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 Post subject: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 12:58 pm

Joined: Tue Aug 11, 2009 2:44 pm
Solving TomZ's Curvy Copter II is mostly straightforward, if you can solve a Helicopter. But there is an intriguing complexity.

Nomenclature:

Corner -- obvious
Face -- thin curved triangle that touches a corner piece
Center -- obvious
Offset center -- touches a center and two faces

First, suppose you don't jumble it. Then the corners and faces are just like on a Helicopter. There's a very simple 4-move sequence that permutes only centers and offset centers, from which you can easily build transforms to permute them individually. OK.

But what about jumbling? Although I have not yet made a detailed comparison, at first sight it seems to jumble the same way as a Helicopter. The shapes look a little different, but the same move sequences seem to be possible. Or are they? I am not sure. It is more confusing to me to unjumble than a Helicopter is.

Jumbling means it's more work to solve the faces. Just like on a Helicopter. But... here's the real surprise... jumbling allows you to swap two centers, something you cannot do without jumbling. And it's not at all obvious to me how. When I unjumble it, solve the corners, and observe the center permutation, if it's not even, I'm reduced to re-jumbling and scrambling.

How the hell do you go about building jumbling transforms? I'm reduced to experimentation, whereas ordinarily I can build transforms by reasoning alone. And in this case, my experimentation has not yet yielded a transform.

This is the brave new frontier of twisty puzzles, I believe: can one find general techniques for solving bandaged/jumbling puzzles? (And, as far as solving alone is concerned, is there a difference?) I've also made very little progress with Oskar's More Madness (in which every move jumbles!). And as for TomZ's Helicopter Skewb, well, I made the mistake of letting people play with it before I'd stickered it, and now I can't even unjumble it! I may have to disassemble and reassemble it to sticker it.

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 1:24 pm

Joined: Mon Mar 30, 2009 5:13 pm
Incredible. In terms of solution space, it sounds like the jumbling has opened up a wormhole to travel between parallel universes!

This could be the biggest breakthrough ever in the theory of twistypuzzle cosmology!

_________________
If you want something you’ve never had, you’ve got to do something you’ve never done - Thomas Jefferson

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 1:41 pm

Joined: Tue Aug 11, 2009 2:44 pm
Kelvin Stott wrote:
Incredible. In terms of solution space, it sounds like the jumbling has opened up a wormhole to travel between parallel universes!

Well yes, but that's not really new. Jumbling a Helicopter already permits face permutations that are not reachable otherwise. Still, I was very surprised the first time I solved it except for the centers, used my center-cycle transform a couple times, and then... huh???!

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 1:47 pm

Joined: Mon Jul 21, 2008 4:52 am
Location: Brighton, UK
bhearn wrote:
Although I have not yet made a detailed comparison, at first sight it seems to jumble the same way as a Helicopter. The shapes look a little different, but the same move sequences seem to be possible.
That is my impression too.

bhearn wrote:
But... here's the real surprise... jumbling allows you to swap two centers, something you cannot do without jumbling.
You can swap two centers without jumbling, if you solve the corners into an "odd" overall color orientation, in other words, where it takes an odd number of 90 degree rotations of the cube to get from the solved color arrangement to the new one. For example, from your photo, if you re-solve the corners so the yellow and white faces stay in the same position, but green becomes red (y rotation), the centers will be in a relative odd permutation and will be unsolvable until you re-solve the corners into an even permutation.

It's a great puzzle, I love it!

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 1:51 pm

Joined: Tue Aug 11, 2009 2:44 pm
Julian wrote:
You can swap two centers without jumbling, if you solve the corners into an "odd" overall permutation, in other words, where it takes an odd number of 90 degree rotations of the cube to get from the solved color arrangement to the new one
Oh... of course. My bad! Thanks for saving me from a large, fruitless search for a jumbling transform.

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 2:03 pm

Joined: Mon Jul 21, 2008 4:52 am
Location: Brighton, UK
bhearn wrote:
Thanks for saving me from a large, fruitless search for a jumbling transform.
You're welcome! I was lucky with the centers for my first 3-4 scrambles and solves, a couple of which included jumbling, so when I first saw two swapped centers following a scramble without jumbling, I was amazed and actually felt a bit dizzy, as if logic had been temporarily suspended!

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 2:18 pm

Joined: Mon Mar 30, 2009 5:13 pm
Would this be possible also with Oskar's Mixup Cube?

_________________
If you want something you’ve never had, you’ve got to do something you’ve never done - Thomas Jefferson

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 2:32 pm

Joined: Mon Aug 27, 2007 3:50 pm
Location: Copenhagen, Denmark
This is the one puzzle of my collection, which I am most scared of scrambling!

Did you scramble it up completely? And how long time did it take you to solve it?

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 3:08 pm

Joined: Tue Aug 11, 2009 2:44 pm
Julian wrote:
I was lucky with the centers for my first 3-4 scrambles and solves, a couple of which included jumbling, so when I first saw two swapped centers following a scramble without jumbling, I was amazed and actually felt a bit dizzy, as if logic had been temporarily suspended!
Same here... except that it was on a jumbling scramble I got swapped centers. So, obviously, it must have been due to jumbling! Still, I feel silly for not seeing it until you pointed it out.

Sigurd wrote:
This is the one puzzle of my collection, which I am most scared of scrambling!

Did you scramble it up completely? And how long time did it take you to solve it?
Well I started with a non-jumbling scramble. And I already knew how to solve the Helicopter, so that part was easy. It was pretty quick to find the necessary new transforms, though tedious to position all the offset centers. Slightly faster once I realized that transform (actually two related transforms) was shorter if I didn't try to preserve the centers, and saved them for last. The only slightly challenging part, really, is visualizing the conjugate sequences sometimes needed to position the offset centers for the relevant transforms to work. It's a little harder to do than, say, positioning edges on a Rubik's Cube in preparation for a transform. But maybe this will get easier.

