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 Post subject: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 2:22 am

Joined: Sat Sep 15, 2012 7:42 am
This conversation initially started in this thread regarding Zem's SuperOcto Cube / Mf8's Sun Cube I. The initial suggestion, by Bram, was that the Sun Cube I is an unbandaged Bermuda Cube. This was met with some resistance, at which point definitions were called into question. Oskar's Twistypedia listings for bandaging and unbandaging were the only mildly formal definitions I could find:

Oskar wrote:
Bandaging: Restricting moves of a puzzle. Classical bandaging is achieve by "gluing" pieces together
Oskar wrote:
Unbandaging: Cutting pieces into smaller parts (to enable more types of moves)

I created my own definition of unbandaging based on Oskar's definition of classical bandaging:

TheMathKid wrote:
"If puzzle Y can be bandaged into puzzle X, then X can be unbandaged into Y."

On the other hand, TheCubingKyle has offered the following definition:

TheCubingKyle wrote:
Bandaging: Restricting the turns of a puzzle that would be allowed by the core mechanism. In a bandaged cube, you can state that it is a bandaged 3x3 because the mechanism logically allows each edge and corner to exist individually. A cuboid bandages when the pieces that are extended past the core mechanism are no longer adjacent to other extensions, and now no longer part of a fully cyclical mechanism. Basically, nonextended pieces are blocking the path that the extensions demand.

Unbandaging: Creating new cuts to allow a puzzle to operate fully based on the logical cuts of the original mechanism. The Skewb+2x2 has visible cuts that are turnable until you shaoe-shift and jumble, and then are blocked internally. This puzzle has been unbandaged by creating internal cuts to allow logical turns based on the original cuts that are invisibly bandaged. A 6x6 is NOT an unbandaged 3x3 because the 3x3 mechanism does not logically allow these additional layers.

While I once considered these very basic terms, the discussion has led to many questions about the semantic and mathematical aspects of unbandaging. Bandaging seems to be the easier of the two to conceptualize [probably why it was named before UNbandaging!], but it takes many, many forms: classical, geared, overhang, internal, jumbling, hidden bandaging, exotic 3x3x3 centers ala TomZ and Burgo, and more. There was some discussion on this thread about what actually constitutes bandaging, both physically and mathematically, and I'm sure that we will delve into something similar here, but mostly I am concerned with the following issues:

Issue #1: Meaningful Unbandagings
One of the main concerns was that Oskar's and my definitions allow seemingly silly statements. For example, that the 6x6x6 is an unbandaged 3x3x3. Is this considered correct? Or does it only make sense to discuss the unbandaging of a bandaged puzzle? Should the definition of unbandaged account for such? If so, how? There are trivial bandagings. Can there be trivial unbandagings?

Issue #2: Mechanism's Role
Does the mechanism come into play at all in using these terms? What I mean to so say is: for a correct use of "unbandaging," must the cuts be made in such a way that the new turns 'work' without changes to the mechanism? Otherwise, the cuts are simply made to the puzzle's Jaaps sphere without regard to mechanism [as in the case of 6x6x6 vs. 3x3x3]. For ease of conversation, perhaps we can temporarily distinguish these as "mechanical unbandaging" and "mathematical unbandaging." I would argue that the mechanism should not matter and that only mathematical unbandaging should be discussed, seeing as mechanisms vary widely. In this sense, then, "unbandaged version of" would be used very similarly to "superset of."

Issue #3: Reversibility
Does a bandaging relationship imply an unbandaging relationship? Is is proper to use the terms in an easily reversible manner? I feel my definition aligns very strongly with both intuition and the English language. Again, though, this allows seemingly undesirable statements as with the 6x6x6 and 3x3x3. Are jumblers considered bandaged since they can be unbandaged partially?

Issue #4: Supersets / Order of Unbandaging
Is there a concept of "simplest" unbandaging? The Sun Cube I appears to be the simplest possible shape that can be bandaged into all the Bermuda Cube variants. Can we refine the concept of "simplest" into a definition? Is our intuitive sense of "simplest" related to the idea of doctrine puzzles? Is there a procedural way to unbandage a puzzle to produce the simplest unbandaged version?

