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 Post subject: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 8:17 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
At a recent gathering we got into a discussion with a few people who don't know it already about jumbling, and discussed how conjectural it is in many cases. Bill Gosper, who actually knows a thing or two about these things, asked a few other people, and got the following response from Julian Ziegler Hunts:

Quote:
Putting Cos[x]=p/q into the cosine double angle formula you get a denominator of either q^2 or q^2/2 for Cos[2x]. Iterate. If you started out with a rational angle you must eventually repeat; if q>2 then at this point you have two different smallest denominators for the same number. Unless he means an irrational number rather than an irrational angle, in which case you just use Lindemann-Weierstrass to show that e^(i*p/q) (or e to any nonzero algebraic number) is transcendental, so since Cos[x] is an algebraic function of e^ix it's transcendental (hence !=1/3) for a nonzero rational (or indeed algebraic) argument.

Honestly I don't really follow. Gosper also gave some general incredulousness about whether it's arccos(1/3) instead of arccos(-1/3). I don't know about that either. Could anyone with more of a mathematical bent, and/or someone who's thought seriously about what the jumbling angle on a helicopter cube is, care to comment?

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 9:06 pm

Joined: Fri Dec 20, 2013 11:47 pm
Jumbling is when a piece leaves its normal orbit correct? If so then the helicopter cube definately jumbles. Each edge/center piece on that puzzle is trapped in an orbit with the 5 other color pieces and cannot leave that orbit unless it jumbles, which is why when you do the jumbling moves you can have 2 red pices next to each other when you could not if you did not jumble the puzzle.

Last edited by garathnor on Tue Jul 15, 2014 11:35 pm, edited 1 time in total.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 9:20 pm

Joined: Wed Jan 01, 2014 11:06 am
garathnor wrote:
Jumbling is when a piece leaves its normal orbit correct?

Not quite. Jumbling is moves that cannot be unbandaged or un-shapemodded into a fully doctrinaire set of moves. Changing orbits is often a consequence, however it is not the definition. Since nobody has shown a way to unbandage the helicopter cube, and because as I know it the irrational angles(pi/x where x is irrational) mean jumbling I think the helicopter cube does jumble.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 9:31 pm

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If I remember correctly, the helicopter cube was the first jumbling puzzle, therefore it must jumble as it brought up the need to coin the term. And
rayray_2561, you are correct with your definition, but I have no idea about the math.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 9:58 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
BelcherBoy2000 wrote:
If I remember correctly, the helicopter cube was the first jumbling puzzle, therefore it must jumble as it brought up the need to coin the term. And
rayray_2561, you are correct with your definition, but I have no idea about the math.

We don't yet have a method of proving something jumbles. Bob has shown that some of the 2D puzzles that look like they jumble actually can be fully resolved and don't jumble until they cross some critical cut depth. We don't even have a proof that past this critical cut depth the puzzle really does jumble. The evidence is overwhelming but that isn't a proof. This means we don't KNOW that the Helicopter cube jumbles even though everyone thinks it does.

I'm not sure proving that the Helicopter Cube jumbling angle is irrational helps much either. There are a few hiccups with using irrationality as evidence. First, a circle is 2 pi radians with itself is irrational. I think we really want (2 pi) / (jumbling angle) to be irrational. Even that breaks down though because the 45-degree Rubik's cube seems to jumble which has a jumbling angle of (pi / 4) and (2 pi) / (pi / 4) = 8, a rational number.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 10:13 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
It depends on rather you ask a theorist or an experimentalist. Even if the puzzle can be proven to not jumble yet it would need to be the size of the solar system to resolve the smallest pieces with the human eye... I think all experimentalists would still say it jumbles.

Just my too cents...
Carl

P.S. Even having said that, I'd love to see what the theorists can come up with.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 10:18 pm

Joined: Wed Jan 01, 2014 11:06 am
How I understand the helicopter cube is that because it has an irrational angle(pi/x with x irrational), it jumbles. However I am not saying that irrational angle is the defintion of jumbling, just that you can't have an irrationally angled puzzle that doesn't jumble.

And the theorist in me says that if a puzzle has been proven to not jumble, then it doesn't jumble, even though it may seem like it.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 10:36 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
An irrational angle makes the puzzle have to be jumbling, because repeating the jumbling angle has to all be part of the allowable cuts, and by definition if the angle is irrational then no amount of repeating that rotation can wind up with exactly lining up to where it started.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Tue Jul 15, 2014 11:05 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Bram wrote:
An irrational angle makes the puzzle have to be jumbling, because repeating the jumbling angle has to all be part of the allowable cuts, and by definition if the angle is irrational then no amount of repeating that rotation can wind up with exactly lining up to where it started.

I think your argument is too simplistic.

Take a Helicopter cube. It jumbles at acos(1/3) which is about 70.528779 degrees. You don't just turn an edge 70.53 apply a cut, turn it another 70.53 degrees, apply another cut, etc. Your "no amount of repeating the rotation" seems to imply this is what is done.

The adjacent edges on Helicopter cube can make turns when an edge is at the following angles:
• 0 degrees
• 70.528779 degrees
• 109.471220 degrees (180 - jumbling angle)
• 180 degrees
• 250.528779 degrees (180 + jumbling angle)
• 289.471220 degrees (360 - jumbling angle)

Nowhere here is there (N * jumbling angle MOD 360).

The thing that causes pieces to keep getting cut more and more is that pieces undergo some rotation as they're moved between various edges. Just like how a corner can get twisted 120 degrees on a Rubik's cube even though faces only perform rotations in increments of 90 degrees.

I think what actually needs to be shown is not that the jumbling angle is irrational but that some composition of several turns causes pieces to rotate irrational amounts.