And jumbling, evidently, doesn't add anything new, beyond what jumbling a Helicopter does.

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 9:51 pm

Joined: Tue Aug 11, 2009 2:44 pm
Kelvin Stott wrote:
Would this be possible also with Oskar's Mixup Cube?

I don't think the same reasoning applies to the Mixup Cube. Suppose you try to solve the entire Mixup Cube to an odd color orientation. Well, starting from a solved state, that's actually trivial: just do RL' and a matching center slice. Voilá! But now the centers are in an odd permutation as well, consistent with the odd color orientation, which is not what you wanted. The logic for the Curvy Copter II is that to put the corners in an odd color orientation, you need an even number of moves, so you wind up with an even center permutation, inconsistent with the new coloring. The slice move on the Mixup Cube makes the difference.

However! That doesn't necessarily mean you can't swap two centers on the Mixup Cube. Centers and edges are really the same piece type, and slice moves come in eighth turns. That means they form an 8 cycle, which is an odd permutation, on a single piece type. If there were 8 centers, then one could just put them all along the equator, make an eighth-turn move, and wind up with odd center parity. But there are only 6 centers. Still, no problem: just put two centers adjacent on an equator, make an eighth-turn slice, and then perform a 7-cycle (which can be built out of simple Rubik's Cube edge 3-cycles, via suitable conjugate sequences) that leaves center A (now in B's original slot) alone, puts center B where center A came from, and puts everything else where it originally was. Result: you've swapped center A and center B.

More simply, I guess, the 8 cycles mean that if you solve the corners and edges, you will have a center parity issue in general. I'm sure there's a much shorter transform to swap centers than what I proposed. I don't have a Mixup Cube to try this on, but the logic seems sound. Anybody who has one care to comment?

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 Post subject: Re: Solving the Curvy Copter IIPosted: Wed Dec 29, 2010 10:22 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
bhearn wrote:
This is the brave new frontier of twisty puzzles, I believe: can one find general techniques for solving bandaged/jumbling puzzles? (And, as far as solving alone is concerned, is there a difference?) I've also made very little progress with Oskar's More Madness (in which every move jumbles!). And as for TomZ's Helicopter Skewb, well, I made the mistake of letting people play with it before I'd stickered it, and now I can't even unjumble it! I may have to disassemble and reassemble it to sticker it.

It being difficult to get a puzzle back to square shape is a very interesting property, and I have no idea how one might develop a systematic approach for that.

The jumbleprism, by the way, has the odd property that there's a single weird jumbled position which it's very, very difficult to get out of.

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 Post subject: Re: Solving the Curvy Copter IIPosted: Thu Dec 30, 2010 8:45 am

Joined: Mon Jul 21, 2008 4:52 am
Location: Brighton, UK
bhearn wrote:
Centers and edges are really the same piece type, and slice moves come in eighth turns. That means they form an 8 cycle, which is an odd permutation, on a single piece type. If there were 8 centers, then one could just put them all along the equator, make an eighth-turn move, and wind up with odd center parity. But there are only 6 centers. Still, no problem: just put two centers adjacent on an equator, make an eighth-turn slice, and then perform a 7-cycle (which can be built out of simple Rubik's Cube edge 3-cycles, via suitable conjugate sequences) that leaves center A (now in B's original slot) alone, puts center B where center A came from, and puts everything else where it originally was. Result: you've swapped center A and center B.

More simply, I guess, the 8 cycles mean that if you solve the corners and edges, you will have a center parity issue in general. I'm sure there's a much shorter transform to swap centers than what I proposed. I don't have a Mixup Cube to try this on, but the logic seems sound.
What confuses me about the Mixup Cube is that if we restrict ourselves to slice moves only, there are 9 pairs of pieces, where the pieces of each pair are always poles apart, and by themselves they form a puzzle of 9 pieces which can be in any permutation. Presumably each pair can be reversed/flipped too, like with the Astrolabacus (which I also don't own but have played around with in simulated form). The pairings can only be broken by mixing in regular face turns, which have the restriction that the sign of the perm of the corners and "edges" always matches: both even or both odd. Does the combination of these types of move allow us to cycle all 18 center-or-edge pieces into any permutation, or not?

Is it possible for the Mixup Cube to be solved apart from one center and edge in swapped positions? (I assume it is, following your method above.)

Is it possible for the Mixup Cube to be solved apart from one flipped edge? (I assume it is.)

Is it possible for the Mixup Cube to be solved apart from one edge misoriented out of the cubic shape? (Instinctively I think not, that any out-of-shape edges would come in pairs, but I can't explain why, I'm only guessing!)

I really need to buy a Mixup Cube in 2011.

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 Post subject: Re: Solving the Curvy Copter IIPosted: Thu Dec 30, 2010 10:46 am

Joined: Tue Aug 11, 2009 2:44 pm
Julian wrote:
The pairings can only be broken by mixing in regular face turns, which have the restriction that the sign of the perm of the corners and "edges" always matches: both even or both odd. Does the combination of these types of move allow us to cycle all 18 center-or-edge pieces into any permutation, or not?
It would seem so -- I think I already showed how to switch a single pair of center / edges, so you can build any permutation.

Quote:
Is it possible for the Mixup Cube to be solved apart from one edge misoriented out of the cubic shape? (Instinctively I think not, that any out-of-shape edges would come in pairs, but I can't explain why, I'm only guessing!)
I think so. Put an edge in a center slot (with just a slice move), then apply a transform that rotates two centers in opposite quarter turns, then put it back.

Quote:
I really need to buy a Mixup Cube in 2011.
Me too!

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