Issue #5: Relationships between Sun Cube I / Bermuda Cube / Rubik's Cube
What is the deal with the Bermuda cube? It obviously shares parts from both the Sun Cube I and the Rubik's cube. What is the precise relationship? Is Sun Cube I an unbandaged Bermuda cube? [I would say yes, but my reasoning goes back to issue #3.] Is the Bermuda cube a bandaged Rubik's cube? [I would say no.]

Wow. The first post is clearly a lot of questions! I would love to hear from some of the experts [Bram, Oskar, Brandon, Carl, Andreas, etc.] as well as others. I bet everyone on this forum has had questions about bandaging / unbandaging at one point or another, so I'm sure everyone has something to contribute [even if it's just more questions! ]. My goals for this thread are:

1. Provide a more rigorous definition of bandaging / unbandaging if deemed necessary
2. Once we have a definition, explore the consequences of that definition.
3. Settle the issue of the Bermuda Cube / Sun Cube I
4. Free Tibet

Have at it, y'all!

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Named 3x3x3 Bandaging Patterns

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 2:51 am

Joined: Mon Feb 28, 2011 4:54 am
themathkid wrote:
I'm sure everyone has something to contribute

My contribution is

As usual, that went so far over my head that I've become catatonic. I'm going to have to retreat to solving a 3x3 cube just to remind myself that the world is still ok...

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 4:01 am

Joined: Sun Jul 14, 2013 11:10 pm
I still stand by my definition (The large quoted portion in the original post ) That answers the majority of the issues and I'm eager to hear how I am wrong in it.

Wouldn't the current definition of unbandaging allow for just about any two cubes to be "bandaged" versions of each other? IE Is a curvy copter potentially bandaging rex cube cuts, and is a rex cube bandaging curvy copter cuts? This is clearly the extreme of the 6x6 vs. 3x3 argument (apparently our adopted paradigm for discussion) But how far does that argument actually reach?

As for various types of bandaging, I think my definition is tight enough for classic bandaging and things like cuboid bandaging, and also vague enough to cover what gears and jumbling do to puzzles.

Also from that post:
So. Is the Hyper Octocube an unbandaged Bermuda Cube? I don't have much experience with the Bermuda Cubes but I say no. The mechanism has nothing to do with bermuda cubes other than their functionality can be emulated by bandaging turns. But this is in the same way a 6x6 can be bandaged into a 3x3 but not vice-versa. It's like rectangles and squares definition. A Bermuda mechanism could NOT sustain the amount of cuts and turns as this puzzle, and therefore a Bermuda cube cannot be cut to allow the puzzle to operate fully based on the logical cuts of the hyper octocube.

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 4:16 am

Joined: Mon Mar 30, 2009 5:13 pm
Oskar's definitions are simple, accurate, complete, and unambiguous. Why make it complicated? I don't think this adds any value IMO.

In particular, the following "definition" is completely useless because it's circular logic (2 terms which are defined relative to each other, rather than independently in absolute terms):

themathkid wrote:
"If puzzle Y can be bandaged into puzzle X, then X can be unbandaged into Y."

On that basis, bandaged/unbandaged could mean any reversible transformation, such as solved/unsolved, assembled/disassembled, made visible/invisible, or anything else.

And everything else is just plain confusing. It's not helpful to anyone.

Mind you, it could be interesting to classify all different types of bandaging (end hence unbandaging), based on what type of moves are restricted, and how they are restricted...

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 5:06 am

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
I think bandaging is used as much as a verb as it is a noun: It's how designers describe what they're `doing` to a puzzle (and even solvers how they're `seeing` a puzzle).

An interesting example is the 1face. It exists as both bandaging and unbandaging at the same time (a sort of Schrodinger's cat bandaging ). It's unbandaging Inner Circle Pieces from the centre of a Circle Cube, and `temporarily bandaging` the Inner Circle Pieces to the Outer Collar pieces. So, as you can see, it's not really more than a helpful way to describe things.. it's use is derived from a more specific term that originally meant `joining together`.. and `unjoining`.

I don't see a conflict with it's use in multiple ways to describe various elements of `blocking and unblocking` from different perspectives, mechanically and mathematically. I don't think it's contradictory, I think it's descriptive, and serves a purpose within a context .

