: Mon Oct 03, 2011 2:57 pm

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
I guess you have not intended making the last part unreadable small?
Allagem wrote:
...

And always remember to consider puzzles where the orientation of the "core" cannot be immediately determined by looking at the outside of the puzzle. On a standard 3x3x3 or a curvy copter, you can deduce the orientation of the core by looking at the right pieces (centers and edges respectfully). However, on a void cube or a normal helicopter cube, these pieces are not visible so in general you must consider an additional available move: a rotation of the entire puzzle WITHOUT REORIENTING THE CORE OR RELABELLING ORBITS!!!

So on a helicopter cube (remember to not relabel orbits! In other words, imagine you wanted to make the puzzle look like you rotated it 90 degrees, but without actually doing it. You must get all the pieces to their new locations manually - 8 centers should not move at all...) you get 2 even cycles of corners (even parity) and a grand total of 4 4-cycles of centers (one 4 cycle in each separate orbit), so the parity of every orbit of centers actually is flipped, but this can be countered by the correct edge turns, (ex. UF UB flips the parity of all 4 center orbits without flipping the parity of the corners) So really, no problems. On a void cube, a rotation of the whole puzzle gives 2 4-cycles of corners (even parity) and 3 4-cycles of edges (odd parity!) and since there is no way to counter this with other moves, it can cause huge problems later in the solve....

Peace,
Matt Galla

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 Post subject: Re: Puzzling ParitiesPosted: Thu Oct 06, 2011 7:07 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Thank you so much for your replies, I have given them some time to roll around in my brain. As you know, I understood most of `what was going on`, but I was unsure of what to call it.

Quote:
when you get to the end you could have two edges swapped in an ODD permutation parity. You can think about this issue a few ways. The first way is that you got the whole color scheme rotated by 90 degrees (relative to the invisible centers) and you have to fix it. The second way to look at it is just that there are no centers and a slice move is an ODD permutation. These are different ways to look at the same problem.
Brandon, this in particular was very useful to my understanding, you see, I have been locked in the left brain I think.

I mostly operate on the right brain: I appreciate the `artistic` nature of puzzles and the `creativity` of solving them. But obviously I have a strange balance that lends me to become involved (passionately I might add) with the `extreme left`. But the way you solve puzzles is very different to me.. I need to twist them for a fair time and then, quite suddenly, an idea will go `bang` and the whole solve, beginning to end, will flash before my eyes (even on larger puzzles). You are much more calculated in your approach, breaking it down into parts and assessing their move counts. I find comparing the different solving approaches very interesting, thank you for your videos. And thanks to everyone in this post for contributing.

Cheers,
Burgo.

PS I'll leave you with the Mars Tet parity similar to the Master Skewb ^^above.

 Attachments: Mars Tetrahedron Parity.jpg [ 127.79 KiB | Viewed 3003 times ]

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

Last edited by Burgo on Sat Oct 08, 2011 7:27 pm, edited 1 time in total.
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 Post subject: Re: Puzzling ParitiesPosted: Sat Oct 08, 2011 5:56 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
I want to share another interseting parity that I just got on the Saturn Tetrahedron. I think it makes a nice pattern too (and I was tempted to put it on the pattern thread). But this is one of the reasons I started this thread, so I hope you like it..

Cheers,
Burgo.

 Attachments: Saturn Tetrahedron Parity.jpg [ 69.69 KiB | Viewed 3000 times ]

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)
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 Post subject: Re: Puzzling ParitiesPosted: Tue Jul 10, 2012 1:36 am

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
I want to make a bump here for a few reasons:
1. I believe I have something very interesting to add to the thread.
2. When I started the thread I had intended it to be available for interesting future discussion on particular parities that might come up in the future.
3. I’d expect, as more of a solving type of discussion, that people might feel free to ask future questions to do with parity on this thread.
4. If you understand parity just skip to the end for the interesting part, I am just trying to make it accessible with a preamble because it’s been a while:

Brief summary of Parity:
The origin of the term `parity` in cubing terminology comes from an incorrect assumption. When people first began to develop methods for solving the 4x4x4 cube they assumed that you could reduce it to a 3x3x3, but they came across a few problems. They called these `parity` because they were based on the assumption that the reduced 4x4x4 was a 3x3x3. But it isn’t, it’s a 4x4x4 and simply breaking the paradigm of it being a 3x3x3 and performing 4x4x4 moves can resolve the situations. So `parity` is a name that’s been `butchered` a lot, to the point where it has even been used just to describe `a difficult situation`.

What is it really in terms of twisty puzzles? A scrambled twisty puzzle will have either odd twists or even twists to a solved state. If you try to solve the puzzle by making an even amount of twists when the puzzle requires an odd amount of twists you cannot solve it! Normally you won’t be aware of the `parity` state of the puzzle until you are close to placing all of the pieces. Then you will make a decision based on `parity` even if it’s not a difficult situation.