I'm sure the calculation of how much a piece gets twisted has as much to do with the jumbling angle of the turnable portions as does the angle and relationship the turnable portions have with each other. It is entirely conceivable that the faces of some strange polyhedron cause pieces to be rotation rational amount even though the angle you turn the faces is irrational.

I think this is why even the 45-degree Rubik's cube jumbles. The amount pieces get twisted must be irrational. I haven't worked out the math to show that though.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 2:00 am

Joined: Mon Feb 28, 2011 4:54 am
Bram wrote:
Quote:
Putting Cos[x]=p/q into the cosine double angle formula you get a denominator of either q^2 or q^2/2 for Cos[2x]. Iterate. If you started out with a rational angle you must eventually repeat; if q>2 then at this point you have two different smallest denominators for the same number. Unless he means an irrational number rather than an irrational angle, in which case you just use Lindemann-Weierstrass to show that e^(i*p/q) (or e to any nonzero algebraic number) is transcendental, so since Cos[x] is an algebraic function of e^ix it's transcendental (hence !=1/3) for a nonzero rational (or indeed algebraic) argument.

Honestly I don't really follow.

Thank goodness. If you don't follow, I can relax, and stop thinking how dumb I must be to not grasp these basic concepts.

From a mathematical point of view I would be interested in understanding about jumbling angles and how to find/show them. If anyone wants to PM me, or point me to such a discussion, that'd be cool, unless it belongs in this thread.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 2:14 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
rline wrote:
From a mathematical point of view I would be interested in understanding about jumbling angles and how to find/show them. If anyone wants to PM me, or point me to such a discussion, that'd be cool, unless it belongs in this thread.

The jumbling angle for the Helicopter cube is the the same as the central angle for the cube. See http://www.kjmaclean.com/Geometry/Cube.html for some details and check out this table.
Matt Galla did an amazing job of explaining how this work here.

For the Helicopter cube, you only have to be able to picture three edges that share a corner to understand what's going on.

Imagine the cuts for the UF, UR, and FR edges. If you don't turn UR at all, both UF and FR edges can turn. Jumbling happens when you start to turn UR clockwise and the cut that was made for the FR edge aligns with the cut made for the UF edge. Matt goes into some detail about how this works.

Re: Deeper cut FT tetragonal trapezohedron aka Crystal Trap!
Re: Edge turning dodecahedron jumbling properties.

I wouldn't mind trying to reproduce these results and maybe compute some more but my trigonometry + geometry skills have atrophied a lot.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 2:24 am

Joined: Mon Feb 28, 2011 4:54 am
Brandon Enright wrote:
rline wrote:
From a mathematical point of view I would be interested in understanding about jumbling angles and how to find/show them. If anyone wants to PM me, or point me to such a discussion, that'd be cool, unless it belongs in this thread.

The jumbling angle for the Helicopter cube is the the same as the central angle for the cube....
I wouldn't mind trying to reproduce these results and maybe compute some more but my trigonometry + geometry skills have atrophied a lot.

Thanks a lot Brandon I'll have a good look through them.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 7:06 am

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
Just to make sure we're speaking about the same things:

A rational angle θ is an angle such that there exists a natural number n such that nθ ≡ 0 (mod 2π).
Equivalently, a rational angle θ can be written as θ = qπ for some rational number q.

An irrational angle φ is an angle such that there's no natural number n such that nφ ≡ 0 (mod 2π).
Equivalently, an irrational angle φ can't be written as φ = qπ for any rational number q.

Agreed?

Brandon Enright wrote:
Bram wrote:
An irrational angle makes the puzzle have to be jumbling, because repeating the jumbling angle has to all be part of the allowable cuts, and by definition if the angle is irrational then no amount of repeating that rotation can wind up with exactly lining up to where it started.

I think your argument is too simplistic.

Take a Helicopter cube. It jumbles at acos(1/3) which is about 70.528779 degrees. You don't just turn an edge 70.53 apply a cut, turn it another 70.53 degrees, apply another cut, etc. Your "no amount of repeating the rotation" seems to imply this is what is done.

The adjacent edges on Helicopter cube can make turns when an edge is at the following angles:
• 0 degrees
• 70.528779 degrees
• 109.471220 degrees (180 - jumbling angle)
• 180 degrees
• 250.528779 degrees (180 + jumbling angle)
• 289.471220 degrees (360 - jumbling angle)

Nowhere here is there (N * jumbling angle MOD 360).

The thing that causes pieces to keep getting cut more and more is that pieces undergo some rotation as they're moved between various edges. Just like how a corner can get twisted 120 degrees on a Rubik's cube even though faces only perform rotations in increments of 90 degrees.

I think what actually needs to be shown is not that the jumbling angle is irrational but that some composition of several turns causes pieces to rotate irrational amounts.

I agree with Bram. Let's use a Curvy Copter because the centers are visible. Let's call the jumbling angle α (≈ 70.5°) for short.

Attachment:

jumblingcurvycopter.png [ 27.79 KiB | Viewed 643 times ]

• Turn the right edge counterclockwise by α, so that the red cut aligns with the cut for the bottom edge.
• This blocks the left edge from turning. To enable it again, we need to introduce the blue cut.
• Turn the right edge counterclockwise by α again, and repeat the process.

The angle between two consecutive added cuts is α, and if α is an irrational angle, the Curvy Copter (and Helicopter Cube) jumble. This construction applies to all puzzles that turn in irrational angles.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 9:58 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Coaster1235 wrote:
A rational angle θ is an angle such that there exists a natural number n such that nθ ≡ 0 (mod 2π).
Yep, agreed.

Coaster1235 wrote:
I agree with Bram. Let's use a Curvy Copter because the centers are visible. Let's call the jumbling angle α (≈ 70.5°) for short.

• Turn the right edge counterclockwise by α, so that the red cut aligns with the cut for the bottom edge.
• This blocks the left edge from turning. To enable it again, we need to introduce the blue cut.
• Turn the right edge counterclockwise by α again, and repeat the process.

