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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Fri May 16, 2008 3:03 am 
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I think you misread me. I wasn't saying that the angle must be a rational multiple of 360 degrees. Of course the helicopter cube angle is irrational. I'm not the kind of person to argue for a blatantly false mathematical statement...

What I WAS saying was that I don't think defining jumbleability as "involving rotations which aren't a rational fraction of 360 degrees" is a good definition. I'll elaborate: I don't think it's unreasonable to posit a puzzle that you would consider jumbleable but which would only involve rational angles. Also, rotations of an irrational angle do not necessarily signify a jumbleable puzzle. For example, we could make a square-1 type puzzle with a 2x1 rectangle on top and bottom, each divided along its diagonals (each piece has an irrational angle), so that it is possible to do a turn and keep the top and bottom rectangular. It doesn't seem like you'd be more likely to call this thing jumbleable than the square-1 just because it has some irrational angles in it. Besides, if we use a sqrt(3)x1 rectangle instead the angles become rational but the puzzle's concept is unchanged.

What do you think of my definition of jumbleable, though? Other than the helicopter cube and its dodecahedral cousin, what existing puzzles would you consider jumbleable? And, more interestingly, do you think the helicopter cube's mechanism is the cause of its jumbleability, or is it an inherent characteristic of the puzzle?

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Fri May 16, 2008 4:11 am 
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There may be a puzzle which I'd consider jumbleable but only had rational angles, but for the most part such things can straightforwardly be viewed as bandaged versions of a puzzle with a bunch more pieces by adding cuts, while for truly jumbleable puzzles if you started adding cuts to make the non-jumbled version the cutting would never end.

Of course you can make puzzles which have irrational angles for which the angled behavior is degenerate and non-useful. I'm trying to provide a definition which gets the message across, not one which is necessarily perfect in every case. For example, I'd say the never successfully produced cmetrick AE is a jumbleable puzzle, even though it doesn't even have slices, because if one were to make the complete supergroup of it the result would have an infinite number of tooth holes.

Yes there are quite a few jumbleable puzzles. I mentioned one in this thread earlier today and Robert Webb provided a picture of it, and I gave a whole long list of them way back when this thread started.


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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Fri May 16, 2008 4:13 am 
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I'm not sure what you mean by whether the jumbling is inherent to the helicopter cube. One could imagine a version of it which wasn't allowed to jumble but can do the traditional 180 degree rotations, which isn't a bad puzzle, although I'm not sure what the mechanism for it would be (you could also add a single center slice move to it, which would spice things up quite a bit). There are other, more exotic puzzles which can only be scrambled by jumbling, and are just silly manipulatives if you ban such moves.


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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Fri May 16, 2008 5:34 am 
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Quote:
I mentioned one in this thread earlier today and Robert Webb provided a picture of it,


Now, I'm not 100% sure about this, and it's a wee bit late for me to be checking, but I'm pretty sure that that picture that Robert put up a few posts ago (the truncated tetrahedron with the hexagons on each face) is what you get if you embed the helicopter / bevel mechanism into a tetrahedron shape, making "jumblability" (how's that for a word!) a feature of the 12 axis mechanism.

Hmmm...

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Fri May 16, 2008 12:13 pm 
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Bram wrote:
I'm not sure what you mean by whether the jumbling is inherent to the helicopter cube.


What I'm trying to determine is this: if you really carefully analyzed the theoretical helicopter cube puzzle, without thinking at all about the mechanism used to make it in real life, would you come to the conclusion that it could be jumbled, or not? I'm beginning to think it is necessitated by the idea of the puzzle:
[url]http://www.geocities.com/jaapsch/puzzles/sphere.htm?red=213&symred=4[url]
You can see that if you turn one of the circles by an angle smaller than 180 degrees (although you wouldn't know whether it is rational just by looking at the drawing) you can still turn adjacent faces; that's where the jumbling comes from. I wonder what other puzzles have this characteristic. It might be possible that gelatinbrain 1.2.2 is jumbleable, for instance, since the intersection of two of the non-adjacent corners from one face is asymmetrical.

