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 Post subject: Jumbling Puzzle ChallengePosted: Fri Apr 13, 2012 1:42 pm

Joined: Mon Nov 30, 2009 1:03 pm
[Admin: Split from Puzzles that Jumble, this deserves its own topic]

Hi fans of Twisty Puzzles that Jumble,

Yesterday, I presented "Twisty puzzles that jumble, a challenge looking for a mathematician" at the Dutch Mathematics Congress in Eindhoven. Here is my presentation: PDF.

As one would expect with such an audience, there were more questions than answers. Here are some interesting questions that were raised during and after my talk.
-What constitutes a "state" of a jumbling puzzle?
-Is the number of "states" of a jumbling puzzle always a finite, integer number?
-Is it essential to keep the core fixed when evaluating the number of states?
-If so, how would that work for deep-cut jumbling puzzles?

Prof.dr. Hans Zantema suggested that we start with the underlying group structure of a jumbling puzzle (c.f. Rubik's Cube group) and then consider the "basic operations". Basic operations are what takes a twisty puzzle from one state to an adjacent state (c.f. Rubik: F,B,U,D,L,R).

Food for thought ...

Oskar

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Last edited by Oskar on Sun Apr 22, 2012 12:12 pm, edited 2 times in total.
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 Post subject: Re: Puzzles that jumblePosted: Sun Apr 15, 2012 12:22 pm

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Oskar wrote:
Prof.dr. Hans Zantema suggested that we start with the underlying group structure of a jumbling puzzle (c.f. Rubik's Cube group) and then consider the "basic operations". Basic operations are what takes a twisty puzzle from one state to an adjacent state (c.f. Rubik: F,B,U,D,L,R).
Here begins the problem for all jumbling puzzle like the traditionally bandaged puzzles as well:
On a Rubiks cube the six "basic operations" can be applied on every permutation. On a jumbling or bandaged puzzle the set of possible moves depends on the current state.

It is indeed time that some professionials deal with these puzzles!

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Mon Apr 16, 2012 8:23 am

Joined: Sun Oct 08, 2006 1:47 pm
Location: Houston/San Antonio, Texas
To quickly throw out an idea and make an obligatory post to show I still exist.......

As far as stable-cored (i.e. not puzzles like Mixup Cube) jumbling puzzles go, there are only a finite number of types of pieces that can only be configured in a finite number of shapes. Viewing a puzzle like the Rubik's Cube from the mechanism out, every state is identical. It's not until you add sticker colors that states become distinguishable. Many of the properties that are true on twistypuzzles can be proven because the states form a mathematical group. A puzzle can be the realized form of a group ONLY if every move is available in every state, so there are some good reasons to try to cleanly handle the blocked moves in jumbling puzzles.

Based on the above observations, perhaps where we have considered non-jumbling puzzles as a set of positions and a universal set of moves, we should consider jumbling puzzles as a set of mechanical arrangements, each having its own set of configurations and states. Take the Helicopter Cube for example. When the puzzle is in a cube shape, there are 12 available rotation vectors (or planes, however you prefer to call them) and 24!/(4!)^6*8!*3^7/24 = approx. 1.19 * 10^22 possible sticker states (sticker arrangements). Each of the rotation vectors can stop at 1 of 5 angles, not including its current position, to allow another move to interact with the same pieces altered by the previous move. 1 of these angles (180 degrees) returns you to another state in the same mechanical arrangement. However, the other angles lead you to 1 of 2 different mechanical states per rotation vector. From these different mechanical states, the set of available rotations changes, both in vectors and angles. There are the same number of sticker states (sticker arrangements) although permutations and orientations will have to be redefined somewhat. I conjecture there is about 150 possible mechanical states each with 1.19 * 10^22 sticker states.

By careful defining/tracing out how piece permutations/orientations transfer from mechanical state to mechanical state, we can even prove what parities are possible and what are not. For example, I have scrambled and solved Oskar's Meteor Madness 6 or 7 times and have observed that every "edge" (piece with two stickers) can end up in every edge location in either orientation and every "corner" (piece with 3 stickers) can end up in every corner location in any of 3 orientations. Swapping two edges, swapping two corners, flipping a single edge, and rotating a single corner all appear to be impossible. I think it would be interesting to try to prove that for this puzzle. In fact, it seems to me that any piece-type for which no move changes the orientation but not the position of any single instance of this piece type is restricted by this orientation parity, on both non-jumbling and jumbling puzzles alike. Does anyone know of a counter-example to this? Could there be some subtle law of geometry that forces this?

Anyway that's my idea. Define the set of moves based on the puzzle's mechanical state, allowing multiple definitions for moves throughout a single puzzle. This keeps the number of overall puzzle states (mechanical state * sticker state) finite and integral. I am a huge fan of keeping the core of a puzzle fixed, but with non-jumbling puzzles, it is mathematically sound to fix another piece of the puzzle, or even to fix no piece at all. I BELIEVE if a piece other than the core of a stable-core puzzle (this is important!, ask Carl ) is fixed in space, the number of states remains finite, but the number of mechanical states may increase due to an identical mechanical state occuring while the fixed piece is in a portion of the puzzle that has been jumbled counting as a new state. As for the last question, I have always wondered this but never sat down long enough to think through it. Depending on if we are "discovering" what moves are available or defining them rigorously before hand, we can easily leap into an infinite number of states by rotating R x degrees CW, rotate L x degrees CCW, repeat or have states where a move is available but not "defined" there (a rotation of the whole puzzle however, could make the move available again by rigorous definitions). Perhaps we are forced to fix a non-core piece in this case... that one will require more thought.

Good Questions!

Peace,
Matt Galla

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Mon Apr 16, 2012 10:04 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Nice topic. I don't have too much time at the moment but I wanted to get a few quick answers out there. Great presentation Oskar. I wish I could have been there.
Oskar wrote:
-What constitutes a "state" of a jumbling puzzle?
Starting from the solved state a new state is reached anytime after a turn in made where an additional turn is allowed which moves some of the same pieces moved in the first turn plus some additional pieces not moved in the first turn. Not sure of a better way to word that. Looking at the normal 3x3x3 you need to exclude the case where the top layer is turned by 45 degrees. Without such a definition one could call this a new state as one is still allowed to turn the bottom layer while the puzzle is in this position. Note the plus part is needed to cover the case where the slice and bottom layer are turned together by 45 degrees and the following argument that this is a new state because I can now turn the bottom layer by itself again.

You can use this rule from any state of the puzzle to find new states.
Oskar wrote:
-Is the number of "states" of a jumbling puzzle always a finite, integer number?
Yes. As long as the puzzle has a finite number of pieces.
Oskar wrote:
-Is it essential to keep the core fixed when evaluating the number of states?
It doesn't need to be the core but any "holding point" should work and produce the same number of states. Note a holding point can be virtual but not imaginary. These are terms I've defined elsewhere. If you don't hold something fixed I think you can run into a case where you might find an infinite number of states but many would be just global rotations of the entire puzzle from other states.
Oskar wrote:
-If so, how would that work for deep-cut jumbling puzzles?
Just pick any piece that is in the puzzle and consider that your holding point. Look at this thread where I define the pieces in the Complex 3x3x3 by considering a corner of the 3x3x3 as your holding point. It doesn't need to be the core and by doing this I show how it can be generalized to define the pieces of a Complex 2x2x2 or Complex 4x4x4 both of which would be deep cut puzzles.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Mon Apr 16, 2012 10:19 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Allagem wrote:
I am a huge fan of keeping the core of a puzzle fixed, but with non-jumbling puzzles, it is mathematically sound to fix another piece of the puzzle, or even to fix no piece at all.
I believe you need to fix "something" but as I said it doesn't always need to be a piece in the puzzle. It could be a "virtual piece" which is defined based on the geometry of the puzzle itself. Think of a Skewb. It has two virtual cores. One is made physical in the actual construction of the mech which holds the puzzle together but you can imagine you are holding on to the other core by keeping the other corners not attached to the core fixed in position and just allowing them to rotate.
Allagem wrote:
I BELIEVE if a piece other than the core of a stable-core puzzle (this is important!, ask Carl ) is fixed in space, the number of states remains finite, but the number of mechanical states may increase due to an identical mechanical state occuring while the fixed piece is in a portion of the puzzle that has been jumbled counting as a new state.
I need to think about this. I tend to think that if the fixed piece (aka holding point) hasn't moved how did it jumble? But when I imagine a Helicopter Cube being held by a corner I think I see what you mean. It may very well be the case that some holding points are considered "better" then others.