The angle between two consecutive added cuts is α, and if α is an irrational angle, the Curvy Copter (and Helicopter Cube) jumble. This construction applies to all puzzles that turn in irrational angles.

Nice images. Very useful.

We disagree on the scope of cuts that should be added for unbandaging. If cuts were added like you propose the proof would be trivial like above.

It seems like you want to add a cut any time anything is blocked. When you turn the right edge -α the cut that was applied by the left edge moves into the bottom and you can turn the bottom edge. This follows naturally from the geometry. What doesn't follow for me is that you want to add the blue cut to re-enable the left edge. When the right edge is in the -α state the left edge isn't naturally turnable in the geometry (but the bottom edge is).

I think a more minimal approach to unbandaging should be taken by following the geometry of the puzzle to determine when we should expect cuts to align.

The rules I've followed for unbandaging cuts are:

1) Whenever an edge is in the 0 or 180 degree turn state the 4 adjacent edges should be turnable (assuming they aren't blocked by other edges they're adjacent to).

2) When an edge is in the +α or 180 + α state the top left and bottom right adjacent edges should still be turnable (assuming other edges don't have them blocked).

3) When an edge is in the -α or 180 - α state the bottom left and top right adjacent edges should be turnable (assuming...).

Following these rules the puzzle maintains its rhombic dodecahedral nature and no amount of jumbling will cause blockage beyond what should be blocked by the geometry.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 3:22 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Brandon Enright wrote:
We disagree on the scope of cuts that should be added for unbandaging. If cuts were added like you propose the proof would be trivial like above.

It seems like you want to add a cut any time anything is blocked. When you turn the right edge -α the cut that was applied by the left edge moves into the bottom and you can turn the bottom edge. This follows naturally from the geometry. What doesn't follow for me is that you want to add the blue cut to re-enable the left edge. When the right edge is in the -α state the left edge isn't naturally turnable in the geometry (but the bottom edge is).

I think a more minimal approach to unbandaging should be taken by following the geometry of the puzzle to determine when we should expect cuts to align.
You need to unbandage the puzzle into a doctrinaire one. True... doctrinaire puzzles can have stored cuts, the Bubbloid122 is an example. But what I don't see here is how this is a doctrinaire puzzle if that blue cut isn't put in to re-enable the left edge.

Let's assume the puzzle is a sphere. After any and all turns the puzzle should return to its doctrinaire state. If it doesn't... its either jumbling or its still bandaged.

Carl

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 5:11 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
Let's assume the puzzle is a sphere. After any and all turns the puzzle should return to its doctrinaire state. If it doesn't... its either jumbling or its still bandaged.

I agree with this statement but I'm saying that the 4 twist angles that aren't either 0 or 180 degrees should always be viewed as a mid-turn state and all of the ones not at 0 or 180 should be turned back to 0 or 180 before the puzzle is evaluated to see if its doctrinaire (with stored cuts).

In Coaster1235's image the edge has been turned by -α and this moves the portion of the cut that was made by the left edge into the portion turnable by the bottom edge. This is naturally allowed by the rhombic dodecahedral geometry so we'd expect the bottom edge to be enable when the right edge is twisted by -α. The left edge isn't enable though and we wouldn't expect it to be based on the rhombic dodecahedral geometry. Adding the blue cut is unnecessary and arbitrary unbandaging.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 6:37 pm

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Wed Jul 16, 2014 8:33 pm

Joined: Tue Aug 11, 2009 2:44 pm
Brandon Enright wrote:
wwwmwww wrote:
Let's assume the puzzle is a sphere. After any and all turns the puzzle should return to its doctrinaire state. If it doesn't... its either jumbling or its still bandaged.

I agree with this statement but I'm saying that the 4 twist angles that aren't either 0 or 180 degrees should always be viewed as a mid-turn state and all of the ones not at 0 or 180 should be turned back to 0 or 180 before the puzzle is evaluated to see if its doctrinaire (with stored cuts).

Well with that view, then it trivially doesn't jumble. If I turn a Rubik's Cube face 45°, a mid-turn state, the adjoining faces are now blocked, but we don't call that bandaged.

Carl and I had a discussion along these lines several months ago; I'm not sure I could dig up the thread now. My point then was that before we call a puzzle bandaged (let alone jumbling), we have to have a specification of what the possible moves are. For the helicopter to be considered jumbling, we have to have those α and -α angles for each edge in the set of possible states, therefore moves. Brandon is right that those states do emerge naturally from the geometry, but this is incidental to the design of an edge-rotating cube, not an intentional aspect of desired moves. So from a certain perspective, the puzzle may let you move that way, but those are not "legal" moves, so it shouldn't be considered bandaged. On a practical level, those moves are different from turning a Rubik's Cube face by 45°, because they enable further moves. So we want to consider those states as part of the definition of legal moves. But then we are talking about something different from a 180°-edge rotating cube.

So -- I guess I'm saying I agree with Carl and Coaster1235 here. If the Helicopter Cube is to be considered jumbling at all, you want to enable a move to any state (0, 180, α, -α, 180 + α, 180 - α) on any edge in any configuration. But more generally, I'm saying there are subtle issues with the definition of jumbling.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 1:17 pm

Joined: Tue Aug 11, 2009 2:44 pm
To continue -- I remember now some of my concerns in that earlier discussion with Carl. For e.g. the Helicopter Cube to be considered jumbling, we have to not only specify those additional turn angles; we have to incorporate all the current turns angles into our puzzle state. This seems inelegant. Also, if we remove the stickers, then every single position does *not* look the same. Of course, it's not a doctrinaire puzzle, so that's not surprising, but even if we were to "infinitely unbandage it", into jumble dust, it *still* wouldn't be doctrinaire (unlike the case with jumbling 2d circle puzzles). I find this disturbing. So I repeat, IMO there are still subtle issues with the concept of jumbling.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 1:30 pm

Joined: Tue Aug 11, 2009 2:44 pm
bhearn wrote:
Carl and I had a discussion along these lines several months ago; I'm not sure I could dig up the thread now.