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Fri May 16, 2008 12:31 pm 
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the.drizzle wrote:
Now, I'm not 100% sure about this, and it's a wee bit late for me to be checking, but I'm pretty sure that that picture that Robert put up a few posts ago (the truncated tetrahedron with the hexagons on each face) is what you get if you embed the helicopter / bevel mechanism into a tetrahedron shape, making "jumblability" (how's that for a word!) a feature of the 12 axis mechanism.


Yep. It's a weird 12 axis mechanism though, with spider arms corresponding to the vertices of a truncated tetrahedron, so it would have to be a from scratch build. There are other, better groups to do the same trick with, because they have fewer vertices and less sharp angles.


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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Fri May 16, 2008 12:34 pm 
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qqwref wrote:
What I'm trying to determine is this: if you really carefully analyzed the theoretical helicopter cube puzzle, without thinking at all about the mechanism used to make it in real life, would you come to the conclusion that it could be jumbled, or not?


I came up with the same mechanism independently and probably wouldn't have noticed it was jumbleable until actually building it if another person hadn't made it first and posted pictures of the jumbling. I believe both people to have successfully built it had the jumbling fall out naturally of the mechanism design as well, without it being an a priori design goal.


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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sat May 17, 2008 1:28 am 
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There probably needs to be another word for puzzles which change shape. What we're talking about here appears to be a newish phenomenon, not like bandaged puzzles or shape-changing puzzles like Square-1 which have been around for ages. We don't want to include those in this new category. The new category seems well-served by my proposed definition that slices may be rotated by angles that don't divide rationally into 360 degrees. I'm not sure that "jumbleable" is such a good word for this though, since it sounds like it just means that it can be mixed up, like any twisty puzzle. Maybe "disturbable"? Or "wonkable"??

Anyhoo, back to the helicopter cube specifically, I've been thinking about theoretical mods that could be made. There are some limits on the shape-changing possibilities. For example, any extension above the cubes face would cause two slicing planes to pass through each other, thus requiring a change to the mech to work. It can't be turned into a rhombic dodecahedron shape unfortunately nor an octahedron. So shaving down seems the way to go.

It could be shaved down into a cuboctahedron, but this would add nothing to the solution, and would in fact make the solution easier since corner pieces could now have any orientation. I think the best concept would be to shave it down to a rhombicuboctahedron, like this:

Image

I don't think the shaving down would interfere with the inner mech. Now the solution changes. With the colouring above (the 12 rhombic squares all the same colour), the original corner pieces are no longer orientable, but the order of the original four central pieces on each face now matters. But wait, we can do better:

Image

With this colouring, all 8 triangles and 6 cubic squares can be the same colour, but all 12 rhombic squares (in the face planes of a rhombic dodecahedron, ie the ones that share edges with both squares and triangles) would be different colours. Now orientation and order of all pieces matters.

Note: the rhombuses at the centre of these 12 squares rotate, but can't move from their starting positions, like the face centres on a 3x3x3. I imagine solving would be pretty similar, except for the whole jumbleable thing, but I can't tell how much that changes things.

Rob.


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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sat May 17, 2008 3:18 am 
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What would a Rhombic Dodecahedron version of the Helicopter Cube look like?

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sat May 17, 2008 6:58 am 
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It'll look like just the mechanism--to make a rhombic dodecahedron from a cube, one would make the cuts along the same line that define the slices for the helicopter cube.

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sat May 17, 2008 11:59 am 
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Robert Webb wrote:
There probably needs to be another word for puzzles which change shape.


How about 'shape-changing'?

Robert Webb wrote:
What we're talking about here appears to be a newish phenomenon, not like bandaged puzzles or shape-changing puzzles like Square-1 which have been around for ages. We don't want to include those in this new category. The new category seems well-served by my proposed definition that slices may be rotated by angles that don't divide rationally into 360 degrees.


Wasn't that my proposal?

