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 Post subject: Avoiding parity on the barrel cube
PostPosted: Tue Apr 09, 2013 9:19 pm 
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I had a Rubik's barrel type (octoganal prism) puzzle when I was a kid and remember it having the same color on all four truncated surfaces. I solved the top layer, then the bottom layer and didn't bother with any F2L or really any middle layer thinking at all since the color scheme made all that irrelevant. That puzzle is long gone, lost in the mess of life somewhere.

Fast forward to 2013 and I find both the Maru Barrel cube (which I recently bought) and simulators online that have truncated surfaces of four different colors, making a ten-colored puzzle! Out of habit I solve the ten-colored puzzle the same way I solved the seven-colored puzzle from the 80s and now find strange parity issues that I never saw. (These are well-known and already discussed in other threads.)

Curiously, in the way I solve the barrel, the parity manifests itself in the middle layer with two un-exchangable edges (with the rest of the puzzle fully solved).

Attachment:
BARRELPARITY.jpg
BARRELPARITY.jpg [ 27.43 KiB | Viewed 1492 times ]


It's not hard to undo the parity but it got me thinking about how helpless one is facing parity on other puzzles: it cannot be predicted or avoided in almost all situations. But on the ten-colored barrel puzzle, one can intentionally select the arrangement of the truncated sides to avoid the parity error.

There are 24 possible arrangements of the truncated edges around the barrel; only 12 of them have "even" parity and correspond to a solved state The rest I'll call "odd" parity.

So how would one intentionally avoid the "odd" parity? I tried picking one known "even" state and decided to memorize the color order of the truncated sides (Silver, Lime, Pink, Black) and simply decided to always use that state. Well that worked, but was inconvenient if I already had a good position for one of the other 11 solved states.

I started noticing a pattern in the "even" states and "odd" states. Starting from the 12-oclock position (just pick one of the non-truncated edges as 12-oclock, on the Maru cube I used the white-blue edge on U) and going counter-clockwise, here are the 12 "even" parities on the Maru cube that correspond to a solved state:

Code:
silver  lime    pink    black
silver  black   lime    pink
silver  pink    black   lime
black   pink    lime    silver
black   lime    silver  pink
black   silver  pink    lime
lime    silver  black   pink
lime    black   pink    silver
lime    pink    silver  black
pink    lime    black   silver
pink    black   silver  lime
pink    silver  lime    black


If you have a non-Maru puzzle, just think of these as ABCD and substitute A B C D for any colors you have on your puzzle.

I'm not one to memorize this kind of chart so I started to look for patterns.

Things one notices:
-If ABCD is an even parity that ABDC (the last two swapped) is odd. In fact if you're curious what the 12 "odd" states are just take any of the "even" states above and swap the last two colors.
-If ABCD is even parity then BCDA is odd (take the A off the front and put it at the end).

But memorizing the even parity rules are more interesting:
1. If ABCD is even then CDAB is even. (last two before first two - this is equvalent to starting the sequence from the 6-o'clock position instead)
2. If ABCD is even then DCBA is even (reversed order)
3. If ABCD is even then DBAC is even (this one is harder to perceive but will come to you if you stare at the chart for long enough)

...and combinations of the above rules to derive other combinations.

Honestly rules 1 and 2 are the only ones practical to quickly apply. So in the example given above for Silver Lime Pink Black:
SLPB (memorized)
PBSL by rule 1
BPLS by rule 2
LSBP by rules 1 and 2

and one can see that now I have one "even" parity state starting with each color. In fact, each of the four "even" states above have one color in all of the positions so no matter which color you decide to start with at any of the clock positions one may proceed. So there, just memorize one "even" parity state and two rules and you can avoid "odd" parity completely on the barrel cube.

A long post for a barrel cube, I know, but avoiding parity is kind of like the holy grail of twisty puzzles so I thought it might be interesting. I wonder if anyone else has any methods to avoid parity on this puzzle, or other puzzles.