Found it, here: http://twistypuzzles.com/forum/viewtopic.php?p=309625#p309625

Quote:
... The Fractured Cube, as I understand it, essentially allows 60° turns at each vertex, when not blocked by bandaging. The Helicopter Cube has these additional, irrational-angle turns. When you have made one, that edge is in an inherently different state. You can't turn by that angle again. So, you cannot simply characterize a Helicopter Cube as a particular instance of a circle puzzle on a sphere, with given center points, R, and N. But you *can* (I believe) so characterize a Fractured Cube. Thus, in order to speak of the Helicopter Cube as jumbling, it's necessary to define exactly what it is in a somewhat more complicated way. And the same seems to be true for most jumbling puzzles.

Basically, it comes down to what configurations should naturally be considered as bandaged. You want to consider any position you can physically reach as "valid", thus any moves made to get there should count as moves. Except that partial moves that don't enable other moves are not really moves (like turning a Rubik's Cube face 30°)... except when they are. I can imagine a bandaged position in which only one turn axis is unblocked, when one would "expect" others to be unblocked. (I think More Madness has states like this.) So then, it seems to me a little fuzzy, still, defining a position as bandaged, without explicitly specifying the desired unbandaged states (which in the case of the Helicopter Cube means adding state, a kind of parity, describing the allowed and current angles of each edge). Therefore, I'm not 100% sure it's even meaningful to describe a puzzle as bandaged without reference to some subjectively defined notion of what one should be able to do with it, above and beyond what is directly implied by the physical object itself. Perhaps one could come up with a definition. Maybe it's already out there. But if this issue doesn't seem familiar to you (I haven't been on the forums much lately), maybe not.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 1:33 pm

Joined: Tue Aug 11, 2009 2:44 pm
Bram wrote:
At a recent gathering we got into a discussion with a few people who don't know it already about jumbling, and discussed how conjectural it is in many cases.

The funny thing is that this discussion started so we could explain jumbling to Tom Rokicki, so I went and grabbed a Helicopter Cube. Later I remembered that I had borrowed *his* Helicopter Cube at G4G to use as a prop explaining jumbling in my talk on circle puzzles. I guess I must not have explained it very memorably!

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 2:34 pm

Joined: Wed Jan 01, 2014 11:06 am
I think the biggest discrepancy here is about whether or not all axes need to be able to turn after every move, and what constitutes a legal move. I personally think that all axes should be able to turn. These are some starting definitions that others can object to or offer changes to:

Legal Move: Any twist of any axis that allows at least 1 axis that it previously shared at least 1 piece with to move.

Jumbling: A puzzle jumbles if legal moves are possible that block adjacent axes and are unable to be unbandaged into a puzzle where all axes can freely turn both before and after any legal move.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 3:44 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bhearn wrote:
but even if we were to "infinitely unbandage it", into jumble dust, it *still* wouldn't be doctrinaire (unlike the case with jumbling 2d circle puzzles). I find this disturbing. So I repeat, IMO there are still subtle issues with the concept of jumbling.
You have to elaborate here a bit. Why is the infinitely unbandaged helicopter cube not doctrinaire? If it doesn't return to the doctrinaire state after a single turn I'd be inclined to say you haven't completed the infinite unbandaging yet. Seeing as this is an infinite process and it is never really complete your statement really puzzles me.

Carl

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 4:03 pm

Joined: Tue Aug 11, 2009 2:44 pm
wwwmwww wrote:
You have to elaborate here a bit. Why is the infinitely unbandaged helicopter cube not doctrinaire? If it doesn't return to the doctrinaire state after a single turn I'd be inclined to say you haven't completed the infinite unbandaging yet. Seeing as this is an infinite process and it is never really complete your statement really puzzles me.

Because not every reachable position looks the same.

You agreed with me back in November.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 4:22 pm

Joined: Tue Aug 11, 2009 2:44 pm
rayray_2561 wrote:
I think the biggest discrepancy here is about whether or not all axes need to be able to turn after every move, and what constitutes a legal move. I personally think that all axes should be able to turn. These are some starting definitions that others can object to or offer changes to:

Legal Move: Any twist of any axis that allows at least 1 axis that it previously shared at least 1 piece with to move.

Jumbling: A puzzle jumbles if legal moves are possible that block adjacent axes and are unable to be unbandaged into a puzzle where all axes can freely turn both before and after any legal move.

This is not bad... there is just one annoying factor, that I mentioned above:

bhearn wrote:
You want to consider any position you can physically reach as "valid", thus any moves made to get there should count as moves. Except that partial moves that don't enable other moves are not really moves (like turning a Rubik's Cube face 30°)... except when they are. I can imagine a bandaged position in which only one turn axis is unblocked, when one would "expect" others to be unblocked. (I think More Madness has states like this.)

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 4:38 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bhearn wrote:
Because not every reachable position looks the same.

You agreed with me back in November.
I need to go back and see what I agreed to. On first pass I agreed that there were some fundamental differences between 3D and 2D puzzles. I'm not sure I realized this was one of them.

So if the Jumble Dust Helicopter Cube has positions which look different don't you just keep adding cuts until they do all look the same? At some point you have infinitely many essentially zero volume pieces and I would think all states would look the same.

Granted I'm getting over a bug I had yesterday and it wouldn't take much to convince me the grass was blue and the sky was green at the moment... so maybe I'm missing something that was obvious to me back in November that is escaping me now.

