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 Post subject: Helicopter cube solution
PostPosted: Sat Feb 11, 2012 4:13 pm 
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I'm just posting my helicopter cube solution since it's a little different from ones I've seen. Besides that I encountered a problem which the Web failed to enlighten me on. I'm pretty certain that it's possible to trade two corners *without* using jumbling moves, because I've had two corners traded on a computerized copter cube which didn't even allow jumbling. Yet I haven't figured out how to trade just two corners without jumbling. If anyone knows how to do this it would be nice!

My general method, unlike others I've seen, is to solve the bottom layer and then immediately solve all the centers, followed by the top corners. The algorithms I use are:

180 turn of an edge is simply called, eg, UF for edge shared by upper & front face; clockwise jumbling turns are called, eg, UF1.

Changing a center piece's "orbit": UR1 FL1 UF and then return to square (UR1' FL1'). This exchanges the two center pieces sitting to the left and right of the edge UF, but additionally flips some pieces of UF around its axis.

Corner orientation: I can rotate three corners clockwise without affecting anything else by (UF UR UB UL)^3. The corner which is carried around the whole time, ULF, is the unaffected one. Obviously when one corner needs reoriented and I don't mind messing up the layer I use a single UF UR UB UL.

Permuting three corners: Exchanging all but the ULB corner can be done by (UF UL UB UL)^2.

Permuting two corners: Doing the orbit change move and then doing its mirror image orbit change flips everything on UF but the corners; so UR1 FL1 UF UR1' FL1' UL1 FR1 UF UL1' FR1' UF will do nothing but exchange two corners.

All these moves are ones I did not see elsewhere and feel 'aesthetic' to me (I prefer pleasant/symmetrical solving methods to fast ones).

EDIT: Well some or all of these algs are present in youtube tutorials. An internet traditionalist like myself is slower to notice when stuff isn't in plain text :P It seems to me the popularity of youtube makes searching slower since it's harder with videos to skip to the part you want but I suppose search technology will eventuall catch up and videos will have automated transcriptions and other metadata to make things easier.

Oops I ranted a bit. Anyway I still don't know how two corners can be traded without jumbling! If anyone does that would be cool.


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 Post subject: Re: Helicopter cube solution
PostPosted: Sun Feb 12, 2012 8:44 pm 
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It's definitely possible to swap two corners without jumbling. It's just a lot harder. :)

Well, technically, it's not possible to just swap two corners without jumbling. What's really happening is that the four face centers around the equator where the two swapped corners are have been rotated 90° from their "real" solved position. It's impossible to tell visually because there's no fixed center of reference, so it looks like the faces are correct but actually 4 of the faces are 90° from where they're supposed to be. So to fix this, move the 4 face centers 90° from where they are, then re-solve the corners.


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 Post subject: Re: Helicopter cube solution
PostPosted: Sun Feb 12, 2012 9:58 pm 
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quickfur wrote:
What's really happening is that the four face centers around the equator where the two swapped corners are have been rotated 90° from their "real" solved position. It's impossible to tell visually because there's no fixed center of reference, so it looks like the faces are correct but actually 4 of the faces are 90° from where they're supposed to be. So to fix this, move the 4 face centers 90° from where they are, then re-solve the corners.


That's actually incorrect. A rotation of the entire puzzle 90° about a face (i.e. moving 4 face centers each 90° over) results in all 24 face pieces switching orbits (overall 6 4-cycles -> even parity) and two 4-cycles of the corner pieces -> also even parity. Therefore puzzle reorientation is a parity-preserving operation for the helicopter cube. What you are describing is a situation in which puzzle reorientation is not a parity-preserving operation, such as for the void cube, but is not true on the helicopter cube.

Long story short, if you had a special helicopter cube which allowed you to see the "core orientation", a completely solved helicopter cube except with the core rotated 90° about a face is a legal position and has nothing to do with corner permutation parity.

