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 Post subject: Deeper-than-Origin Puzzles
PostPosted: Sat Jul 21, 2012 3:50 pm 
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Hi Everyone,

Since there have now been two puzzles created that are deeper-than-origin turning (RotoPrism 2 and Trapentrix), it has got me pondering a few questions...

Does the twisty community agree that these are actually deeper-than-origin turning, or should these geometries be explained or described in a different way?

Must any deeper-than-origin turning puzzle have internal voids? Or is it just coincidence that the first two geometries discovered share this trait?

How should "deeper-than-origin turning" be defined? Is it simply that for the puzzle to maintain the orientation of its axis system, a slice that includes the origin point must be turned?

Are there other possible geometries for deeper-than-origin turning puzzles? It seems that there must be, but now that I'm actively looking for them, they're not immediately obvious (at least not yet).

Is there a finite set of deeper-than-origin turning puzzles, or is this a subset with infinite possible geometries?

One of the concepts for a new deeper-than-origin puzzle that I've been toying with is to imagine a cube that looks much like a standard 3x3x3, but with only two axes. One entire edge (two corner pieces plus an edge piece) would be bandaged together. The two axes would be immediately adjacent to this bandaged edge block (see image).
Attachment:
2 axis cube.JPG
2 axis cube.JPG [ 52.11 KiB | Viewed 7597 times ]
In the image, only the slice planes highlighted in green would be available for turning. Note that the portion of a slice that must be turned in order to prevent the other axis from shifting 90 degrees is the portion that contains the origin of the cube. If it could be implemented, I think this might also qualify as a deeper-than-origin turning puzzle. However, I may be missing something obvious, but I don't see any way to bandage the cubies to restrict it from turning on the outer (non-highlighted) axes. Does anyone else? Of course, this principle could also be applied to a 3-axis cube, in which case it would be visually identical to a 3x3x3 cube, although it would only be able to turn on three axes rather than six. Do you think these cubes would also qualify as deeper than origin?

Any other ideas, questions, or comments?

Dave

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Jul 21, 2012 4:39 pm 
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That's an easy one :) There are two planes where the second layer turns with the first layer. On the two 'deeper than origin' faces, glue the center to the core and VOILA, two layer turns ! =)

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Jul 21, 2012 4:44 pm 
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RubixFreakGreg wrote:
glue the center to the core and VOILA, two layer turns ! =)


Not VOILA. Think one more time. :wink:

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Jul 21, 2012 6:51 pm 
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Aleh wrote:
RubixFreakGreg wrote:
glue the center to the core and VOILA, two layer turns ! =)


Not VOILA. Think one more time. :wink:

Yeah, that wouldn't work...


I was thinking it might be possible to internally bandage the mechanism of a Void Cube or Treasure Chest, perhaps by adding some kind of extension from the bandaged section to the opposite edge of the puzzle. I'm not really sure if that would work, though. Could Oskar's HandiCube be bandaged to block those moves?

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Jul 21, 2012 7:44 pm 
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These are very interesting puzzles. In effect though they are just bandaged forms of their normal versions. Take a 3x3x3 for example and paint one corner cubie gold. Now the rule is you can only rotate the face if the cubie is on it. This puzzle would act like a deeper than origin fused cube but be a bandaged 3x3x3. By thinking of these puzzles this way they are easier to understand and simpler to solve. Plus by thinking this way I might have figured out a mechanism. I'll post it soon.

Edit: Yes. The puzzle is made with a hollow core. The locking piece extends a bit into this void. On each axis a locking piece is pushed up by a spring. When the extended piece is on that face its bump pushes the locking piece down allowing the center to rotate. This is actually simple.

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Jul 21, 2012 8:29 pm 
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Attachment:
Super deep core.JPG
Super deep core.JPG [ 63.05 KiB | Viewed 7485 times ]

Attachment:
Super deep 2.JPG
Super deep 2.JPG [ 60.1 KiB | Viewed 7485 times ]

Attachment:
Super deep 3.JPG
Super deep 3.JPG [ 41.02 KiB | Viewed 7485 times ]


This very very quickly made design shows my concept. It could be used to make the puzzle in the first post or any other deeper than origin puzzle. I'll try printing it in a couple weeks. Do you guys understand?

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Jul 22, 2012 7:21 am 
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PuzzleMaster6262 wrote:
This very very quickly made design shows my concept. It could be used to make the puzzle in the first post or any other deeper than origin puzzle. I'll try printing it in a couple weeks. Do you guys understand?

It took me a while but now I understand it. That's a very good idea. :)

The one thing I can't tell from your pictures is how you ensure that the regular pieces won't push or accidentally slide through the blocked centers.

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Jul 22, 2012 7:36 am 
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My bad, didn't think enough, I'd have to try designing that to find the solution...