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 5:16 am

Joined: Mon Mar 30, 2009 5:13 pm
It's also interesting that a bandaged (e.g., 5x5x5) form of one puzzle (7x7x7) can be an unbandaged form of another (3x3x3).

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 6:25 am

Joined: Thu Jan 19, 2012 12:04 pm
Bandaging as a term is so overused that it's becoming meaningless. In practice, the meaning is typically "My puzzle won't turn like I think it should, therefore bandaging!". Which is essentially Oskar's definition of "Restricting moves of a puzzle". But the scope of what "restricting" means that that isn't a definition that can be used practically... it's just a very generalized abstraction, so much on the metalevel that you have to specify what subset you're talking about before you can do anything. Personally, I find "restriction" as a good enough term for that already.

So for me, I limit my use of "bandaging" to along the lines of the classical definition... pieces physically attached to each other. If they only act like they are (ie inner corner and edge pieces on a circle cube), that's virtual bandaging. But the important thing for me is that the bandaging here is static... the pieces maintain their relationship to each other across permutations. Which brings up the possibility of dynamic bandaging, which would be when special switches or buttons can be pressed to change which pieces are stuck together. Note that things like the circle on a 1face is definitely not dynamic bandaging here, that would lead to having to say that the centers on a 5x5x5 are dynamically bandaged (the squareness of the pieces cause turning the outer edge and corner pieces to push the center pieces along). The key difference here is that you can just do a regular perpendicular move and break the pieces apart... dynamic bandaging is just for convenience for talking about things like Tom Z's Switch Box or Rubik's Clocks, where it's useful because it's changeable static bandaging.

For most other things I find it's just easier to leave out the bandaging. For example, "geared" is enough... the idea that gears are "bandaging" faces by tying them together is pretty much inherent, as gears attach things to things with ratios. Adding "bandaging" there just doesn't add adding, it's like it's only standing in to clarify that things are attached to each other. But we don't go around saying that the cubies on a regular 3x3x3 cube are bandaged together, the fact that's the pieces are attached together in a certain relationship is really part of the cube being a twisty puzzle instead of an assembly one, so it goes unsaid.

Other that that, I tend to prefer "lock" and "block" for things like the inner circle on a 1face. Ie it's locked to the outer collar via the center... not a "bandaging" face, in fact it's an un-virtual-bandaging one when you add it to a regular circle cube... the 0center is the bandaged one (attached physically to the core).

themathkid wrote:
"If puzzle Y can be bandaged into puzzle X, then X can be unbandaged into Y."

That's also what I think of as the definition of unbandaging.

TheCubingKyle wrote:
Bandaging: Restricting the turns of a puzzle that would be allowed by the core mechanism. In a bandaged cube, you can state that it is a bandaged 3x3 because the mechanism logically allows each edge and corner to exist individually. A cuboid bandages when the pieces that are extended past the core mechanism are no longer adjacent to other extensions, and now no longer part of a fully cyclical mechanism. Basically, nonextended pieces are blocking the path that the extensions demand.

Nonextending pieces are bandaging here because they can be unbandaged so they don't block, with circles like on the Super 3x3x5s. But that's a mechanical approach.

From a more abstract point of view, my definition for fully functional cuboid only requires that a cut between two fully formed layers work, so bandaging isn't a consideration because there's no layer to turn. Allowing turns without a full layer would be "fully functioning plus" (and in practice, mechanical puzzles tend to be, because centers don't block).

TheCubingKyle wrote:
Unbandaging: Creating new cuts to allow a puzzle to operate fully based on the logical cuts of the original mechanism. The Skewb+2x2 has visible cuts that are turnable until you shaoe-shift and jumble, and then are blocked internally. This puzzle has been unbandaged by creating internal cuts to allow logical turns based on the original cuts that are invisibly bandaged. A 6x6 is NOT an unbandaged 3x3 because the 3x3 mechanism does not logically allow these additional layers.