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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Tue Apr 09, 2013 9:38 pm 
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I'm no expert, and correct me if I'm wrong, but I believe making your final move count on a puzzle "even" or "odd" or a multiple of 2, 3, 4 etc. can create or avoid parity issues (obviously this mainly depends on the puzzle). In other words parities are related to the move count. Am i right?

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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Tue Apr 09, 2013 11:06 pm 
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Hello rplass

You and benpuzzles indeed have the right idea. Parity is a property of permutations that many experienced solvers on the forum know a great deal about. Basically every permutation of n elements is either of even or odd parity. The key point here is that a permutation cannot be both, it is either one or the other. There is also a (somewhat) quick way of telling whether or not a given permutation is even or odd. A given permutation has even parity if it takes an even number of swaps to change the permutation to the solved permutation. It has odd parity if it takes an odd number of swaps. For example: 4321 is even because:
4321 -> 3421 -> 3412 -> 3142 -> 1342 -> 1324 -> 1234 took an even number (6) of swaps. Actually you can do better:
4321 -> 1324 -> 1234 Notice this i still even (2). No matter how you change this permutation it must take an even number of swaps. See http://en.wikipedia.org/wiki/Parity_of_a_permutation if you want a more in depth explanation.

Using this as a tool, we can identify permutations that are impossible to reach on various puzzles. Your example is bit tricky because parity is playing into this situation a little differently than most situations. On a Rubik's Cube (or barrel), there is only one type of move: rotating a face by 90 degrees. Note that this move send the edge 1234 to 4123 and the corners ABCD to DABC. Notice that both of these permutations are odd. Therefore every time a 90 degree move is made on the Rubik's cube, if the edges were in an even parity permutation, they get sent to an odd parity permutation, and the same goes with the corners. It isn't hard to see that if the edges and corners were already in odd parity permutations, they get sent back to even parity permutations. Just like adding odd and even numbers, an odd parity applied to an odd parity will result in even and applied to an even parity result in an odd parity. Note that we can now prove that not all permutations of edges and corners are achievable on a Rubik's Cube. In the solved state, both the edges and corners are in even parities (they require 0 swaps to fix). Every time we make a move, BOTH parities flip. Therefore it is impossible to have the edges in an even parity and the corners in an odd parity at the same time. This is why it is impossible to have everything solved on a Rubik's cube except to corners swapped with each other. (Note ORIENTATION is a completely different matter, we're only discussing the locations of the pieces here)

You can do this analysis on many different puzzles and get results unique to each puzzle. A megaminx has only one type of move that preserves the parity of both piece-types so only even permutations are possible. A helicopter cube using 180 degree only turns is an interesting case because it has only 2 piece-types but the centers come in 4 orbits, so it is more useful to examine the parity changes on not just the different piece types, but the different orbits to. In that case, every move flips the parity of the corners and 2/4 center orbits. Careful analysis of these parities will explain why i is possible to swap 2 corners on a Helicopter Cube using only 180 degree turns (something which is near trivial if jumbling moves are allowed). Another strange case is the Void Cube. We claimed that the Rubik's Cube has only one move type, but this meant we held the centers fixed and knew which color was to end up where - in other words we had full knowledge of exactly what permutation we were looking at. With the void cube, you must guess where the centers go. This is equivalent to allowing global rotations of the puzzle WITHOUT moving the centers as a second move type (because you naturally consider the centers to be solved no matter what, since you can't see them). This can actually be constructed from rotating two opposite sides and the center slice. The two opposite sides we know flip the parities of both the edges and corners twice, so the new move, rotating the center slice, sends the edges in the middle layer from 1234 to 4123, an odd parity without flipping the parity of the corners. Therefore it IS possible to have two edges swapped on a Void Cube. This is caused by having a global rotation of the puzzle without moving the centers. Fix this by rotating the whole cube 90 degrees in any direction then solving it back to the color scheme you were going for previously without using slice moves (alternatively rotate any center slice 90 degrees and solve from there without using slice moves, assuming the corners are correct)