I do agree that two states which only differ by a global rotation of the entire puzzle shouldn't be counted as seperate states.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sun Apr 22, 2012 12:31 pm

Joined: Mon Nov 30, 2009 1:03 pm
Hi Jumble-specialists,

Prof. Zantema suggests that we first properly define jumbling in 2D before going to 3D. Doug Engel wrote this excellent overview of 2D twisty puzzles (some of which beg to be implemented in the physical realm).

I copied some graphs from Doug Engels work below. To you expert opinions, which of these 2D twisty puzzles jumble and which don't?

Oskar
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Line periodic circle puzzles.jpg [ 72.27 KiB | Viewed 6461 times ]

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Non-periodic circle puzzles.jpg [ 47.09 KiB | Viewed 6461 times ]

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Dual offset grid circle puzzles.jpg [ 60.69 KiB | Viewed 6461 times ]

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Penrose circle puzzles.jpg [ 76.19 KiB | Viewed 6461 times ]

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Partially symmetric circle puzzles.jpg [ 64.3 KiB | Viewed 6461 times ]

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Mon Apr 23, 2012 11:27 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
I tried hard to give a useful answer but this all I could find:

The puzzles in the first images do not jumble if they are restricted to 180Â°-turns.

Why don't you use the example you and Bram used for your article in CFF?

Maybe we can construct a jumbling 2D-puzzle from scratch?
It could be puzzle with 2 circles. Beside the usual (?) turns of 120Â° (or any other rational number) a move must be possible after a turn of a irrational angle.

EDIT: Corrected minor language mistakes.

Last edited by Andreas Nortmann on Tue Apr 24, 2012 12:14 pm, edited 1 time in total.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Mon Apr 23, 2012 1:01 pm

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
Here are my guesses whether the puzzles jumble or not. I guess it's pretty useless as it's based on no deeper analysis than looking at the puzzles.

Line-periodic:
The two puzzles that contain many small circles jumble, except if turned by 180 degrees only as Andreas said. The other two don't jumble.

Non-periodic:
I'm very sure that thing jumbles. It would at least be a mess unbandaged anyway

Hybrid circle puzzles:
The three of these look like they're combinations of six-fold and four-fold cuts, and can be unbandaged to 12-fold.

Offset circle puzzles:
Again those jumble unless there's some specific angle used.

Dual offset grid circle puzzles:
I'm rather sure that it jumbles, unless some specific offset value is used. It's two four-fold cut patterns offset.

Penrose circle puzzles:
Since the prototiles have rational angles, the circles are rotated in rational angles, but that puzzle unbandaged would be very scary.

Partially symmetric circle puzzles:
The first one is identical to one of the hybrid circle puzzles, and the other doesn't jumble.

I don't really know if this is worth anything, but that's my view of them

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Mon Apr 30, 2012 1:34 pm

Joined: Mon Nov 30, 2009 1:03 pm
Hans Zantema implemented Doug Engel's Binary Bisect 5 puzzle (Slide Rule Duel) in software. You can see his webpage here. You can also directly download the .exe executable. Hans made this implementation to study the concept of jumbling.

Enjoy and respond!

Oskar
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Doug Engel's Binary Bisect 5 - programmed by Hans Zantema.jpg [ 38.51 KiB | Viewed 6305 times ]

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Mon Apr 30, 2012 1:59 pm

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
I'm a bit confused with why the Slide Rule Duel is considered jumbling, it unbandages fine. Is it because one could slide the half-circle to some random position and expect it to turn from there aswell?
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unbandagedslideruleduel.png [ 33.42 KiB | Viewed 6290 times ]

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Mon Apr 30, 2012 10:11 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Coaster, THANK YOU for that picture. I've been wondering the same thing since Oskar's first article.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Tue May 01, 2012 10:58 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
Hans made this implementation to study the concept of jumbling.
I don't expect this to help him very much. As Coaster1235 points out, this puzzle doesn't jumble. If 45 degree turns are allowed then its a bandaged puzzle.
Oskar wrote:
Prof. Zantema suggests that we first properly define jumbling in 2D before going to 3D.
I'm not sure this is the best approach. I tend to think of the 2D puzzle as being on the surface of a very large (r=infinite) sphere. So the circles you see are just very shallow cuts into the sphere. In that sense the 2D puzzles are a subset of the 3D puzzle and any approach to define jumbling in 3D should be applicable to 2D as well. However if you start with 2D you are looking at infinitely shallow cuts in an infinitely large sphere and you are looking at paterns which in principle could go on forever. If you start with a helicopter cube for example everything is finite and there are a finite number of cuts.

So let's say a puzzle is bandaged if it requires only a finite number of cuts to totally unbandage it into a doctrinaire puzzle. And we can also say a puzzle jumbles if it requires an infinite number of cuts to totally unbandage it into a doctrinaire puzzle.

Now this is exactly where you run into an issue with 2D. Let's look at Doug Engel's Binary Bisect 5 puzzle again. This is a 2D puzzle and as presented its bandaged. You can count the cuts Coaster1235 had to add above. However let's imagine a version of Doug Engel's Binary Bisect 5 puzzle where the top of the patern is copied infinitely many times. The puzzle now has infinitely many pieces. I'd argue that the puzzle can still be unbandaged and its easy to see how but it now requires infinitely many cuts so does it jumble?

I think it should be easier to start with 3D puzzles (say a sphere of radius=1) cut with a finite number of 2D surfaces with depth>0. This is a case where I think cataloging all possibilites in 2D will actually be harder then dealing with the 3D problem.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Tue May 01, 2012 11:52 am

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
Oskar wrote:
Prof. Zantema suggests that we first properly define jumbling in 2D

Haven't we done this already? A 2D circle-based puzzle jumbles if a circle has to be turned an irrational amount of degrees. Wasn't that how Bram defined jumbling?

Moving to the 3D twisties, I think the correspondent definition goes: if a puzzle's face (when visualized as faceturning) is turned less than it's rotational symmetry (sorry I don't know a better wording for that) and subsequent turns can be made, the puzzle jumbles. Obviously from this we can see that if the faces of a puzzle have no rotational symmetry, it purely jumbles. Can someone think of a counterexample?

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Tue May 01, 2012 12:59 pm

Joined: Mon Nov 30, 2009 1:03 pm
Coaster1235 wrote:
I'm a bit confused with why the Slide Rule Duel is considered jumbling, it unbandages fine.
I obviously made a mistake. Your illustration proves to me that Slide Rule Duel does not jumble.
Coaster1235 wrote:
A 2D circle-based puzzle jumbles if a circle has to be turned an irrational amount of degrees.
I disagree. The Offset Circle puzzles do not jumble, despite their irrational angle. I call "stored cuts".

Hans Zantema posed an interesting conjecture in an email to me today.
Hans Zantema wrote:
If all states of the puzzle have exactly the same number of neighbouring states, then the puzzle does not jumble.
Of course, this criterion is insufficient to prove that a puzzle jumbles. And his conjecture may even be incorrect. Who knows a counterexample of a twisty puzzle that definitely jumbles, but where each state still has the same number of neighbouring states?

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Tue May 01, 2012 1:21 pm

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
Oskar wrote:
Coaster1235 wrote:
A 2D circle-based puzzle jumbles if a circle has to be turned an irrational amount of degrees.
I disagree. The Offset Circle puzzles do not jumble, despite their irrational angle. I call "stored cuts".

Looking at it from your point of view makes it not jumbling, looking at it from my point of view makes it jumbling. To be able to turn all the circles you'd have to introduce new cuts, and because of the irrational offset angle the puzzle jumbles. Of course solving the puzzle is like it had stored cuts if any unbandaging isn't done, kind of like the Bermuda cubes (they also jumble).
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Last edited by Coaster1235 on Tue May 01, 2012 1:49 pm, edited 1 time in total.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Tue May 01, 2012 1:34 pm

Joined: Mon Nov 30, 2009 1:03 pm
Coaster1235 wrote:
To be able to turn all the circles you'd have to introduce new cuts, and because of the irrational offset angle the puzzle jumbles. Of course solving wise it acts like the puzzle has stored cuts ...
Interesting point. I argue that two puzzles should fall in the same category, if they are identical from a solving perspective. A mathematician would probably rephrase my argument in terms of mapping state spaces onto each other.