Carl

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 4:39 pm

Joined: Wed Jan 01, 2014 11:06 am
bhearn wrote:
rayray_2561 wrote:
I think the biggest discrepancy here is about whether or not all axes need to be able to turn after every move, and what constitutes a legal move. I personally think that all axes should be able to turn. These are some starting definitions that others can object to or offer changes to:

Legal Move: Any twist of any axis that allows at least 1 axis that it previously shared at least 1 piece with to move.

Jumbling: A puzzle jumbles if legal moves are possible that block adjacent axes and are unable to be unbandaged into a puzzle where all axes can freely turn both before and after any legal move.

This is not bad... there is just one annoying factor, that I mentioned above:

bhearn wrote:
You want to consider any position you can physically reach as "valid", thus any moves made to get there should count as moves. Except that partial moves that don't enable other moves are not really moves (like turning a Rubik's Cube face 30°)... except when they are. I can imagine a bandaged position in which only one turn axis is unblocked, when one would "expect" others to be unblocked. (I think More Madness has states like this.)

I'm not quite sure what you are getting at here... I'm sorry if I'm missing something, but what is your "annoying factor" specifically? Is it that I'm not allowing physical states to be reached with my definitions? Or something else?

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 4:43 pm

Joined: Tue Aug 11, 2009 2:44 pm
wwwmwww wrote:
So if the Jumble Dust Helicopter Cube has positions which look different don't you just keep adding cuts until they do all look the same? At some point you have infinitely many essentially zero volume pieces and I would think all states would look the same.

The 0° position and the α° position will never look the same. From 0, you can turn α, -α, 180, 180 - α, or 180 + α degrees. You can't turn those same angles from the α position.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 4:45 pm

Joined: Tue Aug 11, 2009 2:44 pm
rayray_2561 wrote:
I'm not quite sure what you are getting at here... I'm sorry if I'm missing something, but what is your "annoying factor" specifically? Is it that I'm not allowing physical states to be reached with my definitions? Or something else?

It's this: there can be cases where this definition of legal move excludes things that would normally be considered moves. Consider a cube where in some configuration, only one face can turn, and only in one position are other moves possible. Then the canonical 90° turns of that face are not considered legal moves. Whereas intuitively, these are legal moves that result in bandaged states.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 4:51 pm

Joined: Wed Jan 01, 2014 11:06 am
bhearn wrote:
rayray_2561 wrote:
I'm not quite sure what you are getting at here... I'm sorry if I'm missing something, but what is your "annoying factor" specifically? Is it that I'm not allowing physical states to be reached with my definitions? Or something else?

It's this: there can be cases where this definition of legal move excludes things that would normally be considered moves. Consider a cube where in some configuration, only one face can turn, and only in one position are other moves possible. Then the canonical 90° turns of that face are not considered legal moves. Whereas intuitively, these are legal moves that result in bandaged states.

Is spamurai's 1x1x2 roadblock a bandaged puzzle or just a 1x2x3 with the solved state in mid-turn? Just because it appears that faces should turn in increments of 90° or because you are used to them doing so does not make them automatically need to be legal moves.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:00 pm

Joined: Tue Aug 11, 2009 2:44 pm
... also, and this is in line with my answer to Carl, "a puzzle where all axes can freely turn both before and after any legal move" is not the same as Bram's definition of a doctrinaire puzzle. What would one call such a puzzle, say a hypothetically fully unbandaged Helicopter (though we know that can't be done)?

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:04 pm

Joined: Wed Jan 01, 2014 11:06 am
bhearn wrote:
... also, and this is in line with my answer to Carl, "a puzzle where all axes can freely turn both before and after any legal move" is not the same as Bram's definition of a doctrinaire puzzle. What would one call such a puzzle, say a hypothetically fully unbandaged Helicopter (though we know that can't be done)?

free-turning. That's my suggestion.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:09 pm

Joined: Tue Aug 11, 2009 2:44 pm
rayray_2561 wrote:
Is spamurai's 1x1x2 roadblock a bandaged puzzle or just a 1x2x3 with the solved state in mid-turn?

I would say the latter. The difference from my scenario is that in the case I proposed, in some positions turning a face by 90° is a legal move, and in others it's not.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:09 pm

Joined: Tue Aug 11, 2009 2:44 pm
rayray_2561 wrote:
bhearn wrote:
... also, and this is in line with my answer to Carl, "a puzzle where all axes can freely turn both before and after any legal move" is not the same as Bram's definition of a doctrinaire puzzle. What would one call such a puzzle, say a hypothetically fully unbandaged Helicopter (though we know that can't be done)?

free-turning. That's my suggestion.

That seems reasonable.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:11 pm

Joined: Wed Jan 01, 2014 11:06 am
bhearn wrote:
rayray_2561 wrote:
Is spamurai's 1x1x2 roadblock a bandaged puzzle or just a 1x2x3 with the solved state in mid-turn?

I would say the latter. The difference from my scenario is that in the case I proposed, in some positions turning a face by 90° is a legal move, and in others it's not.

I'm sorry but I still don't understand. Could you show some pictures that demonstrate the functioning of the puzzle in your scenario?

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:15 pm

Joined: Wed Jan 01, 2014 11:06 am
bhearn wrote:
wwwmwww wrote:
So if the Jumble Dust Helicopter Cube has positions which look different don't you just keep adding cuts until they do all look the same? At some point you have infinitely many essentially zero volume pieces and I would think all states would look the same.

The 0° position and the α° position will never look the same. From 0, you can turn α, -α, 180, 180 - α, or 180 + α degrees. You can't turn those same angles from the α position.

As shown by Coaster1235's pictures, the center does get cut. I think you'd end up with 12 circles surrounded by dust for the fully unbandaged version. These circles would look the same in the 0° position and the α position.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:16 pm

Joined: Tue Aug 11, 2009 2:44 pm
rayray_2561 wrote:
I'm sorry but I still don't understand. Could you show some pictures that demonstrate the functioning of the puzzle in your scenario?