I introduce the Void Cube because the barrel cube has a similar problem. There are no clues as to what order the edge colors go in when solved, similar to how there are no clues as to which "center" is which on the Rubik's cube, even though mathematically, only certain patterns will ultimately work. In the case of the barrel, by allowing any order of the edge colors, you are allowing a magical operation that will swap all 3 pieces of one edge with all 3 pieces of another edge (going from the solved state to another potentially correct solved state). Notice that this move swaps to pairs of corners (1234 -> 2143 is even!) but only one pair of edges. The corner permutation keeps its old parity while the edge parity is reversed. This is why you will see situations where only two edges are swapped when you attempt to solve the puzzle into a state where the order of the edge colors are different from intended. Notice, by the way, that if you label one of the legal edge orderings as correct, then getting to all of the other legal orderings requires an odd number of swaps between the edge colors, each of which flips the parity of the edges only. Doing an odd number of these swaps causes the parity of the edges to flip and thus only even swaps, even parity permutations of the edge colors, if you will, result in solvable positions.

So to answer your question, yes it is always possible to avoid parity problems on every puzzle. Just imagine in your head what you want the solved state to look like and count how many individual swaps it would take to get all the pieces on the puzzle to their correct spots in the solved state. This gives you the current parities of the permutations of each orbit. Then look at the available moves, if some combination of the available moves will produce the correct pattern of parities among the orbits, the solution you have visualized in your head is indeed legal and you will not encounter parity problems. However most people don't bother doing this because it is very easy to lose track of how many swaps it would take and it is typically quicker to just guess and correct it later when the permutation in question is close enough to solved that counting the number of swaps is trivial (usually just a single swap)

I should probably point out that this is not the usual way parities are used to solve puzzles. The "parities" people talk about when attempting to solve puzzles usually comes from the fact that they have the correct solution envisioned but they didn't analyze the effects on parities of each move type or determine the current parities of the orbits beforehand and they have come across a situation that needs to use some or maybe all of the move types an odd number of times to correct the parities of all orbits simultaneously, which violates the rules of the method of using conjugates to construct commutators that affect select pieces that so many solvers use. This sometimes results from assuming the incorrect parity of a given orbit because the pictured solution actually had two identical pieces swapped leading to an incorrect conclusion that the parity of that orbit in the supposed solved state is even. The puzzle itself can still be solved into the exact position you have pictured, just not without flipping the parity of one or more of the orbits.

Did that help? :)

Peace,
Matt Galla

tl;dr: For the barrel cube, some positions that would look solved actually have sets of parities among the piece types that are impossible to reach. The parities in this case can be avoided by counting the number of swaps from a scrambled state to the proposed solve state and making sure that the available moves can indeed create this set of parities.

For some other puzzles with situations that people may call a "parity", some positions that are very close to solved have an odd parity in the permutation of one or more of the piece types. These puzzles are solvable so do not fit in the first situation, but require a flip in the parity of the permutation of the appropriate piece orbits. The only reason this is considered an issue is because solvers are constructing commutators to move select pieces and commutators require an even number of moves of all move types, keeping the parity fixed. This can be avoided by again counting the number of swaps from a scrambled puzzle to the solved state and make the necessary moves before hand to flip all parities to even. This typically requires longer time than it actually takes to guess that all parities are correct and moving forward until you reach a point where you can easily tell the parities are indeed odd and correcting for this later

To make things extra confusing, the Rubik's Barrel actually has both types, but you have evidently figured out how to deal with the second one (even if you didn't realize you were!), so I attempted to discuss the first one more. Just FYI most people actually mean the second case when they talk about parity so this conversation has the potential to get extremely confusing :)


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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Wed Apr 10, 2013 6:24 am 
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I have learnt a lot from past parity discussions, e.g. in this thread and especially from Matt's reponses. :) Still, I want to add my two cents here. (OK, at the end it is closer to two dollars :lol: )
I guess cubers still tend to use the word parity in a pretty sloppy way pretty often. (Some days ago, somebody talked abou a "parity of 3" in a different thread.)