Compare 2 ways that Sune can be used to cycle edges on a Rubiks cube: a Sune has even twists (RU R’U R U2 R’), let’s analyse it: (R R’R R’) no parity change, and (U U U2) 4 twists of the U layer returns it, even twists, no parity change, it creates a 3cycle of edges. Now add a U: (RU R’U R U2 R’ U) and you have an odd twist put into the U layer changing the parity of the puzzle and making a swap of 2 edges. This is an example of how you are considering `parity` even in an easier situation.

It’s the situations where you `choose` to set the `parity` of the puzzle at a relatively early stage (by making an action or decision- that you may not even be aware of) that create a true `parity error`.

1. 2 edges switched is not a `true parity`. The fix: Uu2 F2 R2 u2 R2 F2 Uu2 is an even permutation that just uses 444 moves, breaking the `idea of it being a 333` with the u2 twist.

2. the single flipped reduced RC edge is an actual parity error situation that requires changing the parity of the puzzle to solve it. The inner 222 is actually in an odd permutation, that’s why you don’t see `this situation` on the Circle 4x4 II, because you solve the inner 222 first.

Let’s pull apart the most common fix algo for the 444 `single flipped reduced edge`- remember from our analysis of `parity` we will expect the single twist to be in the inner 222 (so: `a 444 slice layer`, which is obvious for another reason- it’s a 444 twist not a 333 twist):

r2 B2 U2 l U2 r’ U2 r U2 F2 r F2 l’ B2 r2

F2 F2 = 0
B2 B2 = 0
U2 U2 U2 U2 = 0
l l’ = 0
r2 r’ r r r2 = r <<< there it is!

What is parity in the mixup Plus series:

A single flipped edge on the 333 Mixup (or Mixup Plus) is not a true parity. It just has to do with the nature of the Mixup cube as compared to a Rubiks cube. A single flipped edge on a mixup requires [a setup, an R2, an undo setup]. [Setup + Undo will always be even] + [R2=even], no parity change. It’s a normal part of the solve because in this case the edge is `the same as` a centre (it looks like an RC, but it’s not an RC).

A 90* twist in a single edge: is trickier, it requires a `hidden twist` but it still requires [a 90* twist in a centre] + [a 90* twist in the edge] which is the same as a 180* twist = no parity change.

There is only 1 true parity error in the Mixup 333 and it can be manifested as `2 RC edges swapped` or `2 corners swapped` (or 2 swapped centres or an edge and centre swapped- although I can’t imagine a solve where you would end up with this). To understand how this is the same situation, try this:
1. physically remove 2 edges at UL and UR on your Rubiks Cube and swap their places, keep the orientation of the U layer preserved (this creates parity that you can't remove by twisting).
2. perform (R2 U R2 U' R2) U'D (R2 U' R2 U R2) D'
If you perform this a few times it will just demonstrate to you that the parity is the same thing in edges or corners, by exchanging from one situation to the other.

The `twist` that is causing this true parity error is a Mixup + or - twist in a slice layer. The parity error has occurred in the moment of choosing `the armature of the puzzle`. The moment you chose `the Mixup's parity` is when you line up the E layer `relative to the placed` D face edges+centre positions and say: I'm putting an E layer `centre or edge` HERE, when it could go `next door`. And the manifestation is in the 2 RC edge switch or 2 corner switch later.

A very interesting parity situation in the Mixup Plus series:

Andrea has made an interesting post in the Patterns thread here: viewtopic.php?f=8&t=16088&start=474
It demonstrates `parity` and `an impossible situation on the RC`, it’s an amusing situation! Thanks for sharing it Andrea.

I want to write about another very amusing thing in the Mixup Plus 444:

In the Mixup Plus 444 you can have a single edge flipped which is a parity on the 444, but not on the Mixup Plus 333!! So if you solve it as a reduced Mixup333 (or better said: using Mixup twists) you don’t have parity.

The 2 edges swapped situation is a parity on the Mixup 333, but it is not a parity on a 444 Cube!! This is really cool.

You could solve the Mixup Plus 444 seeing 2 parities or only seeing 1 or even NONE at all, and as such really: it has none, even though the most clever solve is to use the 444 `single edge flip` parity fix (which is a real parity change, but it's not needed).

Mind you, I’m not claiming to be an expert on this, I’m just calling it as I see it and sharing what I `think` I know.
I just wanted to share that.. don’t you just love puzzles.

Cheers,
Burgo.

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: Puzzling ParitiesPosted: Tue Jul 10, 2012 10:05 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Hi Burgo,

thank you for sharing your thoughts.