It is technically (as you say) completely 100% possible to swap two corners without jumbling and without moving any face pieces.
For example:
FR UF UR UF UR UF UR FR UR UF UR FR UR UF UR FR UF UR UF FR UF UR UF
will swap the UFR and DFR corners without moving any face centers.

Some invisible pieces DID move, so your intuition is correct: something "invisible" is changing: the piece that is moved only by UF was also rotated 180°. This piece is physically inside the helicopter cube, but has no stickers because it is an internal piece. The same is true for the corresponding pieces moved by only UR and only FR

If you are curious, the proper analysis of this puzzle without jumbling is there are 5 orbits of pieces: 4 orbits of centers and 1 orbit of corners. Every move swaps 2 pairs of center pieces (in different orbits!) and 2 corners. This flips the parity of 3 out of 5 orbits: 2 center piece orbits and the corner orbit. Two corners swapped by themselves is a parity in the corner orbit only. This is indeed possible using helicopter moves! If we label the four center orbits A, B, C, and D and the corner orbit E, note that if we find moves that change the parity on: {ABE, BCE, and ACE}, this will preserve parity on all four center piece orbits (because each of A, B, C, and D appeared an even number of times) but change the parity on the corner orbit (because E appeared an odd number of times). 3 such moves are: UF, UR, and FR and you will notice in the sequence above each of these is used an odd number of times, thereby reversing the parity on the corner orbit while preserving the parity on all 4 center piece orbits. Don't you just love math? 8-)

I don't mean to slam you here, you made an excellent attempt at an explanation and I have been studying upper-level abstract math for several years now. Let me know if you have any questions :)

Peace,
Matt Galla

PS Thanks for using the ° symbol in your text. It made it very easy to copy and paste into mine :wink:


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 Post subject: Re: Helicopter cube solution
PostPosted: Sun Feb 12, 2012 10:36 pm 
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Thank you Matt, that made perfect sense and I'd never thought about breaking down the puzzle into how many orbits total there are like that. :D I think you are making better use of your "upper-level abstract math" than me, or maybe you are just further along. Anyway I hope people who google the same things I was will now find your explanation!


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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 12:34 am 
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Allagem wrote:
[...] If you are curious, the proper analysis of this puzzle without jumbling is there are 5 orbits of pieces: 4 orbits of centers and 1 orbit of corners. Every move swaps 2 pairs of center pieces (in different orbits!) and 2 corners. This flips the parity of 3 out of 5 orbits: 2 center piece orbits and the corner orbit. Two corners swapped by themselves is a parity in the corner orbit only. This is indeed possible using helicopter moves! If we label the four center orbits A, B, C, and D and the corner orbit E, note that if we find moves that change the parity on: {ABE, BCE, and ACE}, this will preserve parity on all four center piece orbits (because each of A, B, C, and D appeared an even number of times) but change the parity on the corner orbit (because E appeared an odd number of times). 3 such moves are: UF, UR, and FR and you will notice in the sequence above each of these is used an odd number of times, thereby reversing the parity on the corner orbit while preserving the parity on all 4 center piece orbits. Don't you just love math? 8-)

Ah, cool, it never occurred to me to analyze the helicopter cube by its orbits. :) Thanks for the correction & explanation, it's very clear.

Quote:
I don't mean to slam you here, you made an excellent attempt at an explanation and I have been studying upper-level abstract math for several years now. Let me know if you have any questions :)

Actually what I wrote was wrong. I was actually ignorant of the fact that you can swap two corners without cycling the face centers, which is what I did in the past when I found swapped corners. Not exactly the smartest thing to do, I'll admit, but then my usual method of solution is to use commutators, which works for pretty much every twisty puzzle out there. (The only exceptions are even cubes that sometimes require special sequences to "fix" the parity of a paired edge, and apparently in this case where a corner parity must be "fixed" by taking advantage of its orbital structure (pun intended -- I used to be a chemistry geek)). I'm not sure exactly how cycling the face centers cause the corner parity to flip; I guess something about the way I move the face centers introduced an extra flip in the corners orbit.