Quote from the earlier thread:
Konrad wrote:
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. ...
Allagem wrote:
...For example: 4321 is even because:
For clarity, we could add here that the original, fixed ordering is 1234. We count transpositions of elements (=swaps of pieces) depending on the target configuration. The usual target for twisty puzzling is a solved state or a pattern.
Allagem wrote:
Notice, by the way, that if you label one of the legal edge orderings as correct, then getting to all of the other legal orderings requires an odd number of swaps between the edge colors, each of which flips the parity of the edges only.
Is this a typo and "odd" should be "even"?
Allagem wrote:
Doing an odd number of these swaps causes the parity of the edges to flip and thus only even swaps, even parity permutations of the edge colors, if you will, result in solvable positions.
The "even" here seems to be contradicting with the "odd" above. Going from one solved states of the Barrel to another means an even count of swaps of edge colours as in your last sentence.
Allagem wrote:
I should probably point out that ....
Was this font size intended? I could not read it and had to enlarge it first. :wink:
Allagem wrote:
...tl;dr: For the barrel cube,
Was tl;dr: a typo or what does it mean?
Allagem wrote:
some positions that would look solved actually have sets of parities among the piece types that are impossible to reach. The parities in this case can be avoided by counting the number of swaps from a scrambled state to the proposed solve state and making sure that the available moves can indeed create this set of parities.

For some other puzzles with situations that people may call a "parity", some positions that are very close to solved have an odd parity in the permutation of one or more of the piece types. These puzzles are solvable so do not fit in the first situation, but require a flip in the parity of the permutation of the appropriate piece orbits. The only reason this is considered an issue is because solvers are constructing commutators to move select pieces and commutators require an even number of moves of all move types, keeping the parity fixed. This can be avoided by again counting the number of swaps from a scrambled puzzle to the solved state and make the necessary moves before hand to flip all parities to even. This typically requires longer time than it actually takes to guess that all parities are correct and moving forward until you reach a point where you can easily tell the parities are indeed odd and correcting for this later

To make things extra confusing, the Rubik's Barrel actually has both types, but you have evidently figured out how to deal with the second one (even if you didn't realize you were!), so I attempted to discuss the first one more. Just FYI most people actually mean the second case when they talk about parity so this conversation has the potential to get extremely confusing :)
A good example of a situation where twisty puzzlers talk about "parity", while there is none is the swapped edges situation on a 4x4x4
Image
If you view this as a reduced 3x3x3 (each pair of edges glued together) this would be an odd parity - unreachable by legal turns starting from a solved state.
Actually, it is a double swap of single edge pieces of a 4x4x4. And 2 swaps = 2 transpositions mean even parity.

I know only one real "parity" (meaning "odd parity"; a situation requiring an odd number of transpositions to reach the solved state), a swap of two edges on a 4x4x4. My picture shows it on a Supercube and you can see that no other piece type is permuted.
Image
On the reduced 3x3x3 it would be "wrong orientation of one edge", what it is indeed "swap of one pair of 4x4x4 edges".

My last picture shows odd parity of two piece groups of the 3x3x3:
Image
The overall count of transpositions needed to reach the solved state is even: odd + odd = even.
I guess, Matt means this when he says
Quote:
but you have evidently figured out how to deal with the second one (even if you didn't realize you were!),

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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Thu Apr 11, 2013 5:11 am 
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Konrad wrote:
Quote from the earlier thread:
Konrad wrote:
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. ...