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Tue May 01, 2012 2:14 pm

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
Coaster1235 wrote:
Moving to the 3D twisties, I think the correspondent definition goes: if a puzzle's face (when visualized as faceturning) is turned less than it's rotational symmetry (sorry I don't know a better wording for that) and subsequent turns can be made, the puzzle jumbles. Obviously from this we can see that if the faces of a puzzle have no rotational symmetry, it purely jumbles. Can someone think of a counterexample?

I just thought of one: cuboids. By my definition, a 3x4x5 jumbles but that is not true. What do you think of that?

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Tue May 01, 2012 4:40 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Coaster1235 wrote:
By my definition, a 3x4x5 jumbles but that is not true.
What is your definition again? I think I'm missing something.

Also let's go back and look at Oskar's statement here again:
Oskar wrote:
I disagree. The Offset Circle puzzles do not jumble, despite their irrational angle. I call "stored cuts".
When I think of "stored cuts" I think of puzzles like the Fuzed Cube. Its doctrinaire as after every turn it returns to the same state if stickers aren't considered. Another example would be my doctrinaire Deep Uniaxial 3x3x3. It also makes heavy use of stored cuts and is a doctrinaire puzzle.

However these two 2D puzzles (the Offset Circle Puzzles) are NOT doctrinaire. So you can either say they are bandaged or that they jumble. If you try to "unbandage" these puzzles you end up cutting things to dust in the process of trying to turn these into doctrinaire puzzles, provided we are talking about irrational offset angles. So I'm not sure how one would say these don't jumble.

But how does that imply a 3x4x5 jumbles? You can view the 3x4x5 as a bandaged 60x60x60 and if you go to unbandage that you can get to a doctrinaire 60x60x60 with a finite number of cuts. Not sure that is the best way to prove it but I'd say the 3x4x5 is a bandaged puzzle and not one which jumbles.

Back to the Offset Circle Puzzles... again as drawn I'd say they jumble. However, does the "offset" really add anything to these puzzles? Could removing the "offset" be viewed as a form of unbandaging? In which case the puzzle on the left moves the bottom circle to a position in line with one of the top 2 circles and the puzzle on the right becomes a strait line. If I do that have I really removed anything from these puzzles? In this case these puzzles ARE doctrinaire and don't jumble and DO make use of stored cuts. If we are allowed to move these elements around then I guess I would consider the puzzles as drawn shape modifications of doctrinaire puzzles. In 3D this would be equivalent to moving an axis of rotation which generally isn't allowed if one is expected to produce the same puzzle but here in 2D it appears that is an option in some cases.

So now after saying all that I now think I agree with Oskar. These two Offset Circle Puzzles are shape modifications which make use of stored cuts. Actually don't ALL 2D puzzles make use of stored cuts? If you didn't have any stored cuts I think you'd have an infinite puzzle.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Wed May 02, 2012 12:07 am

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
wwwmwww wrote:
What is your definition again? I think I'm missing something.

Coaster1235 wrote:
Moving to the 3D twisties, I think the correspondent definition goes: if a puzzle's face (when visualized as faceturning) is turned less than it's rotational symmetry (sorry I don't know a better wording for that) and subsequent turns can be made, the puzzle jumbles. Obviously from this we can see that if the faces of a puzzle have no rotational symmetry, it purely jumbles.

On the 3x4x5 the 3x5 sides (which are rectangles, twofold symmetry) can make 90 degree turns, which is not two-fold. But I think the definition holds for many puzzles. I don't know if my definition is that useful though. What's your opinion on the Bermuda cubes? They can be viewed as a heavily bandaged cube whose faces can turn 45 degrees (and would jumble), but it could also be viewed as if it had stored cuts, in a way.

wwwmwww wrote:
Back to the Offset Circle Puzzles... again as drawn I'd say they jumble. However, does the "offset" really add anything to these puzzles? Could removing the "offset" be viewed as a form of unbandaging? In which case the puzzle on the left moves the bottom circle to a position in line with one of the top 2 circles and the puzzle on the right becomes a strait line. If I do that have I really removed anything from these puzzles? In this case these puzzles ARE doctrinaire and don't jumble and DO make use of stored cuts. If we are allowed to move these elements around then I guess I would consider the puzzles as drawn shape modifications of doctrinaire puzzles.

To allow for this the definition would need some addendum, such as "...and if the puzzle can't be readjusted into a doctrinaire puzzle without changing the solving process." Though, that is pretty cumbersome in my opinion, so I'd still want to view it as a jumbling puzzle. Solvingwise the jumbling doesn't add anything new though.

wwwmwww wrote:
So now after saying all that I now think I agree with Oskar. These two Offset Circle Puzzles are shape modifications which make use of stored cuts. Actually don't ALL 2D puzzles make use of stored cuts? If you didn't have any stored cuts I think you'd have an infinite puzzle.

Indeed!

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Wed May 02, 2012 1:51 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
You can view the 3x4x5 as a bandaged 60x60x60 and if you go to unbandage that you can get to a doctrinaire 60x60x60 with a finite number of cuts. Not sure that is the best way to prove it but I'd say the 3x4x5 is a bandaged puzzle and not one which jumbles.
I agree. We should not get confused by the shapes of those beautiful cuboids. For the unbandaging process, the only thing that matters is the puzzle's equivalent Jaap's Sphere.
wwwmwww wrote:
However these two 2D puzzles (the Offset Circle Puzzles) are NOT doctrinaire. So you can either say they are bandaged or that they jumble. If you try to "unbandage" these puzzles you end up cutting things to dust in the process of trying to turn these into doctrinaire puzzles, provided we are talking about irrational offset angles. So I'm not sure how one would say these don't jumble.
I wonder how the state diagram of an Offset Circle Puzzle (and hence the solving experience) relates to the state diagram of the associated "Non-Offset Circle Puzzle". I suspect that those state diagrams trivially map onto one another, in which case I would argue that both puzzles should be classified the same, i.e. both are doctrinaire.

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Wed May 02, 2012 2:36 pm

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
Oskar wrote:
I wonder how the state diagram of an Offset Circle Puzzle (and hence the solving experience) relates to the state diagram of the associated "Non-Offset Circle Puzzle". I suspect that those state diagrams trivially map onto one another, in which case I would argue that both puzzles should be classified the same, i.e. both are doctrinaire.

Since all it takes is a little bit of extra rotation to align the center circle to either of the groups connected, they do map. That means our definition gets its first addendum!

A 2D circle based puzzle jumbles if a circle has to be rotated an irrational amount of degrees to line up cuts AND the puzzle can't be rearranged into a doctrinaire puzzle without affecting the solving experience.

I find that to be a bit cumbersome.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Wed May 02, 2012 2:59 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
I wonder how the state diagram of an Offset Circle Puzzle (and hence the solving experience) relates to the state diagram of the associated "Non-Offset Circle Puzzle". I suspect that those state diagrams trivially map onto one another, in which case I would argue that both puzzles should be classified the same, i.e. both are doctrinaire.
Before I finished my above post I had changed my stance myself. I now MOSTLY agree with you. However I would still not call the Offset Circle Puzzles doctrinaire. To see why lets go back and look at my favorite post to refer to on this topic.

Bram was the first to define doctrinaire here.

The Non-Offset Circle Puzzles clearly are doctrinaire BUT these Offset Circle Puzzles I'd call shape mods also defined by Bram in the same post. Note if you "remove all the coloration then every single position would [NOT] look exactly the same" so these aren't doctrinaire.

These fall in the same boat as the Fisher Cube which Bram also mentions. All of its states can be trivially mapped to a Rubiks Cube yet here you can tell states apart even after the coloration is removed.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Wed May 02, 2012 11:54 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
Bram was the first to define doctrinaire here.
I don't like it that shape-mods would be considered non-doctrinaire, as it confuses the discussion. How about "classic doctrinaire" versus "shape-modded doctrinaire"?
Coaster1235 wrote:
A 2D circle based puzzle jumbles if a circle has to be rotated an irrational amount of degrees to line up cuts AND ...
Maybe, geometry is not the essence. We have identified several jumbling puzzles, both in 2D and 3D, that have rational angles. Based on my discussions with Prof. Zantema, I wonder whether we can have definitions that ignore geometry and only consider the state space of the puzzle.

First of all, note that a Rubik's Cube has an infinite number of states, if you include all mid-turn states. My conjecture is that there exists an unambiguous algorithm to reduce such an infinite state space into the finite state space that we all consider the state space of the Rubik's Cube. Let's call this algorithm call "reduction of the state space", and let's call the result the "reduced state space". Most likely, there exist proper mathematical terms for these.