Quote:
PS Thanks for using the ° symbol in your text. It made it very easy to copy and paste into mine :wink:

I can make that symbol quite easily using a compose-key combination on my key mapping. (I'm also a Linux geek who uses the XKB international keymap :lol: )


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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 2:03 am 
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dranorter wrote:
Thank you Matt, that made perfect sense and I'd never thought about breaking down the puzzle into how many orbits total there are like that. :D I think you are making better use of your "upper-level abstract math" than me, or maybe you are just further along. Anyway I hope people who google the same things I was will now find your explanation!

Glad I could help! It appears I can still explain things on occasion :lol: I love this mathematical stuff and the helicopter cube in particular is one of my favorite puzzles due to its unusual properties :)

quickfur wrote:
Allagem wrote:
I don't mean to slam you here, you made an excellent attempt at an explanation and I have been studying upper-level abstract math for several years now. Let me know if you have any questions

Actually what I wrote was wrong.

Well I never said your attempt was correct now did I? :wink:

quickfur wrote:
I was actually ignorant of the fact that you can swap two corners without cycling the face centers, which is what I did in the past when I found swapped corners. Not exactly the smartest thing to do, I'll admit, but then my usual method of solution is to use commutators, which works for pretty much every twisty puzzle out there. (The only exceptions are even cubes that sometimes require special sequences to "fix" the parity of a paired edge, and apparently in this case where a corner parity must be "fixed" by taking advantage of its orbital structure (pun intended -- I used to be a chemistry geek)). I'm not sure exactly how cycling the face centers cause the corner parity to flip; I guess something about the way I move the face centers introduced an extra flip in the corners orbit.

An excellent solving method! And just FYI, what many people call a parity on even cubes is actually a position with even parity on all orbits of all pieces. The parity "error" comes from the assumption that you have grouped the pieces into a legal position for a lower symmetry of the puzzle (i.e. grouping centers and edges and then solving as a 3x3x3). What appears to be a 3x3x3 may or may not be in a legal 3x3x3 position - what can go wrong is the parity of the grouped edges can not match the parity of the corners (on a 3x3x3, edges and corners both flip parity every quarter-turn, but always TOGETHER). Depending on exactly how you are pairing up edges, if you run into a state where two groups of edges are swapped, you can break your "it's just a 3x3x3 now!" assumption and find commutators to perform three cycles on the edge pieces alone - implementing the use of either slice turns or double layer turns, etc. - switching every piece from one grouped edge to the other group using standard commutators. Then you don't even have to memorize long sequences for even cubes! :wink: The only skills really needed to solve 99% of all twisty puzzles is the ability to recognize orbits, identify parity flips induced by the available moves on these orbits, and construct hierarchical commutators for each individual piece :)

Peace,
Matt Galla

PS The other type of "parity" people refer to on even cubes (occurs on odd cubes as well in a slightly different form!) in which a single group of edges is flipped in place as a whole really is a legitimate parity any way you look at it. Then you have to notice that a slice move changes the parity of a certain orbit of edges as well as some center orbits* - but since there's identical copies of those, parities can be disguised by swapping identical pieces 8-) Then just do the appropriate slice move and use commutators to put everything back. :)

Edit: *I probably should have specified WHICH center orbits... By center, I meant any piece with only one sticker on it, and specifically the orbits that flip parity in addition to the edge pieces are those in line with the very center of the puzzle, which are apparently called T-centers (didn't know that). So on an even nxnxn cube these don't exist - so yes just swapping two edges without changing the permutation of any other piece on the physical puzzle is completely possible (I have miswritten this fact at least once before in another thread). On a 5x5x5 for example, the pieces immediately adjacent change parity whenever a "non-middle" slice move is made, as well as the "non-middle" edges. So the parity of the "non-middle" edges matches the parity of the T-centers (provided the corners/middle edges are in the correct parity). The other centers, those not in the same horizontal or vertical slice as the very center piece, are affected by slice moves too, but in two sets of 4-cycles. A 4-cycle is an odd parity, which would mean problems (i.e. parity flips), except the fact that there are two of them fixes that, so it is a non-issue. Of course, sinlge layer face turns DO flip the parity of every piece on the puzzle - but that would cause every piecetype to be in odd parity, and most people have learned 3x3x3 solving methods that automatically recognize this fact and correct it. Also the corner or middle-edge (if they exist) parity is a dead give-away to whether this has happened as these piece-types are not affected by slice moves.