Seems far to vague for me. A 3x3x3 could be built on a 6x6x6 mech. At which point we could point to it and say that it can be unbandaged to a 6x6x6, because that 3x3x3 does logically allow those layers (so suddenly we have an unbandaging grue problem... puzzles being unbandaged or not based on whether we're talking before or after the date someone bothers to make such an artifact... it's either that or we get No True Scotsman (that's not the True mech for building this puzzle, the one I choose is)). I think the problem here is that it's too mechanical a definition... it might be a useful definition for when talking about building puzzles, but it isn't very good for talking about puzzles (where in the abstract, the internals do not matter... the puzzle is defined by the pieces and the moves, and it doesn't even matter if a puzzle can even be built).

themathkid wrote:
Issue #1: Meaningful Unbandagings
One of the main concerns was that Oskar's and my definitions allow seemingly silly statements. For example, that the 6x6x6 is an unbandaged 3x3x3. Is this considered correct? Or does it only make sense to discuss the unbandaging of a bandaged puzzle? Should the definition of unbandaged account for such? If so, how? There are trivial bandagings. Can there be trivial unbandagings?

I have absolutely no problem with these. It is useful to talk about a 6x6x6 as an unbandaged 3x3x3. If I glue two pieces of a cube together, I have a bandaged cube. But I can also just never make a move that breaks those two pieces apart, thus only using a subset of available space that's equivalent to the bandaged cube. I see the 6x/3x thing similarly, you can easily simulate a 3x3x3 on a 6x6x6, and reducing a 6x6x6 to a 3x3x3 is a way to solve it. I expect similar of other bandaged/unbandaged puzzles... that the unbandaged one can simulate the bandaged, and that reduction of the unbandaged to a legal state of the bandaged can be followed by the bandaged solution to solve it.

Are you going to make a point of the 6x6x6 being an unbandaged 3x3x3 regularly... probably not. But it is still a relationship I see them sharing.

Quote:
Issue #2: Mechanism's Role
Does the mechanism come into play at all in using these terms? What I mean to so say is: for a correct use of "unbandaging," must the cuts be made in such a way that the new turns 'work' without changes to the mechanism? Otherwise, the cuts are simply made to the puzzle's Jaaps sphere without regard to mechanism [as in the case of 6x6x6 vs. 3x3x3]. For ease of conversation, perhaps we can temporarily distinguish these as "mechanical unbandaging" and "mathematical unbandaging." I would argue that the mechanism should not matter and that only mathematical unbandaging should be discussed, seeing as mechanisms vary widely. In this sense, then, "unbandaged version of" would be used very similarly to "superset of."

Well, as you can guess, I'm just fine with bandaging be treated as a proper subset of an unbandaged version. The distinction between mechanical and mathematical is also probably useful... from a puzzle point of view, as we see above, it makes sense to talk about unbandaging by adding cuts and breaking up internal pieces. Although, one could argue that this is still mathematical... adding/subtracting internal bandages just adding rules to what can be moved and when, making for a different puzzle, because the puzzle's space is different.

Quote:
Issue #3: Reversibility
Does a bandaging relationship imply an unbandaging relationship? Is is proper to use the terms in an easily reversible manner? I feel my definition aligns very strongly with both intuition and the English language. Again, though, this allows seemingly undesirable statements as with the 6x6x6 and 3x3x3. Are jumblers considered bandaged since they can be unbandaged partially?

I certainly like it. As for jumblers, the mathematical approach doesn't have the same problems with requiring an infinite number of cuts.

Quote:
Issue #4: Supersets / Order of Unbandaging
Is there a concept of "simplest" unbandaging? The Sun Cube I appears to be the simplest possible shape that can be bandaged into all the Bermuda Cube variants. Can we refine the concept of "simplest" into a definition? Is our intuitive sense of "simplest" related to the idea of doctrine puzzles? Is there a procedural way to unbandage a puzzle to produce the simplest unbandaged version?

Issue #5: Relationships between Sun Cube I / Bermuda Cube / Rubik's Cube
What is the deal with the Bermuda cube? It obviously shares parts from both the Sun Cube I and the Rubik's cube. What is the precise relationship? Is Sun Cube I an unbandaged Bermuda cube? [I would say yes, but my reasoning goes back to issue #3.] Is the Bermuda cube a bandaged Rubik's cube? [I would say no.]

This is tricky. A simplest unbandaging? The über-puzzle that all other puzzles are bandaged from? That's clearly only a mathematical abstraction because of jumbling.