With these terms defined, we could now define
1) "A puzzle is doctrinaire if all states of its reduced state space are equivalent".
This definition would make discussions about shape-modding irrelevant.

Also the process of bandaging could be defined in these terms
2) "Puzzle A is a bandaged version of puzzle B, if the reduced state space of puzzle A is equivalent to a sub-set of the reduced state space of puzzle B".
This definition would make discussions about stored cuts irrelevant. It also removes the concept of unbandaging from the physical realm.

Subsequently, the definition of jumbling would become
3) "A puzzle jumbles if its reduced state space cannot be extended into one where all states are equivalent".

I wonder how well these definitions hold for twisty puzzles in the physical realm. Science is about falsifiability.
Do we know doctrinaire puzzles that do not satisfy 1)?
Do we know bandaged puzzles that do not satisfy 2)?
Do we know jumbling puzzles that do not satisfy 3)?

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Thu May 03, 2012 3:26 am

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Oskar wrote:
Based on my discussions with Prof. Zantema, I wonder whether we can have definitions that ignore geometry and only consider the state space of the puzzle.

I doubt it. If you're willing to allow extraordinary amounts of fudging, getting into smushing and stretching, it's amazing what can be made doctrinaire.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Thu May 03, 2012 11:51 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
wwwmwww wrote:
Bram was the first to define doctrinaire here.
I don't like it that shape-mods would be considered non-doctrinaire, as it confuses the discussion. How about "classic doctrinaire" versus "shape-modded doctrinaire"?
Bram was the first to propose the definition and I think it has served the community very well for nearly 3 years so I'm not really wanting to see it changed but as Bram has joined this discussion he's more then welcome to comment on this topic. What you appear to be after is a term which encompases doctrinaire and shape mods. I'm happy to have two such groups as I believe all here would say the Fisher Cube is a different puzzle from a Rubik's cube even though at their core they are the same basic idea. It also allows us to talk about the doctrinaire shapes for certain puzzles. The 3x3x3 can be a cube, or a sphere, etc. For the Mixup Cube you could put 18 face centers on the puzzle and see a possible doctrinaire shape it could take and so on. How about if we come up with a new term for the union of the sets of doctrinaire and shape mod puzzles? Or maybe we could just say the equivalent Jaap's Sphere is doctrinaire. I'm certainly open to ideas.
Oskar wrote:
Also the process of bandaging could be defined in these terms
2) "Puzzle A is a bandaged version of puzzle B, if the reduced state space of puzzle A is equivalent to a sub-set of the reduced state space of puzzle B".
This definition would make discussions about stored cuts irrelevant. It also removes the concept of unbandaging from the physical realm.
I see problems here. The Fuzed Cube is a sub-set of the 3x3x3 yet its not considered a bandaged puzzle. It is doctrinaire. However the terminology here certainly could be improved as one could argue that its possible to make a Fuzed Cube by bandaging a 3x3x3. So the verb bandage and the adjective bandaged don't really mean the same thing. In fact I'm sure there are many (just yesterday I was tempted to start a thread asking how many and I many yet) doctrinaire subsets of the 3x3x3. For another example see this puzzle.

http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1582

Its four copies of the same doctrinaire subset of the 3x3x3 fuzed into one puzzle. Where Andreas calls it "non-bandaged" I would have said doctrinaire.

If you are proposing that we do call these puzzles bandaged and not doctrinaire then you run into another problem. You'd have to consider the 3x3x3 itself a bandaged puzzle as its equivalent to a sub-set of the 5x5x5. And that leads to a never ending can of worms.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Thu May 03, 2012 12:51 pm

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Oskar wrote:
wwwmwww wrote:
Bram was the first to define doctrinaire here.
I don't like it that shape-mods would be considered non-doctrinaire, as it confuses the discussion. How about "classic doctrinaire" versus "shape-modded doctrinaire"?
Now I can contribute at least something small:
I do not see a problem with defining shape-mods as non-doctrinaire. Shape-mods are nothing more than shape-mods. They are not solved any different (at least it math is considered) and can be completely thrown out of the discussion.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Thu May 03, 2012 1:21 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
I see problems here. The Fuzed Cube is a sub-set of the 3x3x3 yet its not considered a bandaged puzzle. It is doctrinaire. However the terminology here certainly could be improved as one could argue that its possible to make a Fuzed Cube by bandaging a 3x3x3.
What is the problem? I see no contradiction between the following statements.
1) Fuzed Cube is doctrinaire.
2) Fuzed Cube is a bandaged version of Rubiks Cube.
So we may need to distinguish "non-doctrinaire bandaged puzzles" from "doctrinaire bandaged puzzle".

Oskar wrote:
With these terms defined, we could now define
1) "A puzzle is doctrinaire if all states of its reduced state space are equivalent".
Prof Zantema informed me that the proper mathematical term is vertex-transitive graph.
Andreas Nortmann wrote:
I do not see a problem with defining shape-mods as non-doctrinaire.
wwwmwww wrote:
Bram was the first to propose the definition and I think it has served the community very well for nearly 3 years so I'm not really wanting to see it changed
The definition of a "planet" had been stable for centuries. Recently, scientific considerations and debate resulted in a sharper definition, and Pluto losing its status as a planet. I call that scientific progress.

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Thu May 03, 2012 3:46 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
The definition of a "planet" had been stable for centuries. Recently, scientific considerations and debate resulted in a sharper definition, and Pluto losing its status as a planet. I call that scientific progress.
I'm not against progress and I'm not against making the definitions more mathematically accurate. What I was hoping to avoid was what has already happened here in these forums with words like "order". I'm aware of at least 3 different ways its has been defined and there are groups here which hold on to each. I can follow conversations as long as I know which definition is being used and I understand the reasons behind the different mindsets, so I won't go so far as to say any of them are wrong. I can also follow your definitions. Its just that the groups you are defining are different from the groups Bram defined with these terms. Here are a few Vinn diagrams which I hope show where you are making changes.
Attachment:

Venn.png [ 15.87 KiB | Viewed 5709 times ]

Bram has set up each definition such that all puzzles (maybe I should say all twisty puzzles) fall into one and only one of these bins. Your structure is much different and using your definition of bandaging ALL puzzles are bandaged. I don't see how that buys us anything. Sure its a possible definition and its writen more mathematically but I'm not seeing the progress.

Some examples:

The Curvy Copter is a jumbling puzzle. But its also a bandaged Curvy Copter Plus.

The Fisher Cube is a shape mod of a 3x3x3. Its also doctrinaire using your definition. And its a bandaged puzzle because all its states make up a subset of the 5x5x5.

I do agree that making a statement like the Fuzed Cube can be made from bandaging a 3x3x3 and at the same time saying the Fuzed Cube is NOT a bandaged puzzle can be confusing. No one has yet offered a better word or definition that I'm aware of. If you want to say that the 3x3x3 isn't a bandaged puzzle how do you get around the statement that the 5x5x5 can be bandaged into a 3x3x3?

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Thu May 03, 2012 4:19 pm

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The way I understood Bram's definitions was that the two main categories are doctrinaire and jumbling (since they are mutually exclusive by definition), and bandaged and shapemod would be subcategories found in both categories (jumbling+bandaged I imagined to be purposefully bandaged).

In my opinion I see both my interpretation of Bram's view and Oskar's view to collapse to the same thing, just grouped differently (Oskar's system sees doctrinaire puzzles to be a special case of bandaged instead of vice versa).

The Fused Cube can be put to many categories (either a bandaged 3x3x3 or a 2x2x2 with altered cut depths), as can many others as well, for example the Offset Skewb (bandaged Master Skewb or Skewb with altered cut depths) and Carl's Uniaxial Cube (bandaged 5x5x5 or a 3x3x3 with altered cut depths). Is that a problem? Solving wise this category is pretty diverse.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Thu May 03, 2012 5:26 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Coaster1235 wrote:
The way I understood Bram's definitions was that the two main categories are doctrinaire and jumbling (since they are mutually exclusive by definition), and bandaged and shapemod would be subcategories found in both categories (jumbling+bandaged I imagined to be purposefully bandaged).
Let me repost Bram's definitions here.
Bram wrote:
I think I've figured out how to explain what's jumbling in a well-defined manner. For the sake of simplicity, I'm going to be skipping a discussion of puzzles with gaps.