Last edited by Allagem on Mon Feb 13, 2012 9:00 pm, edited 1 time in total.

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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 2:10 am 
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Allagem wrote:
If you are curious, the proper analysis of this puzzle without jumbling is there are [...]
Great analysis Matt. Speaking of orbits and orientations of the core, on the Helicopter CUBE the center triangles are in 4 orbits of 6 pieces just as you describe but each orbit has exactly one of each color in it. This allows the center triangles to be solved in any orientation relative to the core.

On the octahedral shape mod of the Helicopter cube (the Helicopter OCTAHEDRON, similar to Gelatinbrain's 4.3.1) each orbit has 6 unique colors in it but there are 8 colors on the puzzle. Because the pieces are restricted to their orbits there are only 2 orientations of color scheme (relative to the core that never moves) that allow all of the triangles to come together to form solid faces. One of the color schemes is the correct one and the other is the inverse where the colors of opposite faces have been swapped. The corners won't match up with the inverted color scheme though.

The Helicopter DODECAHEDRON (6 obits of 10 pieces each) has the same global orientation issue based on how the orbits are intertwined.

Edit:
And when is that thesis of yours going to be ready? I seem to recall you saying it would be VERY interesting to us mathy twisty puzzlers. I've been looking forward to it :-)

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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 5:11 am 
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bmenrigh wrote:

Edit:
And when is that thesis of yours going to be ready? I seem to recall you saying it would be VERY interesting to us mathy twisty puzzlers. I've been looking forward to it :-)


Hahaha! I'm impressed you remembered! Yes I have been working on that still - that and a job and research and earning two degrees (BS Engineering and BA Mathematics, dont agree with the Arts label, but not my choice and works better for me anyway :lol: ) in my senior year of undergrad. That's what's pulled me away from twistypuzzles so much for the last... oh what has it been =... 2-3 years? Man, I wish I had the time to be on here more often.... I left right as boublesizing (or however we are spelling it) was being discovered. I need to get back into things!! :cry: I still glance in here as often as I can, but don't really have the time to post or research as much as I want. (I only respond to questions or problems or misunderstandings that I happen to catch and know how to answer... like this one :wink: )

Anyway, that thesis is for my Mathematics Degree so is due in May (don't worry - I don't need to be succesful to get credit for working on it), and I'm getting close to finalizing/proving/finishing it. Just a few gaps and loopholes and one surprisingly stubborn case left.... (hopefully I'll find that one last breakthrough soon...) Then I gotta slap it into some computers and let it do its thing... Anyway, I don't wanna spoil it or jynx it or set the expectation too high :wink: so just wish me luck and I will inform you and everyone in May... :D


PS: BTW Carl, if you happen to read this, I just realized I never told you, I didn't get that one award last year :? That's why I'm trying to accomplish what I said and show them I deserved it! :D Thank you very much for all your contribution though! :mrgreen:

PPS: I know there's a PM but it's a little more fun to thank you publicly and be slightly mysterious by not fully explaining what I'm taling about. It's sort of the theme of this post 8-) Anyways, enough hijacking, let's talking about solving the Helicopter Cube :wink: That's a fun puzzle right? :lol:


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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 11:43 am 
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Allagem wrote:
[...]
quickfur wrote:
[...]
Actually what I wrote was wrong.