Robert Webb wrote:
I'm not sure that "jumbleable" is such a good word for this though, since it sounds like it just means that it can be mixed up, like any twisty puzzle. Maybe "disturbable"? Or "wonkable"??


'disturbable' and 'wonkable' just aren't as catchy :-)

Regardless, jumbling is definitely something far stranger than simple shape modding, and should be recognized as such.


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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sat May 17, 2008 7:20 pm 
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I got a chance to play with a Bevel cube made by Okomato, and was shown the 'jumbable' abilites. Is the helicopter cube, made by Puzzlemaster42, 'jumbable'? I havn't show any pictures where it is.

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sat May 17, 2008 7:49 pm 
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joey wrote:
I got a chance to play with a Bevel cube made by Okomato, and was shown the 'jumbable' abilites. Is the helicopter cube, made by Puzzlemaster42, 'jumbable'? I havn't show any pictures where it is.

Yes it is.

I have played with one of Adam's Helicopter Cubes (Thanks again Bryan), and it does indeed 'jumble'.

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sun May 18, 2008 4:20 am 
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Noah wrote:
What would a Rhombic Dodecahedron version of the Helicopter Cube look like?


See below.

the.drizzle wrote:
It'll look like just the mechanism--to make a rhombic dodecahedron from a cube, one would make the cuts along the same line that define the slices for the helicopter cube.


Well, there wouldn't be much point getting to a rhombic dodecahedron by slicing it down using the twisting planes, or the puzzle wouldn't move. Instead, we can get to the same shape by adding an appropriately sized pyramid on each face of the cube. Then we get this:

Image
(Made using Great Stella).

But as I mentioned, it couldn't use the same mech.

Bram wrote:
Robert Webb wrote:
The new category seems well-served by my proposed definition that slices may be rotated by angles that don't divide rationally into 360 degrees.


Wasn't that my proposal?


Maybe another case of great minds thinking alike. The earliest mention of "360" on this thread was when I wrote "twists happen at angles that don't factor into 360 an integral number of times". You then wrote: "How about if we define jumbleability as involving rotations which aren't a rational fraction of 360 degrees?".

You're right about dividing rationally rather than integrally though.

Bram wrote:
'disturbable' and 'wonkable' just aren't as catchy :-)


No, they weren't great suggestions, but then is "jumbleable" so catchy? Sounds pretty awkward to me. Not sure the "e" should be in there. And I can easily imagine someone saying they jumbled their Rubik's cube. Just sounds like scrambled to me. But if others are happy I'm not fussed :)

Rob.


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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sun May 18, 2008 4:53 am 
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Just thought of another couple of helicopter mods that might work. Both would be cut down from the original cube form:

Image

It's quite a close shave though. There may not be enough of each piece left to house the mechanism. Here's another possibility:

Image
(Both created with Great Stella again)

Rob.


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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sun May 18, 2008 5:39 am 
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Robert Webb wrote:
Image


Wow, that would make a great puzzle.

Thomas

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sun May 18, 2008 5:42 am 
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It may have already been concieved. And being built...

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sun May 18, 2008 7:02 am 
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the.drizzle wrote:
It may have already been concieved. And being built...

:O

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sun May 18, 2008 1:40 pm 
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Robert Webb wrote:
Image


Thank you! I was wondering what it would look like. I was sure something of that nature could be done.

Drizzle, come on. Lets see it. :wink: You know you want to show us this secret project.

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sun May 18, 2008 3:43 pm 
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Noah wrote:
Drizzle, come on. Lets see it. :wink: You know you want to show us this secret project.


Give me a week or two...

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 Post subject: Re: New Puzzle: Helicopter Cube
PostPosted: Sun May 18, 2008 3:53 pm 
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the.drizzle wrote:
Noah wrote:
Drizzle, come on. Lets see it. :wink: You know you want to show us this secret project.


Give me a week or two...

A week or two won't cut it the.drizzle. I want it on my desk by monday. This is your last chance, you've been slack too often.

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