Actually this definition is equivalent to the one I used, just not very helpful in this case. I think what Tom was referring to is when a solver says "I got a parity/ parity error" when only two pieces are swapped near the end of a puzzle solve. That sentence really makes no sense. In some cases they have one or more orbits of pieces that is currently in an odd parity and cannot be solve by pure commutators. In other cases, they are just plain wrong and have a 3-cycle (even parity) with two identical pieces and a third piece. In yet other cases they have a single piece rotated. This can only happen on certain puzzles and is a completely different concept. In the first case, there is a situation involving an odd parity so they are close, but simply saying I have parity is just silly. Every permutation has a parity - either an even or an odd one.
The original origin of the term "parity error" actually WAS used correctly, but many cubers have incorrectly adopted that term to mean anything really close to solved :lol: . A parity error originally meant one was solving a 4x4x4 cube by reducing to a 3x3x3 cube but the reduced 3x3x3 had an even parity in the corners but an odd parity in the edges (or vice-versa). Since the edges and corners of a 3x3x3 always flip parity together, this will never occur naturally on a 3x3x3. You have reduced the 4x4x4 to an illegal 3x3x3 position because there is an error in one of the parities under the reduction assumption. Hence, parity error. In the 4x4x4 world, this is actually a double swap among the edges which is even parity. The term really only applies to the fact that you would like to treat the puzzle as a 3x3x3. I want to emphasize that I DO believe it is useful to talk about odd parities in orbits for other puzzle, even if that was not the reason the terminology originally came up.
Konrad wrote:
Allagem wrote:
Notice, by the way, that if you label one of the legal edge orderings as correct, then getting to all of the other legal orderings requires an odd number of swaps between the edge colors, each of which flips the parity of the edges only.
Is this a typo and "odd" should be "even"?
Allagem wrote:
Doing an odd number of these swaps causes the parity of the edges to flip and thus only even swaps, even parity permutations of the edge colors, if you will, result in solvable positions.
The "even" here seems to be contradicting with the "odd" above. Going from one solved states of the Barrel to another means an even count of swaps of edge colours as in your last sentence.

ARG! Yes :roll: This is one of those cases where I started out saying one thing, but then decided it would be clearer to say the contrapositive, but forgot to edit what I had already said. Let me try again:
If you label one of the legal edge orderings as correct then getting to the ILLEGAL ordering requires an odd number of swaps between the edge colors. Each of these swaps flips the parity of the edges only. Since 3x3x3 moves can only flip the parities of the edges and corners simultaneously, having the edge parity flipped without the corner parity flipped (ie odd edge parity and even corner parity) cannot be solved and this explains why these orderings are illegal (i.e. unsolvable/not in the group). The second quote from me is correct as is
Konrad wrote:
Allagem wrote:
I should probably point out that ....
Was this font size intended? I could not read it and had to enlarge it first. :wink:

8-) Yes, I love making text that isn't directly pertinent smaller for three reasons: 1) It allows for readers to skip over it without missing the points I'm trying to make especially if they are only interested in the specific question I'm explaining. 2) If I think someone might try to cause confusion by adding some points to clear up technical dilemmas, I can prove that I was aware of the ambiguities or debates or whatever the case may be :P and 3) It's fun! Consider it a technical footnote :)
Konrad wrote:
Was tl;dr: a typo or what does it mean?

tl;dr means too long didn't read. I tend to type way to much :( This is basically the short answer for anyone who doesn't want to read my long posts (I love math too much!)
Konrad wrote:
A good example of a situation where twisty puzzlers talk about "parity", while there is none is the swapped edges situation on a 4x4x4
Image
If you view this as a reduced 3x3x3 (each pair of edges glued together) this would be an odd parity - unreachable by legal turns starting from a solved state.
Actually, it is a double swap of single edge pieces of a 4x4x4. And 2 swaps = 2 transpositions mean even parity.

Yes you have said this all correctly. Like I said above, THIS is the originally "parity error". The 3x3x3 has a parity error. The 4x4x4 is in fact an even parity which isn't even a problem. Funny how in the original usage this is the only correct case, but in the evolved usage it is an incorrect case even though parity is a well defined concept. It's the term "parity error" that is ill defined :wink:

Konrad wrote:
I know only one real "parity" (meaning "odd parity"; a situation requiring an odd number of transpositions to reach the solved state), a swap of two edges on a 4x4x4.