Well I never said your attempt was correct now did I? :wink:

Ah yes, always be ultra-precise with your words when talking to a mathematician. ;)

Quote:

quickfur wrote:
I was actually ignorant of the fact that you can swap two corners without cycling the face centers, which is what I did in the past when I found swapped corners. Not exactly the smartest thing to do, I'll admit, but then my usual method of solution is to use commutators, which works for pretty much every twisty puzzle out there. (The only exceptions are even cubes that sometimes require special sequences to "fix" the parity of a paired edge, and apparently in this case where a corner parity must be "fixed" by taking advantage of its orbital structure (pun intended -- I used to be a chemistry geek)). I'm not sure exactly how cycling the face centers cause the corner parity to flip; I guess something about the way I move the face centers introduced an extra flip in the corners orbit.

An excellent solving method! And just FYI, what many people call a parity on even cubes is actually a position with even parity on all orbits of all pieces. The parity "error" comes from the assumption that you have grouped the pieces into a legal position for a lower symmetry of the puzzle (i.e. grouping centers and edges and then solving as a 3x3x3).

Yes, I understand that. It's not actually possible to get a 4x4x4 into an odd parity (well, you can have an even number of orbits in odd parity, I suppose, but as a whole, the puzzle is always in even parity). I was using "parity" in the pedestrian sense of "it has odd parity when interpreted as a 3x3x3". Obviously, the 4x4x4 itself is still in even parity in such positions, strictly speaking.

Quote:
What appears to be a 3x3x3 may or may not be in a legal 3x3x3 position - what can go wrong is the parity of the grouped edges can not match the parity of the corners (on a 3x3x3, edges and corners both flip parity every quarter-turn, but always TOGETHER). Depending on exactly how you are pairing up edges, if you run into a state where two groups of edges are swapped, you can break your "it's just a 3x3x3 now!" assumption and find commutators to perform three cycles on the edge pieces alone - implementing the use of either slice turns or double layer turns, etc. - switching every piece from one grouped edge to the other group using standard commutators.

I was referring more to the case where a single paired edge is "flipped". In that case, you need to swap a pair of face centers in order to be able to swap the flipped pair. I already know that the case of two pairs of edges being swapped can be easily fixed by commutators.

Quote:
Then you don't even have to memorize long sequences for even cubes! :wink: The only skills really needed to solve 99% of all twisty puzzles is the ability to recognize orbits, identify parity flips induced by the available moves on these orbits, and construct hierarchical commutators for each individual piece :)

I find that the ABCB'A'BC'B' commutator usually works very well on any kind of puzzle, for suitable values of A, B, and C. A few puzzles need some other combination of ABA'B', but usually I unconsciously incorporate that in the first stage of my solving since at that stage it's simple enough that I can reason about each turn intuitively.

Quote:
[...] PS The other type of "parity" people refer to on even cubes (occurs on odd cubes as well in a slightly different form!) in which a single group of edges is flipped in place as a whole really is a legitimate parity any way you look at it. Then you have to notice that a slice move changes the parity of a certain orbit of edges as well as some center orbits - but since there's identical copies of those, parities can be disguised by swapping identical pieces 8-) Then just do the appropriate slice move and use commutators to put everything back. :)

Yes, this was the kind of "parity" that I was referring to. I invented a rather long (but highly repetitive and therefore easy to remember) sequence for "swapping two edge pieces". There are definitely better algos out there for this, but I suck at memorization and so rather stick with something easy to remember. (Which is why I use commutators on just about everything: they are very easy to remember.)


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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 2:53 pm 
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I do not want to derail this thread further, but those reading here about `parity` might find this thread interesting.
The only real parity a came across so far is swapping two edges on a 4x4x4 (the Super shows that no other pieces have moved)
Image
The shortest sequence to fix it is l2 D2 F2 r F2 l´ F2 l F2 U2 l U2 r' D2 l2 (15 moves)

Julian has given a very ingenious explanation how this sequence might have been found. I'm not sure where this was.