Let's define a 'doctrinaire' puzzle as one where if you were to remove all the coloration then every single position would look exactly the same. The Rubik's Cube is a doctrinaire puzzle, as is the Skewb and Megaminx. Also the Sphere Xyz, Chromo Ball, Puck puzzles, and a bunch of other puzzles which don't have slices like a Rubik's Cube but still have permutations.

A shape mod is a non-doctrinaire puzzle which can be shape modded to a doctrinaire puzzle. The Fisher Cube is a shape mod, as is the Mixup Cube.

A bandage puzzle is a non-doctrinaire one where by cutting the pieces into smaller parts it's possible to transform it into a doctrinaire puzzle.

A jumble puzzle is one which is non-doctrinaire but where it isn't possible to shape mod or unbandage it into a doctrinaire puzzle. Examples include the Helicopter Cube, 24-cube, Jumbleprism, Uncanny Cube, and Battle Gears.

The 24-cube is a somewhat confusing case because if one were to make an identical-looking puzzle which was physically blocked from doing anything but the 180 degree moves then it would be a doctrinaire puzzle, but as it is it's a jumble puzzle, and it's surprisingly difficult to figure out a way of keeping it from jumbling.
So by definition a shape mod is not a doctrinaire puzzle. Neither is a jumbling puzzle.

Another issue is the two meanings of the word bandage. The definition of "bandage puzzle" is NOT a puzzle which has been bandaged, i.e pieces have been glued together to restrict movement.
Coaster1235 wrote:
In my opinion I see both my interpretation of Bram's view and Oskar's view to collapse to the same thing, just grouped differently (Oskar's system sees doctrinaire puzzles to be a special case of bandaged instead of vice versa).
That is not what I'm seeing.
Coaster1235 wrote:
The Fused Cube can be put to many categories (either a bandaged 3x3x3 or a 2x2x2 with altered cut depths), as can many others as well, for example the Offset Skewb (bandaged Master Skewb or Skewb with altered cut depths) and Carl's Uniaxial Cube (bandaged 5x5x5 or a 3x3x3 with altered cut depths). Is that a problem? Solving wise this category is pretty diverse.
All depends on one's set of definitions. Using Bram's definitions the Fused Cube and my Uniaxial Cube are NOT bandaged puzzles. Both can be made by bandaging other puzzles but since they are doctrinaire, they are not bandaged by definition. I agree the group "bandaged puzzle" could have a better name to avoid this confusion.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Fri May 04, 2012 9:48 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Oskar wrote:
wwwmwww wrote:
I see problems here. The Fuzed Cube is a sub-set of the 3x3x3 yet its not considered a bandaged puzzle. It is doctrinaire. However the terminology here certainly could be improved as one could argue that its possible to make a Fuzed Cube by bandaging a 3x3x3.
What is the problem? I see no contradiction between the following statements.
1) Fuzed Cube is doctrinaire.
2) Fuzed Cube is a bandaged version of Rubiks Cube.
So we may need to distinguish "non-doctrinaire bandaged puzzles" from "doctrinaire bandaged puzzle".
I do not have a problem with considering the Fused Cube as doctrinaire and bandaged. I tried to coin the term "restricted" for all subgroups which are still doctrinaire this but it didn't stick so far.
BTW: The Fused cube is a popular example of a bandaged doctrinaire puzzle. Another example is the Siamese Cube.
And to further irritate you: Gear Cube and Gear Cube Extreme are doctrinaire too. Both represent subgroups of the 3x3x3.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Fri May 04, 2012 12:09 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Let me propose this idea...

Attachment:

Venn2.png [ 12 KiB | Viewed 5617 times ]

The groups Bram described are the same. The Vertex-Transitive puzzles are the union of the doctrinaire and shape mod puzzles.

In other words let's define a Vertex-Transitive puzzle as one where if you were to remove all the coloration AND shape differences between mathematically-alike-pieces then every single position would look exactly the same.

I'm also not opposed to calling a Fused Cube a restricted 3x3x3 but for this term to be useful you need a definition that seperates the 3x3x3 as an unrestricted puzzle and I don't know an easy way to do that. You can say the Fused Cube is restricted because its a subset of the 3x3x3 but as the 3x3x3 is a subset of the 5x5x5 how to you keep from saying all puzzles are restricted. You could try to define "order" again and somehow use that but then also note the 3x3x3 is a subset of the Complex-3x3x3.

To me these puzzles aren't bandaged in the same way as the group of bandaged puzzles which Bram defined. Let's look at the turns which are allowed on the Fused Cube. You can say your independant layers of rotation are F,U,R and these 3 faces can ALWAYS be turned... they are never blocked at any time. So even the term restricted sounds odd to me when talking about these puzzles.

So since every puzzle is a subset of some greater puzzle creating a new label/classification seems pointless to me. So going back to mathematical terms maybe instead of calling the Fused Cube a restricted or bandaged 3x3x3 we should simply call the Fused Cube a subset of the 3x3x3 and acknowledge that all sets can be subsets of some greater set and not use restricted/bandaged as adjectives to classify these puzzles. Though I'm still curious to see if restricted can be defined in a meaningful way.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Fri May 04, 2012 2:08 pm

Joined: Mon Nov 30, 2009 1:03 pm
Carl,

Thank you for the Ven diagrams. I like your proposal of three categories. However, whereas all "doctrinaire+shape-mod" puzzles have a vertex-transive transition system, there may exist vertex-transitive transition systems for which there does not exist an associated "doctrinaire+shaped-mod" twisty puzzle.

My conjecture is that it would be possible to define three categories of transition systems, such that
1) All doctrinaire puzzles and their shape mods have a "category 1" transition system.
2) All non-doctrinaire bandaged puzzles and their shapemods have a "category 2" transition system.
3) All jumbling puzzles and their shapemods have a "category 3" transition system.
(Note that none of these definitions say "and vice verse")

Category 1 seems to be the vertex-transitive transition system, do you agree? As for categories 2 and 3, there is still work in progress. At least they need proper mathematical definitions and names

Carl, you are correct that it makes no sense to distinguish "non-doctrinaire bandaged puzzles" from "doctrinaire bandaged puzzle". I withdraw that suggestion.

Andreas, I agree that Gear Cube and Gear Cube Extreme have vertex-transitive transition systems. However, as long as a Fisher Cube is not called "doctrinaire", I see no reason to call Gear Cube or Latch Cube "doctrinaire" either. Since Gear Cube changes shape (watch the edge gears), some people might call it a "shape mod".

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Fri May 04, 2012 4:19 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
However, as long as a Fisher Cube is not called "doctrinaire", I see no reason to call Gear Cube or Latch Cube "doctrinaire" either. Since Gear Cube changes shape (watch the edge gears), some people might call it a "shape mod".
Yes, using Bram's definitions I'd call the Gear Cube a shape mod. And I find mention of the Latch Cube very interesting. I think it breaks Bram's definition of doctrinaire.

"remove all the coloration then every single position would look exactly the same."

It fits that bill BUT I don't believe every single position behaves exactly the same. So I'd consider it a bandaged puzzle. Granted I don't yet have a latch cube so maybe I'm missing something but I think we need to change the definition of doctrinaire to say:

"remove all the coloration then every single position would look (and function) exactly the same."

Also thinking a bit about Oskar's catagories:

Catagory 1: Maybe we could call these MonoState puzzles. All the states are equivalent without consideration of shape or color.

Catagory 2 and 3: Together maybe we could call these PolyState puzzles. And without thinking too hard I'm wondering if Catagory 2 puzzles are the PolyState puzzles where each state has the same number of neighbor states and Catagory 3 puzzles may be PolyState puzzles where different states can have a different number of neighbors.

We can define neighboring states as two states of a puzzle which differ by only a single twist.

Any obvious counter examples to the above conjecture? Actually the Latch Cube may be one. Hmmm...

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Fri May 04, 2012 6:54 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
... Latch Cube very interesting. I think it breaks Bram's definition of doctrinaire: "remove all the coloration then every single position would look exactly the same."
I agree. Latch Cube does not have a vertex-transitive transition system, as not all of its states are equivalent. So it should not be called doctrinaire.

I suspect that Latch Cube has a Category 2 transition system, as cutting the latches results in a regular 3x3x3.

I'll leave it to the geometrically-categorizing experts to decide whether "bandaged" would be a proper geometric category for Latch Cube, or that Category 2 houses more than only "bandaged" puzzles.
wwwmwww wrote:
Catagory 1: Maybe we could call these MonoState puzzles. All the states are equivalent without consideration of shape or color.
Sorry, I have already defined Category 1 as "puzzles with a vertex-transitive transitioning system". Your definition would serve well as the layman's explanation of mine, but in case of ambiguity the formal mathematical definition should prevail.