I think it is not easy to see that the case of two swapped 4x4x4 edges (flipped Rubik's Cube edge) is a true parity in the mathematical sense.
It needs a 4x4x4 Supercube to convince youself.
The same situation on a 5x5x5 is not a true parity, because centres are swapped as well.
(odd parity = an odd number of transpositions of pieces ("transposition" is the mathematical term for swapping two pieces on a puzzle) is needed to reach the intended `solved state`)

I'm still uncertain how mathemticians will judge the Mixup Plus parity of two swapped RC edges:

On the Mixup Plus an E+ turn means
1. 4-cycle of outer edges = odd number of permutations within this set of elements
2. 4-cycle of centres = odd number of permutations
3. 8-cycle of inner edges = odd number of permutations
This indicates an odd number of permutations of the three sets of elements together.
Is this view correct?
I can view the centres and outer edges as one set of elements.
In this view I count over all twice an odd number of permutations, which is overall an even number of permutations.

On the Rubik's Cube I can have an odd parity in the set of edges only, if at the same time I have an odd parity in the set of corners.
Would that not indicate an overall even parity in the case of the Mixup Plus parity?
BTW, if we look at DKwan's post of a perfect swap
Quote:
(Rw E' S+ U2 S- E Rw' E+) x8 Rw U2 Rw' (perfect swap UL with FR)
we recognize an overall even move count.

I would need a Super Mixup Plus (to distinguish the inner edges) to convince myself that we can view the discussed case as a a `true parity`.

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 Post subject: Re: Puzzling ParitiesPosted: Tue Jul 10, 2012 11:33 am

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany

Thank you for the picture of 4x4x4 supercube.

It needs a 4x4x4 Supercube to convince youself.
The same situation on a 5x5x5 is not a true parity, because centres are swapped as well.

Yes. There are 2 definitions of parity. One is a odd number of single permutations. The other is, that you must destroy a shape or subgroup to solve the situation.

Burgo: thank you for the pictures. I think this is a different definition of parity. The 3 corners ( skewb, tetrahedron, etc ) causes an even permutation. But It's parity, too. Is this correct ?

On 4x4x4 is the parity solution the slice turn r.
4 cycle of edges are odd parity.
2 4 cycles of center permutation is solveable with 3 cycles.
So it's possible to permute 2 edges without change the centers.
On 5x5x5 there is a middle-center, too. This causes one more 4 cycle, odd permutation.
So more centers must be permuted to swap 2 edges.
On Mixup Plus turning one edge is like turnung a center on Rubik's Cube.

I don't know if is possible to exchange one center with one edge.

Cheers,
Andrea

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 Post subject: Re: Puzzling ParitiesPosted: Tue Jul 10, 2012 1:28 pm

Joined: Sun Jun 13, 2010 1:00 am
FF Siamese cubes can get some scary parity of two corners being swapped and/or two edges being swapped.

Like with other cases this is actually caused by the 6 edge pieces and their clones. Making the puzzle a super cube fixes this. I'll need to check but I think it can get center piece parity too for the same reason.

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 Post subject: Re: Puzzling ParitiesPosted: Tue Jul 10, 2012 10:07 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
BTW, if we look at DKwan's post of a perfect swap
DKwan wrote:
(Rw E' S+ U2 S- E Rw' E+) x8 Rw U2 Rw' (perfect swap UL with FR)

we recognize an overall even move count.
DKwan made a typo: It should read (Rw E' S+ U2 S- E Rw' E+) x9 Rw U2 Rw' (perfect swap UL with FR)
So you can see the extra move of E+
I would need a Super Mixup Plus (to distinguish the inner edges) to convince myself that we can view the discussed case as a a `true parity`.
Then we must break out the stickers. Still, not a bad idea, I like the idea of a super Mixup Plus. It removes one of the mysteries (the single edge flipped), but we know about it anyway.

That is the funny thing about the Void Mixup: it turns the Mixup parity into a Void Cube parity.

I'm interested to see what the Mathematicians say too Konrad, I'm not one, and I'm pretty much at the edge of my understanding of these things on this thread.
Andrea wrote:
On 4x4x4 is the parity solution the slice turn r.
Yes, Philip Marshall used a Dd’ twist and rebuild to achieve the same in the 444Ultimate Solution.
Andrea wrote:
I don't know if is possible to exchange one center with one edge.
I included a photo to show this interesting manifestation of parity.
PuzzleMaster6262 wrote:
FF Siamese cubes can get some scary parity of two corners being swapped and/or two edges being swapped.
Hi Mike, Which type of Siamese Cubes do you mean. Can we see a picture? I only have the Rubik’s Mate.
I don't think it fits into the definition of `true parity error` to have indistinguishable pieces swapped causing an `apparent parity`. Sorry to be picky, but it's been the nature of this thread to make those distinctions. I still think we should post about `apparent parity due to indistinguishable pieces` here, but we should make the distinction (like you have).

 Attachments: edge corner swap.jpg [ 123.19 KiB | Viewed 1877 times ]

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