EDIT: I could find Julian's post.

EDIT: I changed flipping to swapping. I meant `swapping` in the first place but swapped the words in my non-English brain. :lol:

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Last edited by Konrad on Mon Feb 13, 2012 5:36 pm, edited 2 times in total.

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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 3:47 pm 
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Konrad wrote:
I do not want to derail this thread further, but those reading here about `parity` might find this thread interesting.
The only real parity a came across so far is flipping two edges on a 4x4x4 (the Super shows that no other pieces have moved)
Image[...]
I don't think this is a further derailment since the thread is already off the helicopter cube tracks and on the tracks and on the parity tracks :lol: .

You are correct but not as precise as I think we should state here. Your image does not show "flipping two edges" but rather swapping two edge edge wings. It just happens that on the 4x4x4 the edge wings don't have any orientation. The position they are in determines their orientation.

In your picture it looks like two edge wings have been flipped but actually two have been swapped. You can swap any two edge wings on the puzzle, not just two adjacent ones. It is because one swap is odd that we call this an "odd parity".

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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 3:51 pm 
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Huh, I never knew you could swap a pair of edges on a 4x4x4 without touching anything else. I had always thought that you needed to swap a pair of face centers as well.

When I get home today I'm going to try this algo on the 5x5x5 to see if I can understand what's going on.


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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 4:08 pm 
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quickfur wrote:
Yes, I understand that. It's not actually possible to get a 4x4x4 into an odd parity (well, you can have an even number of orbits in odd parity, I suppose, but as a whole, the puzzle is always in even parity). I was using "parity" in the pedestrian sense of "it has odd parity when interpreted as a 3x3x3". Obviously, the 4x4x4 itself is still in even parity in such positions, strictly speaking.
The edge wings are all in one orbit and they can be in an odd permutation irrespective of the state of any other pieces in the puzzle.

quickfur wrote:
I was referring more to the case where a single paired edge is "flipped". In that case, you need to swap a pair of face centers in order to be able to swap the flipped pair. I already know that the case of two pairs of edges being swapped can be easily fixed by commutators.
Yes a single pair flipped is the same as a single pair swapped. It's actually impossible to change the parity of a permutation with commutators alone. Any sequence to change parity must involve at least 1 move that isn't part of any commutators. In the case of the 4x4x4 edge wings, that 1 move must be a quarter-turn of a slice. After that everything can be resolved with commutators.

quickfur wrote:
I find that the ABCB'A'BC'B' commutator usually works very well on any kind of puzzle, for suitable values of A, B, and C. A few puzzles need some other combination of ABA'B', but usually I unconsciously incorporate that in the first stage of my solving since at that stage it's simple enough that I can reason about each turn intuitively.
Yeah you've described a [1,3] commutator here (using the sequence shorthand of the Gelatinbrain solving thread). Your BCB' sequence is a conjugate.

Overall the sequence is [1,[1:1]] in shorthand. [3,1] commutators are exceptionally useful.


There is a simple proof that commutators alone can never change the parity of any pieces in the puzzle. I'll informally sketch the outline in brief.

First, a commutator is any sequence of the form X Y X' Y' where X and Y are themselves any sequence.

Some axiom: If a sequence Q changes the parity of pieces then Q' undoes that parity change.

Assume X changes the parity of pieces. Then apply the commutator X Y X' Y'. If X changed the parity of pieces then X' undid the parity change (by some axiom). The same goes for Y if Y changed the parity of pieces. Any parity change that occurs is undone later.

You may be wondering if the interaction of X and Y could change the parity even if X and then X' or Y and then Y' alone could not. There is some axiom about parities that shows that can't happen either. The parity of pieces is an overall property. If X changed the parity of some subset of the pieces and Y moved some pieces around and then X' undid the parity of some other pieces not acted on by X, that still doesn't change the overall parity. because ODD + ODD == EVEN and EVEN + EVEN == EVEN.

QED or something like that.