In the more limited sense, it might be useful to talk about a simplest unbandaging of a set class of puzzles. That brings in the somewhat subjective "logical" from above, though... it's easy to interpret a puzzle in many different ways. Which is what issue #5 seems to run into... to start, you need to define where you're coming from, as the Bermuda question becomes moot if you define a Rubik's cube as making 90° turns (which is part of a possible "logical" definition).

Anyways, that's a dump of some of my ideas on the subject... take what you will.

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 6:32 am

Joined: Mon Mar 30, 2009 5:13 pm
@bwross: That's precisely what I meant by over-complicating things.

And yes, "My puzzle won't turn like I think it should" is indeed a form of bandaging, because one can always add more cuts so that the puzzle *does* turn like you think it should. For example, consider a regular 3x3x3 as a bandaged version of a 3x3x3 mix-up.

A geared puzzle is also bandaged, precisely because certain moves are restricted, but in a non-classical manner.

So again, Oskar's definitions are perfectly fine as they are. If they ain't broke, don't try to fix them!

On a general note, any definition must mean the same thing to most people. So it's the opinion of the majority that counts, not just yours, or indeed mine. Therefore a poll would be the best way to resolve this matter.

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 9:50 am

Joined: Thu Jan 19, 2012 12:04 pm
KelvinS wrote:
@bwross: That's precisely what I meant by over-complicating things.

Well, I rambled so it just looks complicated. But I'm really not over complicating things. I'm thinking along the lines of more clarity... where "bandaging" simply means what a person unfamiliar with the term would expect: pieces bound together like with a bandage. Restriction for restrictions in general, and unbandaging is "un-"ing bandaging. Words meaning what they mean. What could be simpler than that?

I'm not a fan of complexities like using a word which is then defined as a synonym of a much better word... my instinct is to simplify that, cut out the weaker word and just use the better one.

Essentially, I'm calling "Bandaging Considered Harmful" here (a reference to Dijkstra's "Goto statement Considered Harmful" letter for those that get it... my reasoning is pretty much the same: using specific names for different things is far more meaningful that extending one name to everything... basically, having "bandaged" regain meaning in the same way the "goto" can, by using always choosing to use meaningful names for the forms of restriction and return "bandaging" to its roots).

Quote:
So again, Oskar's definitions are perfectly fine as they are. If they ain't broke, don't try to fix them!

Note that the definition of unbandaging in the OP is essentially the same as Oskar's, only more abstract in approach). Pieces cut into smaller parts to allow more moves can be glued back together to restrict those moves and rebandage the puzzle, making the cut puzzle an unbandaging.

Quote:
On a general note, any definition must mean the same thing to most people. So it's the opinion of the majority that counts, not just yours, or indeed mine. Therefore a poll would be the best way to resolve this matter.

Let's ask if voting is always the best way to that Colorado senator on the Daily Show last night that got recalled for putting through legislation that 80% of the populace agreed with.

Discussion to try and get some consensus tends to work much better when working on technical specifications. I've see projects go down in flames because people thought that democracy was the best way for everything including programming systems.

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 10:23 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Personally... to me the best solution is to use the terms subset and super set as those terms are already well defined. And I hope they stay that way and no mathematical committees decide to take a vote.
bwross wrote:
I've see projects go down in flames because people thought that democracy was the best way for everything including programming systems.
Oh my favorite example of this... when I was born our Solar System had 9 planets. We now have 8... one has been lost in the voting system someplace. Something tells me that if Pluto could laugh... that is exactly what it would be doing right now. As someone once said... a rose by any other name is still a rose.

Carl

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 5:50 pm

Joined: Sat Sep 15, 2012 7:42 am
TheCubingKyle wrote:
Bandaging: Restricting the turns of a puzzle that would be allowed by the core mechanism. In a bandaged cube, you can state that it is a bandaged 3x3 because the mechanism logically allows each edge and corner to exist individually. A cuboid bandages when the pieces that are extended past the core mechanism are no longer adjacent to other extensions, and now no longer part of a fully cyclical mechanism. Basically, nonextended pieces are blocking the path that the extensions demand.

Your definition is basically identical to Oskar's, but I'll be honest - I don't understand what you're talking about towards the end here. What do you mean by extended? In twisty language, extended usually means adding material to a piece to change its physical shape. Can you clarify why you mean here? Same for cyclical mechanism - I don't know what this means. Perhaps you mean doctrine puzzle?