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Fri May 04, 2012 11:37 pm

Joined: Thu Dec 21, 2006 5:32 pm
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I would agree the definition of doctrinaire needs to be expanded to encompass function to capture the latch cube case. Simple visual similarity doesn't imply function.
I don't know that bandaged is a proper label for it though. The connections and blockages provided by the latches are context sensitive, whereas the traditional definition of bandaging is a permanent fusion of two pieces of an otherwise doctrinaire puzzle. Is this a case to expand the definition of bandaging or something worthy of its own category?

I think the Gear Cube is far more than a shape mod. The edge rotation is only possible because of the interaction of the gears, something added to the doctrinaire 3x3x3 on which it is based. The "edge" turning of the Fisher cube is just the shape mod exposing the unseen (on all but a super cube) rotation of centers that fall out of the doctrinaire 3x3x3. I see these as distinct. So I would call the Gear Cube neither doctrinaire nor a shape mod.

Dave

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sat May 05, 2012 1:00 am

Joined: Mon Nov 30, 2009 1:03 pm
All,

Attachment:

Defining Jumbling.jpg [ 59.15 KiB | Viewed 5506 times ]

DLitwin wrote:
I would agree the definition of doctrinaire needs to be expanded to encompass function to capture the latch cube case.
Are you suggesting that also some Category 2 puzzles could be called doctrinaire?
DLitwin wrote:
I think the Gear Cube is far more than a shape mod.
Thank you

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sat May 05, 2012 11:29 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
DLitwin wrote:
I think the Gear Cube is far more than a shape mod.
Thank you
Arg!!! Now I feel like I insulted the Gear Cube and this highlights another naming issue with Bram's definitions. Many terms that are commonly used here on Twisty Puzzles have several different and in many cases conflicting definitions. Let's go though a few examples of what I mean:

Let's start with the term "order" as its applied to NxNxN cuboids. I assume its fair to call a NxNxN a cuboid in the same sense all squares are rectangles but not all rectangles are squares. To apply "order" to the other cuboids, circle puzzles, higher dimmentional puzzles there are probably even more definitions in use out there.

(1) Order = The number of turning layers per axis of rotation. So for an NxNxN its order is N.

(2) Order = The number of independant turning layers per axis of rotation. Can also be viewed as the number of cuts per axis of rotation. So for an NxNxN its order is N-1.

(3) Order = The number of turning layers per face. So for an NxNxN its order is RoundUp[(N-1)/2].

All are still in use here and I understand the reasons behind each. I generally use definition (2) myself but I've taken part in other discussions where the term is used differently so its important to know which definition is being used.

There are also 3 definitions of bandaged that are commonly used on this site.

(1) Bandage = The process of taking a puzzle with N moving pieces and dividing these pieces into M groups. M<N such that at least 1 group contains 2 or more of the pieces found in the original puzzle. The pieces in a given group are then fused (or glued) together and forced to move as one new piece. Note the two piece which are fused don't need to be in contact in the original puzzle. This is called bridge bandanging.

(2) Bandage = To place resitictions on the movement of a given puzzle. This can include gluing pieces together, the use of gears, latches, etc. Basically anything which removes some of the movement freedom found in the original puzzle.

(3) Bandaged = A group of puzzles which are NOT doctrinaire but which can be transformed into a doctrinaire puzzle with a finite number of cuts.

Note only the last definition doesn't require the knowledge of a given starting puzzle. So I can say something like the Fused Cube is a bandaged 3x3x3 but its NOT a bandaged puzzle. I hope its clear the the definitions I'm using in the same sentence can be easily seen from context. But this is certainly a weakness of Bram's terminology.

And the same thing happens with the term shape mod.

(1) Shape mod = A puzzle created by taking an existing puzzle and changing its external shape.

(2) Shape mod = A puzzle which can be turned into a doctrinaire puzzle by only changing the external shape of the puzzle.

Again here the first definition requires knowledge of a given puzzle. The second is a property of the puzzle itself regardless of the knowledge of any other puzzles.

Some exampes of potentially confusing statements which I could make:

The Gear Change is a shape mod of the Gear Cube. However the Gear Change is NOT a shape mod puzzle, its a doctrinaire form of the Gear Cube. The Gear Cube itself is a shape mod puzzle. The Gear Sphere is another doctrinaire form of the Gear Cube, which was created by shape modding a Gear Cube.

The Big Sphere "Lily Cube" is a shape mod of a Helicopter Cube. However the Big Sphere "Lily Cube" is not a shape mod, its a jumbling puzzle. Neither the Helicopter Cube nor the Sphere "Lily Cube" is a shape mod.

So please know then when I call the Gear Cube a shape mod I'm using definition (2).

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sat May 05, 2012 11:42 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
You need to define what you mean by "extended to vertex-transitive". Take for example the Fracture-6 puzzle. I'd call this a jumbling puzzle but it can be fudged into a Constellation Six puzzle which is doctrinaire. Would this fall within your definition of "extended"? This directly relates to the point Bram was making above in this thread.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sat May 05, 2012 12:13 pm

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Can't most puzzles be extended through fudging into doctrinaire puzzles? I think this can in fact be done with most jumbling puzzles, so how are fudged puzzles like the constellation six defined here? Is this a way of making doctrinaire subsets of jumbling puzzles?

Also... what was the consensus with stored cuts? And are puzzles like the mix-up cube doctrinaire, bandaged or jumbling?

Weren't there also... bobolloid cuts I think they were called?

There seem to be many special cases here that aren't being taken into account. Fudging especially I am very curious about.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sat May 05, 2012 1:13 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
You need to define what you mean by "extended to vertex-transitive". Take for example the Fracture-6 puzzle. I'd call this a jumbling puzzle but it can be fudged into a Constellation Six puzzle which is doctrinaire. Would this fall within your definition of "extended"?
I do not yet have a proper definition of "extend" yet. My conjecture is that "bandaged" and "jumbling" puzzles can be distinguished by their transition system. You make a strong point why my conjecture might be wrong.

By the way, are we getting any closer on achieving consensus on the use of "vertex-transitive"? Or do we call Latch Cube doctrinaire, as Dave suggested?

Oskar

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sat May 05, 2012 1:34 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
By the way, are we getting any closer on achieving consensus on the use of "vertex-transitive"? Or do we call Latch Cube doctrinaire, as Dave suggested?
I didn't take Dave statement here:
DLitwin wrote:
I would agree the definition of doctrinaire needs to be expanded to encompass function to capture the latch cube case.
to mean that he wanted the definition of doctrinaire to include the Latch Cube. I take his stetement to mean that he agrees with me that it needs to be expanded so that it can specifically exclude the Latch Cube. I would call it Bandaged or Catagory 2. So yes, I basically agree with the use of vertex-transitive but I have to defer to your friend Prof Zantema that we are using the term correctly. I can't say I was familiar with the term prior to this discussion.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sat May 05, 2012 1:58 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
elijah wrote:
Can't most puzzles be extended through fudging into doctrinaire puzzles?
I believe the answer to that is yes.
elijah wrote:
I think this can in fact be done with most jumbling puzzles, so how are fudged puzzles like the constellation six defined here?
Using Bram's definitions I'd call the Constellation Six doctrinaire. If you remove the stickers then all states of that puzzle look exactly the same.
elijah wrote:
Is this a way of making doctrinaire subsets of jumbling puzzles?
Yes, that is one application of fudging. I'm not sure its the only one. But I still call the Fracture-6 puzzle a jumbling puzzle. Note you can't "cut" the Fracture-6 puzzle up into the Constellation Six puzzle with a finite number of zero-volume 2D cut surfaces. The fudging process requires the removal of some volume. Granted Bram didn't define "cutting" in his definitions but this is the interpretation I've always taken.
elijah wrote:
Also... what was the consensus with stored cuts?
Stored cuts define a type of cut and they can be present on doctrinaire, shape mod, bandaged, and jumbling puzzles. Its simply a cut surface which doesn't allow turning in a given state of the puzzle.
elijah wrote:
And are puzzles like the mix-up cube doctrinaire, bandaged or jumbling?
The Mixup Cube is a Shape Mod. And there I've just set myself up again as Oskar will thank the next person that says its much more then a Shape Mod. You could change its shape to a sphere for example and note that every state looks the same so it can be made doctrinaire simply by changing its shape.
elijah wrote:
Weren't there also... bobolloid cuts I think they were called?
This term was never very well defined and its been argued its not needed. The diversity of puzzle that came out of those discussion seems best served by a few existing terms that were already around.
elijah wrote:
There seem to be many special cases here that aren't being taken into account. Fudging especially I am very curious about.
Fudging is a process where a puzzle is altered by the removal of volume. And you can have doctrinaire, shape mod, bandaged, and jumbling puzzles which have been fudged... I believe. Not as sure about the jumbling and fudged. It all depends of the process by which we differentiate bandaged and jumbling puzzles. If a puzzle is already fudged with some volume removed does that open us up to fudge it further into a doctrinaire puzzle and thereby consider it bandaged? That's still an open question.