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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 4:13 pm 
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quickfur wrote:
Huh, I never knew you could swap a pair of edges on a 4x4x4 without touching anything else. I had always thought that you needed to swap a pair of face centers as well.

When I get home today I'm going to try this algo on the 5x5x5 to see if I can understand what's going on.
It isn't possible to do on the 5x5x5 because the slice move needed to do it also 4-cycles the + centers (sometimes called T centers) and a 4-cycle is odd. To swap two edge wings you must also swap two + centers. You can't do it pure.

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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 4:36 pm 
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bmenrigh wrote:
[...] There is a simple proof that commutators alone can never change the parity of any pieces in the puzzle. I'll informally sketch the outline in brief.[...]

Thanks for the proof, although intuitively I know that commutators are always even permutations, because it always applies the reverse of each turn it makes at some point, so no matter what turns are being made, even if they have odd parity, their reverse will eventually put it back into even parity.

bmenrigh wrote:
quickfur wrote:
Huh, I never knew you could swap a pair of edges on a 4x4x4 without touching anything else. I had always thought that you needed to swap a pair of face centers as well.

When I get home today I'm going to try this algo on the 5x5x5 to see if I can understand what's going on.
It isn't possible to do on the 5x5x5 because the slice move needed to do it also 4-cycles the + centers (sometimes called T centers) and a 4-cycle is odd. To swap two edge wings you must also swap two + centers. You can't do it pure.

Oh, I know that it's impossible on the 5x5x5; what I was looking for was to find out which "invisible pieces" (in this case "virtual" pieces, i.e., the edges between face centers, not physical pieces) are being moved. Which, thanks to your explanation, now I know are the T centers. :) So that makes sense.

(The reason I thought to try it on the 5x5x5 is because the 4x4x4 group is a subgroup of the 5x5x5 group, and since the 5x5x5 cannot have an odd parity in any of its orbits, the odd parity in the 4x4x4 must arise from "ignoring", or rather "omitting", via the subgroup operation, some parts of the 5x5x5. I wanted to know which part of the 5x5x5 was being "ignored" by this particular odd permutation, or, intuitively speaking, where the odd parity is being "balanced out" -- since the 4x4x4 group is a subgroup of the 5x5x5 group which can't have odd parity, so the odd parity in the 4x4x4 must have been "balanced" by an odd parity in the 5x5x5 group which happens to be ignored.)

Edit: In other words, every twisty puzzle group that admits orbits with odd permutations has a supergroup which only admits even permutations. Intuitively speaking, we can introduce new pieces to the puzzle such that an odd parity in the original puzzle is "balanced" by another odd parity in the newly-introduced orbit. I find that this is a nice way of understanding the structure of the odd parity in the original puzzle.


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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 5:57 pm 
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bmenrigh wrote:
... .

You are correct but not as precise as I think we should state here. Your image does not show "flipping two edges" but rather swapping two edge edge wings. It just happens that on the 4x4x4 the edge wings don't have any orientation. The position they are in determines their orientation.

In your picture it looks like two edge wings have been flipped but actually two have been swapped. You can swap any two edge wings on the puzzle, not just two adjacent ones. It is because one swap is odd that we call this an "odd parity".
I meant `swapping` in the first place but swapped the words in my non-English brain. :lol:

@quickfur: I hope you do not want to say that swapping two edges of a 4x4x4 - and nothing else - is an even permutation?
The same move sequence as above performed on a 5x5x5 does permute centres as well, but a 4x4x4 is NOT equal to a 5x5x5.
A 90 degree slice turn on a 4x4x4 is an odd permutation of the 4 edges and an even permutation of the 8 centres (two 4-cycles), right?
This is different on the 5x5x5, but I cannot see any hidden or virtual 5x5x5 centres on the 4x4x4.

Matt has agreed in the other thread (I copied the picture above from there) that this case - swapping two edges - means a real parity.