Yes. I actually like using the word parity in this way, though as Tom points out, calling this a parity doesn't actually make sense. That's like holding a ruler and saying "this is a distance". Well I guess it DOES make sense but it is a worthless observation... Anyway... Yes! A (super) 4x4x4 has 3 piece types, each of which have only one orbit so 3 orbits. An outer layer 90 degree move flips the parity of the corners and centers but maintains the parity of the edges. An inner layer 90 degree move flips the parity of the edges and maintains the other two. Therefore all legal permutations of a super 4x4x4 must have the parity of the centers and corners the same, but the parity of the edges is free to flip back and forth independent of the other two piece types. I'm sure you already know this but if you rotate one inner layer (I mean slice move here) 90 degrees that will flip the parity of the edges, making all 3 parities even, and allowing the puzzle to be solved from there with commutators.

Other puzzles do have real parities! (See even I want to use the term this way! I mean legal states where some of the parities are odd) Off the top of my head: Helicopter Cube, Super X, Dayan Gems 1-3, and lots more! Basically any puzzle where a face can stop in an even number of positions has the potential for parity flips (precisely why the Megaminx has no parity-related things whatsoever). Technically the 2x2x2 and 3x3x3 have "parities" too, it's just people tend to not think of them that way.
Konrad wrote:
My last picture shows odd parity of two piece groups of the 3x3x3:
Image
The overall count of transpositions needed to reach the solved state is even: odd + odd = even.
I guess, Matt means this when he says
Quote:
but you have evidently figured out how to deal with the second one (even if you didn't realize you were!),

Well yes this is the case I meant. I don't believe that you don't understand this, but I'd like to put it in my own words just in case. This situation is technically a parity in the same way that a 4x4x4 with a single pair of swapped edges is a parity. I see what you are saying by adding the parities together, but I don't think that calculation has much meaning. After all, edges are edges and corners are corners. Edges cannot become corners and corners cannot become edges so the permutation of the combined 20 pieces really loses alot of information. Useful concepts come out when you consider the parity of every separate piece-type (or more specifically orbit in the case of some more complicated puzzles like 180-only Helicopter Cube). In this case the corners have an odd parity. That is different from the solved even parity so the corners are in the wrong parity. The edges are also in an odd parity so they are different from the solved even parity. Both piece types have the incorrect parity so we need to fix both. Fortunately a single move does just that. That is why rotating a face 90 degrees and then solving with pure commutators is possible. You can also guarantee that any solution to the above image will require an ODD number of 90 degree face turns :wink:

Peace (<- does this make me sound too hippy? :shock: I really do like promoting peace and I mean to indicate that I'm never frustrated or mad at anyone!)
Matt Galla


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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Thu Apr 11, 2013 6:21 am 
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Allagem wrote:
...
Konrad wrote:
My last picture shows odd parity of two piece groups of the 3x3x3:
Image
The overall count of transpositions needed to reach the solved state is even: odd + odd = even.
I guess, Matt means this when he says
Quote:
but you have evidently figured out how to deal with the second one (even if you didn't realize you were!),

Well yes this is the case I meant. I don't believe that you don't understand this, but I'd like to put it in my own words just in case. This situation is technically a parity in the same way that a 4x4x4 with a single pair of swapped edges is a parity. I see what you are saying by adding the parities together, but I don't think that calculation has much meaning. After all, edges are edges and corners are corners. Edges cannot become corners and corners cannot become edges so the permutation of the combined 20 pieces really loses alot of information. Useful concepts come out when you consider the parity of every separate piece-type (or more specifically orbit in the case of some more complicated puzzles like 180-only Helicopter Cube). In this case the corners have an odd parity. That is different from the solved even parity so the corners are in the wrong parity. The edges are also in an odd parity so they are different from the solved even parity. Both piece types have the incorrect parity so we need to fix both. Fortunately a single move does just that. That is why rotating a face 90 degrees and then solving with pure commutators is possible. You can also guarantee that any solution to the above image will require an ODD number of 90 degree face turns :wink:

Peace (<- does this make me sound too hippy? :shock: I really do like promoting peace and I mean to indicate that I'm never frustrated or mad at anyone!)
Matt Galla
Yeah, my odd + odd is not really meaningful. What I meant indeed is phrased in your sentences highlighted above.