TheCubingKyle wrote:
Wouldn't the current definition of unbandaging allow for just about any two cubes to be "bandaged" versions of each other? IE Is a curvy copter potentially bandaging rex cube cuts, and is a rex cube bandaging curvy copter cuts? This is clearly the extreme of the 6x6 vs. 3x3 argument (apparently our adopted paradigm for discussion) But how far does that argument actually reach?

The rex cube and curvy copter are not related at all, so I don't understand what you're saying. There is no way to bandage one into the other since their underlying geometries are completely different.

Certainly in the mathematical sense of unbandaging, it's possible to take two puzzles, X and Y, and unbandage them both [in different ways, obviously] until they are identical. This would almost be like a "least common multiple" or "common ancestor." It should be easy to find such a puzzle - just "add" their Jaaps spheres together. This relates back to the Bermuda cubes. The Bermuda cubes and the 3x3x3 have different geometries and are not bandaged versions of each other. But they can both be formed by bandaging the Sun Cube I. So they are more like siblings with the Sun Cube I as a "parent."

KelvinS wrote:
Oskar's definitions are simple, accurate, complete, and unambiguous. Why make it complicated?
KelvinS wrote:
So again, Oskar's definitions are perfectly fine as they are. If they ain't broke, don't try to fix them!

I like Oskar's definitions, too. I even reused part of it. But if they were unambiguous, this conversation would not be happening! Unbandaging seems to be the trickier of the two, and I think it all comes from the word "cutting." Different people seem to have different interpretations of what it means to cut.

KelvinS wrote:
In particular, the following "definition" is completely useless because it's circular logic (2 terms which are defined relative to each other, rather than independently in absolute terms)

I guess I wasn't as clear as I could've been. As part of my definition, I take Oskar's definition of bandaging as a premise. So it's not circular, but unbandaging would be defined solely in terms of bandaging [but not vice-versa]. So to be absolutely clear, my definitions would be:

Bandaging: Restricting moves of a puzzle.
Unbandaging: If puzzle Y can be bandaged into puzzle X, then X can be unbandaged into Y.

The difference between mine and Oskar's is that it avoids the word "cutting" entirely. I think this makes things much simpler, both in regards to English / semantics and the mathematical notions of subset / superset.

KelvinS wrote:
On that basis, bandaged/unbandaged could mean any reversible transformation, such as solved/unsolved, assembled/disassembled, made visible/invisible, or anything else.

This makes no sense to me. None of these would qualify as bandaging under any of the proposed definitions. Perhaps you can clarify?

KelvinS wrote:
Therefore a poll would be the best way to resolve this matter.

"Best" is clearly a matter of opinion. I would much prefer to approach a consensus through discussion with puzzle theorists and builders rather than quickly vote on three undiscussed definitions. You sure seem intent to try to end this matter quickly...

bwross wrote:
This is tricky. A simplest unbandaging? The über-puzzle that all other puzzles are bandaged from? That's clearly only a mathematical abstraction because of jumbling.

That's not what I mean. I'm talking about unbandaging to the point that you have a doctrine puzzle [all positions are equivalent if stickers are removed]. You have to achieve a certain level of symmetry for this to happen, and it seems there should be some minimal / simplest doctrine puzzle that you will hit first. Certainly there can be others beyond that, but what is the simplest doctrine puzzle that can be bandaged to create a given bandaged puzzle?

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Named 3x3x3 Bandaging Patterns

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Tue Nov 12, 2013 6:53 pm

Joined: Sun Jul 14, 2013 11:10 pm
themathkid wrote:
TheCubingKyle wrote:
Bandaging: Restricting the turns of a puzzle that would be allowed by the core mechanism. In a bandaged cube, you can state that it is a bandaged 3x3 because the mechanism logically allows each edge and corner to exist individually. A cuboid bandages when the pieces that are extended past the core mechanism are no longer adjacent to other extensions, and now no longer part of a fully cyclical mechanism. Basically, nonextended pieces are blocking the path that the extensions demand.

Your definition is basically identical to Oskar's, but I'll be honest - I don't understand what you're talking about towards the end here. What do you mean by extended? In twisty language, extended usually means adding material to a piece to change its physical shape. Can you clarify why you mean here? Same for cyclical mechanism - I don't know what this means. Perhaps you mean doctrine puzzle?