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 Post subject: Re: Helicopter cube solution
PostPosted: Mon Feb 13, 2012 6:53 pm 
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Konrad wrote:
[...]@quickfur: I hope you do not want to say that swapping two edges of a 4x4x4 - and nothing else - is an even permutation?

No, of course not. It's clearly an odd permutation.

Quote:
The same move sequence as above performed on a 5x5x5 does permute centres as well, but a 4x4x4 is NOT equal to a 5x5x5.

Obviously. What I'm doing is treating the 4x4x4 group as a subgroup of the 5x5x5 group, and observing the effect of the given operation on the 5x5x5 group, as a way of studying how the parity swapping operation works.

Quote:
A 90 degree slice turn on a 4x4x4 is an odd permutation of the 4 edges and an even permutation of the 8 centres (two 4-cycles), right?
This is different on the 5x5x5, but I cannot see any hidden or virtual 5x5x5 centres on the 4x4x4.

Matt has agreed in the other thread (I copied the picture above from there) that this case - swapping two edges - means a real parity.

I'm not saying this isn't a "real" parity - it's obviously real, and clearly visible.

Perhaps "hidden" or "virtual" centers is a bad name. I'm not suggesting that the 4x4x4 actually has physical pieces between its face centers that get permuted when you perform the algorithm. That would be absurd. What I was trying to do was to understand the permutation structure of the parity-flip algorithm.

Mathematically speaking, the algorithm permutes a number of elements in their respective orbits, and by "element" i'm not just talking about physical pieces, but also the abstract geometric structure of the 4x4x4. Geometrically speaking, the face centers of the 4x4x4 share edges between them (in the physical 4x4x4, it's where adjacent face centers touch each other). In the 4x4x4, these edges do not correspond with any physical piece, but nevertheless, if you treat the 4x4x4 as a 3D wireframe, then the algorithm will permute parts of this wireframe around: some vertices will move, some edges will move, and some cubies (corresponding with cube-shaped groups of 12 edges each) will move.

In this sense, the algorithm does have an effect on more than just the two cubies that got swapped: it has also permuted some of the edges on one of the faces. Of course, this effect is completely intangible because the 4x4x4 doesn't have any physical parts corresponding to these edges. But nevertheless, this effect is part of the characteristics of this particular algorithm.

The reason I referred to the 5x5x5 is because one way of making this effect visible is by mapping the edges that lie on the lines intersecting each face center to actual, physical pieces, that is, to the middle-row pieces on a 5x5x5. In other words, performing this algo on a 5x5x5 makes what was an intangible effect on the 4x4x4 physically visible: now those permuted edges can actually be seen to have been permuted, because they are now actual pieces.

Now, why would I even care in the first place? What's the point of figuring effects that are intangible? The reason is because it tells you, in some sense, how the algorithm "works". The fact that it's the 4 edges between the face centers that got rotated is significant, since, conceivably, another algorithm that flips edge parity on the 4x4x4 could instead rotate the edges between two edge pieces instead (corresponding with the middle edge pieces of a 5x5x5). Or it could have permuted some other group of edges.

Knowing these "virtual effects" allows us to classify different algorithms that perform edge swaps. It's an index into "how they work", so to speak.

As to how this relates to parity: the usual way we define parity for twisty puzzles is the parity of orbits for physical pieces. According to that definition, the permuted edges are not part of the definition, so the algorithm actually flips the parity of the puzzle when it exchanges two edge pieces. We could, however, define a different kind of parity on the puzzle, say by considering not just the moving pieces but also the interfaces between them, or even junctions (or to use geometrical terminology, faces, edges, vertices). Under this expanded definition, there are several more orbits in the 4x4x4, and we see that their aggregate parity is always even. And we see this the case when we build a puzzle in which all of these elements are realized physically, which would be a 9x9x9, which, indeed, always has even parity.

So in this sense, if we abuse the terminology a little, we can say that this particular algorithm on the 4x4x4 achieves an odd parity (swap two edges) by "balancing it with another odd parity on 4 edges".


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