Carl

Note: the difference between jumbling and bandages could be considered mostly mathematical and not physical. For example let's look at a helicopter cube in the shape of a sphere. Now imagine this sphere being a solid sphere of copper with the helicopter cuts projected onto its surface. When you want to make a turn you cut the sphere along the projection, make the turn, and fuse the two copper pieces back together so you again have a solid copper sphere. If you turn one of the helicopter layers by the jumbling angle we can still cut up the copper sphere to allow any of the original turns. You have in effect cut up the helicopter sphere into a doctrinaire puzzle and the copper sphere has a finite number of atoms. So does this mean a helicopter sphere is bandaged? I'd argue no as the puzzle is still fudged at the atomic scale. You can't fill 3-space by packing spheres so there is alot of missing volume inside your copper sphere. But in practice the difference between bandaged and jumbling might get rather fuzzy.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sat May 05, 2012 5:00 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Oskar wrote:
Or do we call Latch Cube doctrinaire, as Dave suggested?

The Latch Cube is most definitely not doctrinaire. It's restricted, and maybe you can call that bandaged depending on how much you're willing to stretch the term 'bandaged'. I've taken to calling things which are bandaged in the classical way 'glued' because there are so many different forms of bandaging which are all referred to using the same term.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sun May 06, 2012 12:18 am

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
Oskar wrote:
Or do we call Latch Cube doctrinaire, as Dave suggested?
My poor choice of words, sorry for any confusion. "capture" should really be "exclude" here. I suppose "capture the concept of function, not just visual similarity, and thereby exclude the Latch Cube" is what I was thinking.

Regarding bandaging, I suppose I tend to think of it in the most traditional sense: The joining of two otherwise separate pieces by means of a physical bandage (often a sticker or tile). Unbandaging, therefore, is the process of separating a single piece into two independent pieces to unblock further movements. Jumbling was originally defined in terms of infinite unbandaging.
I haven't given much thought to more flexible definitions of bandaging.

Dave

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sun May 06, 2012 5:05 am

Joined: Fri Feb 08, 2008 1:47 am
Location: near Utrecht, Netherlands
Here is my attempt at formalizing twisty puzzles:
Attachment:

842ec925d37844c28e29bd9[1].png [ 38.38 KiB | Viewed 5230 times ]

Sorry, I had to use an image here to get the notation right. I'll elaborate a bit on the definitions:
- Definition 1: it is not enough to just look at the various states of a puzzle and the fact that moves take you from state to state: the underlying structure (geometry) of the space a puzzle is built in greatly affects how it works. The set X could be either the (surface of) a sphere or some other geometric object (cube, dodecahedron, etc...) but it could also be more abstract (like simply a set with 27 elements to simulate a Rubik's Cube - but this is not very useful in analyzing jumbling). The bijections f are simply functions which represent the state of the puzzle: the identity function is considered the solved puzzle, and each move is represented by a function that takes the identity and maps it to the puzzle state. This is very much like how symmetric groups work. When analyzing jumbling puzzles, each elementary move could be represented by a function that takes one half space to the identity, and the other half space to a rotation of some number of degrees. The operator set specifies where we can go from each state.
- Definition 2: this is not very interesting, it is a building block towards the state set definition. The problem with my definition of a puzzle is that (f, g) could be an element of the operator set when f cannot be reached by legal moves from the identity so it's just noise.
- Definition 3: the state set gives all the states that can be reached by legal moves from the solved state. We need the state set to define doctrinaire later on.
- Definition 4: The transformation set is crucial in my understanding of jumbling puzzles. For some state s where we can go to state z, we take z(s^-1): this represents the move we are doing - for the Rubik's Cube s is some scrambled position and z is that same position after doing one turn. z(s^-1) simply gives us the transformation (mapping of half spaces) that represents just the particular turn we did.
- Definition 5: this is more or less the analogue of "vertex transitive" for state graphs. A puzzle is state transitive when for each state, exactly the same turns are available.
- Definition 6: for a puzzle to be doctrinaire, it needs to be state transitive but it also needs to have a finite number of states, otherwise the infinite unbandaging of some jumbling puzzle would be doctrinaire which is stupid.
- Definition 7: this tries to capture the process of unbandaging. The state transitive closure is simply the minimal extension of a puzzle in to a vertex transitive one.
- Definition 8: defines bandaging. A jumbling puzzle is not automatically bandaged (but any finite jumbling puzzle is). Let it be noted that puzzles exist that are neither doctrinaire nor bandaged.
- Definition 9: for a puzzle to be jumbling it is not enough for the unbandaging to be infinite. An infinitely long chain of 3x3x3's has an infinite unbandaging, but we can agree that it is not jumbling. By requiring that such a partition does not exist we can circumvent this problem: the chain of 3x3x3's can be partitioned in to doctrinaire puzzles, but the Helicopter Cube cannot: any single point on the puzzle can be rotated (over irrational angles) to infinitely many positions so a partition in to doctrinaire puzzles does not exist. By this definition, an infinite unbandaging is still jumbling, I am not sure if this is appropriate but this could be circumvented by requiring that a jumbling puzzle is bandaged.
- Definition 10: attempts to capture fudging. This definition defines fudging a puzzle as unbandaging it, and then getting rid of the dust. If you defined "fudged" as being the result of fudging, then any doctrinaire puzzle is fudged. I do not think this is a problem as fudging is really just the process of a designing a puzzle that just works because of flexibility of parts and gaps and this is not very mathematical.
- Hypothesis 1: if the fudging is finite it is automatically doctrinaire, but starting with that infinite chain of 3x3x3's we can get an infinite fudging. I hope that if we start with a finite puzzle the fudging will also be finite. edit: unfortunately it does not hold, consider a puzzle on N with operator (id,x->x+1)

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sun May 06, 2012 9:16 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
TomZ wrote:
Here is my attempt at formalizing twisty puzzles:
Ok... consider my bind blown. Tom I haven't seen this side of you before. I have a Ph.D. in physics and have had more math then most but obviously not enough set/group theory as most of this is well over my head. I'm familiar with most of the basic notation but the closest I'm come to working much with some of this was in a two part quantum electrodynamics class I took in grad school where I was introduced to the bra-ket notation. Granted that isn't the notation you are using but it sure reminds me of it. Trust me my mind was blown in that class too. I know you are still in school, may I ask what you are getting your degree in? If you are interested in physics allow me to highly recommed a quantum electrodynamics class. I'm sure you would do well and it was one of the most interesting classes I took in grad school.

Now some questions for the parts of this I understand just well enough to be dangerous...