I'm not sure a picture would help; I don't know if I can create a concrete example. But suppose you have a cube-shaped puzzle, where all the legal moves are 90° or 180° turns of a face. In some positions some axes are blocked. Now, we might have a position where one axis is not blocked, but its face has no legal moves per the definition above, because at any turn angle all adjoining axes are blocked (other than the current, start angle). Yet, in other positions this same face might have 90° turns as legal moves. This the case in which I'm saying it's counterintuitive to not call those 90° turns legal moves (resulting in bandaged configurations). Counterintuitive doesn't mean wrong, though; maybe that's still the most reasonable definition.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:22 pm

Joined: Tue Aug 11, 2009 2:44 pm
rayray_2561 wrote:
As shown by Coaster1235's pictures, the center does get cut. I think you'd end up with 12 circles surrounded by dust for the fully unbandaged version. These circles would look the same in the 0° position and the α position.

Well that's an interesting idea. I guess I would find that surprising; that would mean we would have incidentally enabled the 2α angle as well as the α angle for each axis. My bet is that's not the case, but I'm not sure I can easily prove it. To carry your suggestion to an extreme, we might imagine that we've now enabled turning any axis by *any* angle (and after all, we can get arbitrarily close to any angle with some multiple of an irrational angle). But I would suspect that all dust is not created equal. If my 2d jumbling explorations are any indication, there are still finite-sized pieces embedded in the dust, and those would break the continuous circle symmetry.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:25 pm

Joined: Wed Jan 01, 2014 11:06 am
bhearn wrote:
Counterintuitive doesn't mean wrong, though; maybe that's still the most reasonable definition.

I honestly don't know about this. We could change my "legal move" to something more telling of the definition, like "freely followable move," or we could just keep the definitions as-is. I'd like to see what others think.

(P.S. I get your scenario now. Thanks for the explanation)

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 5:47 pm

Joined: Wed Jan 01, 2014 11:06 am
bhearn wrote:
rayray_2561 wrote:
As shown by Coaster1235's pictures, the center does get cut. I think you'd end up with 12 circles surrounded by dust for the fully unbandaged version. These circles would look the same in the 0° position and the α position.

Well that's an interesting idea. I guess I would find that surprising; that would mean we would have incidentally enabled the 2α angle as well as the α angle for each axis. My bet is that's not the case, but I'm not sure I can easily prove it. To carry your suggestion to an extreme, we might imagine that we've now enabled turning any axis by *any* angle (and after all, we can get arbitrarily close to any angle with some multiple of an irrational angle). But I would suspect that all dust is not created equal. If my 2d jumbling explorations are any indication, there are still finite-sized pieces embedded in the dust, and those would break the continuous circle symmetry.

But irrational angle times [insert needed whole number here] mod 2pi = Any irrational angle. With any irrational angle needing to be able to turn, any finite-sized piece would be broken up by the infinite amount of irrational angles that would put that piece in another slice.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 6:07 pm

Joined: Tue Aug 11, 2009 2:44 pm
rayray_2561 wrote:
But irrational angle times [insert needed whole number here] mod 2pi = Any irrational angle. With any irrational angle needing to be able to turn, any finite-sized piece would be broken up by the infinite amount of irrational angles that would put that piece in another slice.

I was about to explain/argue that we don't have multiples of α, only , α, -α, 180 + α, and 180 - α. But then I went back and looked at Coaster1235's post more closely, and I see I am wrong.

Well, that changes everything for me... hmm.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 6:24 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bhearn wrote:
The 0° position and the α° position will never look the same. From 0, you can turn α, -α, 180, 180 - α, or 180 + α degrees. You can't turn those same angles from the α position.
Granted I was sick yesterday and I'm still not 100% but I'm getting very confused now. Above you stated:
bhearn wrote:
So -- I guess I'm saying I agree with Carl and Coaster1235 here. If the Helicopter Cube is to be considered jumbling at all, you want to enable a move to any state (0, 180, α, -α, 180 + α, 180 - α) on any edge in any configuration.
You do say "any configuration" so aren't the 0° position and the α° position just two different configurations? So yes... you should be able to make all those turns from both. So what you are saying appears contradictory to me... either that or I'm still missing something.
bhearn wrote:
I was about to explain/argue that we don't have multiples of α, only , α, -α, 180 + α, and 180 - α. But then I went back and looked at Coaster1235's post more closely, and I see I am wrong.

Well, that changes everything for me... hmm.
Hmmm... maybe we are on the same page now. I'm so lost at the moment I'm honestly not sure what page I'm on.

Carl

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 6:34 pm

Joined: Wed Dec 14, 2011 12:25 pm
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I'd like to suggest we define an illegal move as such: if after a move no adjacent layer can turn, the move is illegal. Otherwise, the move is legal. (Here, an adjacent layer means a layer that doesn't commute with the layer turned, basically any layer that shares pieces with the turned layer.)

Brandon Enright wrote:
In Coaster1235's image the edge has been turned by -α and this moves the portion of the cut that was made by the left edge into the portion turnable by the bottom edge. This is naturally allowed by the rhombic dodecahedral geometry so we'd expect the bottom edge to be enable when the right edge is twisted by -α. The left edge isn't enable though and we wouldn't expect it to be based on the rhombic dodecahedral geometry. Adding the blue cut is unnecessary and arbitrary unbandaging.

I think I can get behind this school of thought, too. You'd consider the Unbandaged Helicopter/Curvy Copter Plus completely unbandaged (ignoring any overhang bandaging), right?