In cuboids like the 3x3x5, the core is basically a 3x3 with two sides that have been added to with an extension of the idea of that mechanism. When you do a 90 degree turn, you can no longer individually turn that extra layer and it reduces itself to a 3x3 through bandaging. This is restricting the turns of a puzzle that would be allowed by the core mechanism, because the mech of these pieces sticking out is designed to turn, but it's being interfered with because not all pieces on that face are also designed to allow that turn. That's where my definition becomes a bit more over-arching than Oskar's; It explains bandaging that occurs through a new turn rather than through actively and permanently attaching pieces.

themathkid wrote:
TheCubingKyle wrote:
Wouldn't the current definition of unbandaging allow for just about any two cubes to be "bandaged" versions of each other? IE Is a curvy copter potentially bandaging rex cube cuts, and is a rex cube bandaging curvy copter cuts? This is clearly the extreme of the 6x6 vs. 3x3 argument (apparently our adopted paradigm for discussion) But how far does that argument actually reach?

The rex cube and curvy copter are not related at all, so I don't understand what you're saying. There is no way to bandage one into the other since their underlying geometries are completely different.

Other than a face turning 3-axis relationship, the 3x3 and 6x6 don't share a ton of mechanism. You say a 6x6 can be bandaged into a 3x3, and it can. But in order for a 3x3 to be UNbandaged into a 6x6, its pieces must allow for them to be modified directly into the puzzle by modifying the design, and you can't. The only 3x3's that could be cut down or designed into a 6x6 are ones that already include the edge-wings and added center stocks of what used to already be a 6x6x6.

What I said about the rex cube was just a hyperbole of my point; a 3x3 mech can't turn into a 6x6 mech. Again, this is what I consider a personally reasonable definition of unbandaging. I define based on practical puzzle solving and not mathematical principle because I don't play with puzzles that exist on principle.

I don't agree with what you said about if X bandages into Y, Y unbandages into X. I already supported that view by comparing it to rectangles and squares. "If squares have four straight sides and four right angles, all shapes with four straight sides and four right angles are squares." This isn't true because rectangles exist.

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Wed Nov 13, 2013 8:21 am

Joined: Mon Mar 30, 2009 5:13 pm
TheCubingKyle wrote:
I don't agree with what you said about if X bandages into Y, Y unbandages into X. I already supported that view by comparing it to rectangles and squares. "If squares have four straight sides and four right angles, all shapes with four straight sides and four right angles are squares." This isn't true because rectangles exist.

What nonsense logic, your analogy is completely misplaced. All he was implying was that bandaging is reversible, which is true. What does your example have to do with reversibility?

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 Post subject: Re: Defining "Bandaging" and "Unbandaging"Posted: Wed Nov 13, 2013 10:59 am

Joined: Sun Nov 23, 2008 2:18 am
My interpretation of the squares and rectangles comparison:
All squares are rectangles, but not all rectangles are squares.
All physical 6*6*6s can be bandaged into 3*3*3s, but not all physical 3*3*3s can be unbandaged into 6*6*6s.

Of course, this raises the question as to whether mechanism is important to defining bandaging/unbandaging. Afterall, its likewise accurate to say that:
All physical 3*3*3s can be bandaged into a Fused Cube, but not all physical Fused Cubes can be unbandaged into 3*3*3s.

Then again, considering that both the above examples construct one doctrainaire puzzle by bandaging another doctrainaire puzzle, and both of the more restricted puzzles can be constructed with simpler mechanisms, it raises the question:
Are there any Bandaged puzzles that could be constructed with a mechanism that would not allow physical unbandaging into the corresponding doctrinaire puzzle?

That said, I like the idea of restricting bandaging to its classical definition and using Restriction as the name of the umbrella that encompasses bandageing, gearing, internal blocking, overhang blocking, circles, latching, etc.

As for unbandaging, I like to think of it as the process of transforming a bandaged puzzle into its corresponding doctrinaire puzzle with partial unbandaging being the creation of any puzzle between the original bandaged puzzle and the doctrinaire form. This definition ignores mechanical issues and avoids the problem that one doctrinaire puzzle being an unbandaging ov another.

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