TomZ wrote:
The problem with my definition of a puzzle is that (f, g) could be an element of the operator set when f cannot be reached by legal moves from the identity so it's just noise.
Noise meaning hot air? or something mathematical? Can you give a specific example?
TomZ wrote:
Definition 6: for a puzzle to be doctrinaire, it needs to be state transitive but it also needs to have a finite number of states, otherwise the infinite unbandaging of some jumbling puzzle would be doctrinaire which is stupid.
So you are putting the group of puzzles Bram called shape mods in the doctrinaire bin... correct? Oskar is calling this catagory 1.
TomZ wrote:
Definition 8: defines bandaging. A jumbling puzzle is not automatically bandaged (but any finite jumbling puzzle is). Let it be noted that puzzles exist that are neither doctrinaire nor bandaged.
So you have defined jumbling as a subset of bandaged. Is that correct? And an example of a puzzle which was neither doctrinaire nor bandaged would be a jumbling puzzle which had been infinitely unbandaged. Is that also correct? Can you give an example using a puzzle with a finite number of pieces?
TomZ wrote:
but the Helicopter Cube cannot: any single point on the puzzle can be rotated (over irrational angles) to infinitely many positions
This is not true on a Helicopter Cube is it? I trust we are talking about a Helicopter Cube which has been infinitely unbandaged. Correct?
TomZ wrote:
I am not sure if this is appropriate but this could be circumvented by requiring that a jumbling puzzle is bandaged.
I think this means you are calling the normal Helicopter Cube a bandaged version of the infinitely unbandaged Helicopter Cube and if so this seems to answer my above question. Just not sure.
TomZ wrote:
Definition 10: attempts to capture fudging. This definition defines fudging a puzzle as unbandaging it, and then getting rid of the dust. If you defined "fudged" as being the result of fudging, then any doctrinaire puzzle is fudged. I do not think this is a problem as fudging is really just the process of a designing a puzzle that just works because of flexibility of parts and gaps and this is not very mathematical.
So you are saying all fudged puzzles are doctrinaire? I can give a counter example. Look at the Bobble Block. There are pieces which have been fudged into equilateral triangles on the spherical version of this puzzle and that fudging allows them to be rotated into positions unreachable otherwise, yet the puzzle is not doctrinaire. And if you are saying all doctrinaire puzzles are fudged, I'm also confused. It's possible to cut a block into a 3x3x3 without the need to remove volume (at least on paper). Unless we are calling the process of adding tolerances a fudging process.
TomZ wrote:
Hypothesis 1: if the fudging is finite it is automatically doctrinaire, but starting with that infinite chain of 3x3x3's we can get an infinite fudging. I hope that if we start with a finite puzzle the fudging will also be finite.
Is infinite fudging the removal of an infinite amount of volume? If not how do you define infinite fudging? I'm not sure what an infinite amount of fudging means on a finite puzzle. And again I'd say a finite amount of volume has been removed from the Bobble Block and it certainly doesn't appear doctrinaire to me.

Now wishing I had the time to take a set theory class so I could actually follow all of the actual mathematical notation.

Carl

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sun May 06, 2012 10:41 am

Joined: Fri Feb 08, 2008 1:47 am
Location: near Utrecht, Netherlands
wwwmwww wrote:
Ok... consider my bind blown. Tom I haven't seen this side of you before. I have a Ph.D. in physics and have had more math then most but obviously not enough set/group theory as most of this is well over my head. I'm familiar with most of the basic notation but the closest I'm come to working much with some of this was in a two part quantum electrodynamics class I took in grad school where I was introduced to the bra-ket notation. Granted that isn't the notation you are using but it sure reminds me of it. Trust me my mind was blown in that class too. I know you are still in school, may I ask what you are getting your degree in?
I am taking two bachelors: one in Mathematics, the other in Computer Science - it's a special thing where you can get two degrees for the price of 1.4. I am only a second year student so what I have just done is very much mimicking my teachers and trying to sound just like them. It's looks good and sounds real but it surely is way wrong and will probably give Prof. van Zantema a good laugh. Though I certainly hope it will be a stepping stone toward formalizing the discussion, it will still take significant adjustment.

wwwmwww wrote:
TomZ wrote:
The problem with my definition of a puzzle is that (f, g) could be an element of the operator set when f cannot be reached by legal moves from the identity so it's just noise.
Noise meaning hot air? or something mathematical? Can you give a specific example?

Yes, hot air. My definition of puzzle isn't very strict. If (f,g) is in the operator then it means g can be reached from f by a single move, but f may not be reachable in the first place. It's like taking the description of the Rubik's Cube, and adding the move that takes the cube with two edges swapped to the cube with two corners swapped, for instance. Adding this move is absolutely useless since it can never be applied.

wwwmwww wrote:
TomZ wrote:
Definition 6: for a puzzle to be doctrinaire, it needs to be state transitive but it also needs to have a finite number of states, otherwise the infinite unbandaging of some jumbling puzzle would be doctrinaire which is stupid.
So you are putting the group of puzzles Bram called shape mods in the doctrinaire bin... correct? Oskar is calling this catagory 1.
Yes. I am not concerned with the shape of a puzzle. The set X does not need any underlying structure. We could call two puzzles shapemods of each other when there is a bijection between their state sets that preserves the structure of moves (eg: puzzle O is a shapemod of O' (and vice versa) when there is a bijection f such that for any two states s,s' \in S(O), f(s s')=f(s) f(s'))

wwwmwww wrote:
TomZ wrote:
Definition 8: defines bandaging. A jumbling puzzle is not automatically bandaged (but any finite jumbling puzzle is). Let it be noted that puzzles exist that are neither doctrinaire nor bandaged.
So you have defined jumbling as a subset of bandaged. Is that correct? And an example of a puzzle which was neither doctrinaire nor bandaged would be a jumbling puzzle which had been infinitely unbandaged. Is that also correct? Can you give an example using a puzzle with a finite number of pieces?
No, jumbling and bandaging are not comparable. Yes, that example is correct. If a puzzle is finite and jumbling then it is bandaged, since a finite not-bandaged puzzle is doctrinaire. On the other hand, if a puzzle is infinite and jumbling it may or may not be bandaged.

wwwmwww wrote:
TomZ wrote:
but the Helicopter Cube cannot: any single point on the puzzle can be rotated (over irrational angles) to infinitely many positions
This is not true on a Helicopter Cube is it? I trust we are talking about a Helicopter Cube which has been infinitely unbandaged. Correct?

Yes and no. The point I am referring to is to the infinitely unbandaged one, but the argument applies to the normal Helicopter Cube.

wwwmwww wrote:
TomZ wrote:
I am not sure if this is appropriate but this could be circumvented by requiring that a jumbling puzzle is bandaged.
I think this means you are calling the normal Helicopter Cube a bandaged version of the infinitely unbandaged Helicopter Cube and if so this seems to answer my above question. Just not sure.

Yes.

wwwmwww wrote:
TomZ wrote:
Definition 10: attempts to capture fudging. This definition defines fudging a puzzle as unbandaging it, and then getting rid of the dust. If you defined "fudged" as being the result of fudging, then any doctrinaire puzzle is fudged. I do not think this is a problem as fudging is really just the process of a designing a puzzle that just works because of flexibility of parts and gaps and this is not very mathematical.
So you are saying all fudged puzzles are doctrinaire? I can give a counter example. Look at the Bobble Block. There are pieces which have been fudged into equilateral triangles on the spherical version of this puzzle and that fudging allows them to be rotated into positions unreachable otherwise, yet the puzzle is not doctrinaire. And if you are saying all doctrinaire puzzles are fudged, I'm also confused. It's possible to cut a block into a 3x3x3 without the need to remove volume (at least on paper). Unless we are calling the process of adding tolerances a fudging process.
I do not want to define fudging for precisely these reasons. My definition of fudging a puzzle means taking the state transitive closure and finding the maximal not-jumbling sub-puzzle (think constellation six, though it is missing some pieces). Bubble Block is not the result of fudging as per my definition. To me it is simply a jumbling, bandaged puzzle. I do not want to define what it means for a puzzle to be fudged because fudging really just means moving everything around a bit so that it fits and rotates.

wwwmwww wrote:
TomZ wrote:
Hypothesis 1: if the fudging is finite it is automatically doctrinaire, but starting with that infinite chain of 3x3x3's we can get an infinite fudging. I hope that if w start with a finite puzzle the fudging will also be finite.
Is infinite fudging the removal of an infinite amount of volume? If not how do you define infinite fudging? I'm not sure what an infinite amount of fudging means on a finite puzzle. And again I'd say a finite amount of volume has been removed from the Bobble Block and it certainly doesn't appear doctrinaire to me.
No, by infinite fudging I mean that the fudging has an infinite number of states. But then, my definition of fudging has nothing to do with a puzzle being fudged or not.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sun May 06, 2012 12:42 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
One funny thing I haven't seen mentioned anywhere: While it appears to be possible to make any puzzle doctrinaire with sufficient unbandaging and fudging (sometimes a *lot* of fudging) there might be an exception. The never produced, and never really even successfully prototyped cmetrick II allows for 45 degree turns, and I suspect jumbles in an extremely profound way.

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 Post subject: Re: Jumbling Puzzle ChallengePosted: Sun May 06, 2012 12:56 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Bram wrote:
The never produced, and never really even successfully prototyped cmetrick II allows for 45 degree turns, and I suspect jumbles in an extremely profound way.
I take it you aren't talking about the cmetrick too. Do you have a picture or a drawing you can share?

Carl

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