If I'm interpreting you right, unbandaging shouldn't add new rotation states to the centers, yes? In other words, unbandaging shouldn't cut centers. This should probably have a term of its own, I'd suggest 'faithfully unbandaged'. Then again, I'm not sure if that definition is entirely meaningful. What would a faithfully unbandaged Bicube be? Would the term even have any meaning regarding 'artificially bandaged' puzzles? Also, as a side note, the Bermuda Cubes would all be faithfully unbandaged.

rayray_2561 wrote:
bhearn wrote:
rayray_2561 wrote:
As shown by Coaster1235's pictures, the center does get cut. I think you'd end up with 12 circles surrounded by dust for the fully unbandaged version. These circles would look the same in the 0° position and the α position.

Well that's an interesting idea. I guess I would find that surprising; that would mean we would have incidentally enabled the 2α angle as well as the α angle for each axis. My bet is that's not the case, but I'm not sure I can easily prove it. To carry your suggestion to an extreme, we might imagine that we've now enabled turning any axis by *any* angle (and after all, we can get arbitrarily close to any angle with some multiple of an irrational angle). But I would suspect that all dust is not created equal. If my 2d jumbling explorations are any indication, there are still finite-sized pieces embedded in the dust, and those would break the continuous circle symmetry.

But irrational angle times [insert needed whole number here] mod 2pi = Any irrational angle. With any irrational angle needing to be able to turn, any finite-sized piece would be broken up by the infinite amount of irrational angles that would put that piece in another slice.

What happens in your 2d puzzles with higher-fold turning circles? My intuition says those pieces of non-zero size would shrink into equal infinitely small dust as the so-called 'n-foldness' of the turning circle grows infinitely big, as it does with the center. Then again, it's only intuition.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 6:57 pm

Joined: Sun Nov 23, 2008 2:18 am
I think fudged puzzles like the Illegal Cube and Futtminx prove that you can remove the "jumble dust" from the infintely unbandaged version of a jumbling puzzle and still have enough solid pieces left to have a functional puzzle. Then again, both of those have rational jumbling angles, which raises the question if if such is only possible if the jumble angle is rational.

If someone could figure out an algorithm for for having a computer systematically unbanage the helicopter cube, I think it would be interesting to compare say, 1, 10, 100, and 1000 iterations of the algorithm to see if a pattern emerges, perhaps with each iteration removing any piece that would be to small to print in WSF. Logically, the algorithm would either eventually remove all of the cube or reach a point where new cuts would only further sub-divided pieces that have been removedfor being "too small". Of course, it might be more illuminating to use the helicopter ball since it automatically eliminates overhang bandaging and exposes the centers My best thought for such an unbandaging algorithm is something along the lines of: Start with a sphere with circles representing the cuts of the helicopter cube. Rotate a circle until an arc within it completes and external circle. Draw arcs to complete any circles that intersect the rotated circle. copy the new arc to the other eleven of the original circles. blacken out any bounded area of the sphere's curface that drops below a certain threshold size. repeat. No idea how one might go about implementing it though.

As for Free-Turning, while it isn't part of the definition of Doctrinaire as proposed by Bram and agreed on by general concensus, I'm pretty sure all puzzles that are doctrinaire or can be shapemodded to be doctrinaire have this property, and I am pretty sure bandaging and jumbling rule out the property.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 7:05 pm

Joined: Wed Jan 01, 2014 11:06 am
Jeffery Mewtamer wrote:
As for Free-Turning, while it isn't part of the definition of Doctrinaire as proposed by Bram and agreed on by general concensus, I'm pretty sure all puzzles that are doctrinaire or can be shapemodded to be doctrinaire have this property, and I am pretty sure bandaging and jumbling rule out the property.

What about bhearn's example? An internally bandaged 3x3x3 that is doctrinaire but can reach states where doctrinaire moves are possible but followable moves(my previous defintion of legal move) are not.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 7:26 pm

Joined: Wed Jan 01, 2014 11:06 am
My new proposed definitions(feel free to state your opinions on):

Followable move: Any twist of any axis that allows at least 1 axis that it previously shared at least 1 piece with to move.

Freely followable move: A followable move that does not block any axes.

Free-turning puzzle: A puzzle with only freely followable moves.

Jumbling puzzle: A puzzle with states reachable only via followable moves which have blocked axes, where these states are unable to be unbandaged into a free-turning puzzle.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Thu Jul 17, 2014 11:36 pm

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Location: Bay Area, California
Coaster1235 wrote:
If I'm interpreting you right, unbandaging shouldn't add new rotation states to the centers, yes? In other words, unbandaging shouldn't cut centers.
Yes, this was exactly my perspective on it.

It seems like we have two options:

1) Only add unbandaging cuts when the geometry naturally supports it.

2) Anytime something is turned an some cuts align, unbandaging cuts should be added so that everything turns (Coaster1235 blue cuts).

Although I proposed 1 as the right way to do it, it seems that an argument can be made for either. I missed the thread between Bob and Carl but Bob's thinking on it is nearly exactly like mine (edited for size and highlighted for emphasis):

bhern wrote:
Basically, it comes down to what configurations should naturally be considered as bandaged. [...] Except that partial moves that don't enable other moves are not really moves (like turning a Rubik's Cube face 30°)... except when they are. I can imagine a bandaged position in which only one turn axis is unblocked, when one would "expect" others to be unblocked. [...] it seems to me a little fuzzy, still, defining a position as bandaged, without explicitly specifying the desired unbandaged states (which in the case of the Helicopter Cube means adding state, a kind of parity, describing the allowed and current angles of each edge). Therefore, I'm not 100% sure it's even meaningful to describe a puzzle as bandaged without reference to some subjectively defined notion of what one should be able to do with it, above and beyond what is directly implied by the physical object itself. [...]

The main problem with my suggestion that we follow definition 1 above is that it actually eliminates the strange bandaging with jumbling on the Helicopter Cube. I believe a Curvy Copter + is actually a fully unbandaged version of the Helicopter Cube under definition 1.