BTW :wink: :
Regarding peace: I'm all for peace myself. Just the "never" is something I cannot claim for myself. :roll: E.g. I'm not tolerant against intolerant people. But this is tricky, because I define which intolerance I consider being harmful.

Regarding peace for the whole mankind, it is a wonderful utopia. Looking back at the history of the last thousand years, mankind may need another millenium.

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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Thu Apr 11, 2013 10:08 am 
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I think the only way to really avoid any parity issues is to mark ambiguous parts so that you know the correct position and orientation, otherwise it's always going to be a matter of chance (50/50 for edge pieces, 33/67 for corners).

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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Thu Apr 11, 2013 5:50 pm 
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Kevlin S, reminds me of the Superstar with a 6-color sticker configuration, where the owner placed a black dot on the center of each of the stickers of one of the pairs of edges. I suspect that would avoid the parity issue.

http://twistypuzzles.com/forum/viewtopic.php?p=283547#p283547

And to the other posters who have given me quite an education, thanks so much for your fascinating responses!


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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Thu Apr 11, 2013 6:07 pm 
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Makes sense, parity is nothing more than ambiguity, so the only way to resolve/remove that ambiguity is to mark ambiguous parts and configurations so that they can be distinguished.

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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Fri Apr 12, 2013 4:32 am 
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Matt has given the answer above:
Allagem wrote:
So to answer your question, yes it is always possible to avoid parity problems on every puzzle. Just imagine in your head what you want the solved state to look like and count how many individual swaps it would take to get all the pieces on the puzzle to their correct spots in the solved state. This gives you the current parities of the permutations of each orbit. Then look at the available moves, if some combination of the available moves will produce the correct pattern of parities among the orbits, the solution you have visualized in your head is indeed legal and you will not encounter parity problems. However most people don't bother doing this because it is very easy to lose track of how many swaps it would take and it is typically quicker to just guess and correct it later when the permutation in question is close enough to solved that counting the number of swaps is trivial (usually just a single swap)
So, yes you can avoid parity problems by comparing the current scrambled state with the intended solution and do a careful analysis. The real answer is in the higlighted sentence. E.g. on the Void Cube, I correct a single swap of edges on an otherwise solved cube by doing 13 moves.
Probably parity avoidance by analyzing the initial scrambled situation would take much, much longer.

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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Fri Apr 12, 2013 11:10 am 
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Yes, in *theory* one could do this, but in practice one would have to be a savant to do all this in their head!

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 Post subject: Re: Avoiding parity on the barrel cube
PostPosted: Fri Apr 12, 2013 12:13 pm 
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KelvinS wrote:
Yes, in *theory* one could do this, but in practice one would have to be a savant to do all this in their head!

Actually it isn't hard for a small puzzle like the Rubik's cube. I can do it for the edges or the corners of the 3x3x3 and I'm no savant -- many folks on this forum have much more powerful mental machinery. What I can't track is the flip of the edges.

I think I could probably track parity for 30 edges but it would take me a few minutes.

My strategy for the cube is to first mentally swap the 4 U-face edges into place. Then I swap the 4 D-face edges in place. It's pretty easy to track the chain of swaps to figure out where the D-face edges are now located after swapping the U-face edges. The middle-layer edges take most of the mental work but again, they aren't very hard to track.

Every time you do a swap you toggle the parity. I use a finger pointing out or not for the parity.

I think doing this parity calculation is much simpler than a blind-folded solve because that has to track twist too.

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