I thought the jumbling turns would still allow pieces to be rotated irrational amounts and that infinite unbandaging would still be required but after playing with a physical puzzle and thinking about it more, I don't think that's the case.

So my question now is roughly: under definition 1 is the Helicopter Cube simply not deep enough to require infinite unbandaging? For example, if you tried to fully unbandage a Curvy Copter II, would you ever get pieces twisted irrational angles and therefor require infinite cuts?

For something like the 45-degree Rubik's Cube we know it jumbles because pieces get twisted an irrational amount (no I don't have a proof). The 45-degree Rubik's cube sidesteps the issue of having an arbitrary choice between definition 1 and definition 2 so I still think that in the general case the important thing to be looking at is not whether a turnable portion has an irrational jumbling angle, but instead whether pieces that can be twisted can be twisted irrational amounts.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Fri Jul 18, 2014 8:48 am

Joined: Wed Jan 01, 2014 11:06 am
There is something I would like to point out here, and someone else has already said it:
BelcherBoy2000 wrote:
If I remember correctly, the helicopter cube was the first jumbling puzzle, therefore it must jumble as it brought up the need to coin the term.

What are we asking in this thread? Are we asking whether the helicopter cube jumbles? Because the definition of jumbling is still being refined, questioned, and changed. Or are we asking, "Does the helicopter cube jumble by the current definition?" Eithr way, I think the answer should be yes. Not only because of what BelcherBoy2000 said, but also because of another part of the definition of jumbling: "cannot be fully unbandaged [or un-shapemodded] into a doctrinaire puzzle." That means that the "fully unbandaged" version of anything needs to be doctrinaire. Just like Carl said here:
wwwmwww wrote:
So if the Jumble Dust Helicopter Cube has positions which look different don't you just keep adding cuts until they do all look the same?

When you reach a state after a followable move, and this state has blocked axes, you add cuts to the puzzle in the effort to make it doctrinaire.

However, these cuts make all axes always open to turn from a followable state(a state reachable only via followable moves). This essentially is making the puzzle a free-turning puzzle. That's why I think we should change the definition of jumbling to something with free-turning rather than doctrinaire. It would make jumbling easier to prove, and we would no longer be debating whether centers get cut because of our clearer definition. I personally think centers should be cut, because then you can enable a puzzle to be free-turning(and I actually think this means it would also be doctrinaire, but I can't prove it). Others such as Brandon think that centers should not be cut, but ultimately, we just need to agree on something.

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Fri Jul 18, 2014 11:41 am

Joined: Sun Nov 23, 2008 2:18 am
Actually, I think there might be value in considering the end result of both styles of unbandaging. Using 1 for the more conservative style of unbandaging and 2 for the more extreme style(since no one has suggested names for them and I can't think of any), some questions that arise in my mind include:

1. If type 1 unbandaging of a puzzle gives a doctrinaire puzzle, are there any cases where type 2 adds more cuts?

2. Assuming the curvy copter plus(and its non-curvy variant) are an end result of type 1, it would stand as a proof that type 1 can have finite steps, but are there jumbling puzzles where type 1 is infinite?

3. If there are any puzzles where type 1 unbandaging is infinite, does it still produce fewer cuts than type 2? Or perhaps more accurately, are there jumbling puzzles where type 1 produces coarser jumble dust than type 2?

4. Is type 2 always infinite or is there an even more extreme unbandaging method needed to ensure infinite unbandagin of a jumbling puzzle?

Depending on the answers to these questions, we might have a method for subdividing jumbling puzzles by how they unbandage. based on my limited understanding of these things, I would present the following hypothesis's:

Type 1 <= Type 2 in the sense that every cut added by type 1 is also add by type 2 and type 2 can add extra cuts.
Type 1 unbandaging can either produce: 1. A doctrinaire puzzle after finite steps, 2 a Jumbling puzzle that can only be further unbandaged further by type 2 after finite steps, 3. A puzzle from which jumble dust can be removed to form a fudged puzzle, and 4. a pile of jumble dust with no finite pieces left.
Type 2 can produce either 1, 3, or 4 from the type 1 possibilities.

Not that I would have any idea how to go about testing any of these. Of course, the unbandagings of jumbling puzzles that are most interesting are those finite type 1 and the fudged results, at least from a solving/mechanism standpoint.

Now, can anyone prove that the curvy copter plus is the end result of type 1 unbandagin of a curvy coptor? Anyone have any idea what the end result of type 2 might look like?

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 Post subject: Re: Does the Helicopter Cube jumble?Posted: Fri Jul 18, 2014 3:13 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Unbandaging is any way of adding more cuts to a puzzle. It includes such off the beaten path things as adding 45 degree turns to the faces of a Rubik's Cube. What people are discussing here one might call 'necessary unbandaging', as in things which you can show have to be done in order to make a puzzle fully doctrinaire. It's possible to become doctrinaire the other way, but adding restriction, for example in the case of a helicopter cube you could restrict it so that slices are strictly 180 degree turns and jumbling moves are impossible.

Slices which are 'suggested by the geometry' are not necessarily necessary. The straightforward definition is that you do a sequence of defined slices that are allowed by the puzzle, and then you take the slices which are no longer able to slice and add the necessary cuts so they're sliceable. By that definition, any face of any puzzle which has an irrational angle in it will turn completely to dust, except possibly for its center cap which turns into a circle. The Helicopter Cube most definitely falls into this category. It's also true as someone pointed out earlier that in puzzles like the Constellation Six which have rational angles it's sometimes the case that there is dust but not the entire puzzle turns to dust, so you can straightforwardly remove the dusty parts and have a working puzzle. I'm not sure what other conditions might have to be met for that to be the case, or whether it's almost always true when there are rational angles.

Bill Gosper points out that acos(-1/3), which as 180-acos(1/3) also appears on the Helicopter Cube, is the tetrahedral bond angle.

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