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Jul 24, 2012 10:20 am 
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PuzzleMaster6262 wrote:
The puzzle is made with a hollow core. The locking piece extends a bit into this void. On each axis a locking piece is pushed up by a spring. When the extended piece is on that face its bump pushes the locking piece down allowing the center to rotate.
Interesting idea, and it took me a while to see what you're doing, although I must not be comprehending the whole concept. I don't understand how the three locking pieces that contact the "golden corner" (and are thus pushed inwards) will block rotation on the other slice planes. It seems to me like they would block rotation on their own planes instead. They would also need to be designed so that they don't collide with each other as they move inward.

Another possible approach might be a variant of the mechanism from Carl and Tom's Deep Uniaxial 3x3x3. However, I doubt this would work since the offset planes are still shallower than the origin.

Any thoughts on the other questions in my original post?

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Jul 24, 2012 12:15 pm 
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Hi,

I toyed with GAP and found:
Your Wide-Move-Siamese-Cube has 1/2 the permutations of the traditional Siamese cube when you ignore orientation of faces. If you include orientation of faces your variant has 1/32 the permutations of the traditional Siamese Cube.
David Pitcher wrote:
Does the twisty community agree that these are actually deeper-than-origin turning, or should these geometries be explained or described in a different way?
In the past I thought wide moves like you introduced for your Siamese-derivate can always be explained by redefining what a move is. Sadly in this case it does not work without treating your puzzle like a bandaged puzzle.

This can be healed by introduction of the NHPs="Non holding point pieces" aka "imaginary pieces".
That Wide-Move-Siamese-Cube can be treated as a restricted (not truly bandaged!) Complex 3x3x3.
The edges of a traditional 3x3x3 are replaced by the inverted edges.
The faces of a traditional 3x3x3 are replaced by the inverted faces.
The corners stay what they are.
These are the pieces I am talking about: viewtopic.php?p=191464#p191464

Does this makes any sense to anybody?
Andreas


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Jul 24, 2012 12:26 pm 
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Andreas Nortmann wrote:
I toyed with GAP and found:
Your Wide-Move-Siamese-Cube has 1/2 the permutations of the traditional Siamese cube when you ignore orientation of faces. If you include orientation of faces your variant has 1/32 the permutations of the traditional Siamese Cube.
Odd. I would have thought it would have more because there's more pieces. You said this ignores the orientation of the faces, does that include ignoring the orientation of the core too who's orientation is visible because of the face sticker?
What if you allow wide moves on all 3 axis? How does the number of perms compare to a 2x2x2 or the corresponding fused cube?
Andreas Nortmann wrote:
Does this makes any sense to anybody?
Yep.

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Jul 25, 2012 4:07 pm 
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David Pitcher wrote:
Attachment:
2 axis cube.JPG
In the image, only the slice planes highlighted in green would be available for turning. Note that the portion of a slice that must be turned in order to prevent the other axis from shifting 90 degrees is the portion that contains the origin of the cube. If it could be implemented, I think this might also qualify as a deeper-than-origin turning puzzle. However, I may be missing something obvious, but I don't see any way to bandage the cubies to restrict it from turning on the outer (non-highlighted) axes. Does anyone else? Of course, this principle could also be applied to a 3-axis cube, in which case it would be visually identical to a 3x3x3 cube, although it would only be able to turn on three axes rather than six. Do you think these cubes would also qualify as deeper than origin?


That build can be done using the shells approach. On the very inside you have two axes, at a 90 degree angle to each other, opposite where the green slices are. Then there are layers building up until they get deeper than the origin. A novel, if straightforward, application of basic techniques, and probably a very difficult and interesting puzzle.

When I made a comprehensive list of rubik's cube variants, I intentionally didn't include any deeper than the origin cuts, because that would have way more than doubled the already large number of possibilities. There are a bunch of interesting twists, like whether slice moves make the centers move with the center slice or the outer slices. The build described above is probably the simplest of the zany deeper than origin things to do, with making one of the two axes half turn only up next.


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Jul 25, 2012 4:44 pm 
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Bram wrote:
David Pitcher wrote:
Attachment:
2 axis cube.JPG
In the image, only the slice planes highlighted in green would be available for turning. Note that the portion of a slice that must be turned in order to prevent the other axis from shifting 90 degrees is the portion that contains the origin of the cube. If it could be implemented, I think this might also qualify as a deeper-than-origin turning puzzle. However, I may be missing something obvious, but I don't see any way to bandage the cubies to restrict it from turning on the outer (non-highlighted) axes. Does anyone else? Of course, this principle could also be applied to a 3-axis cube, in which case it would be visually identical to a 3x3x3 cube, although it would only be able to turn on three axes rather than six. Do you think these cubes would also qualify as deeper than origin?


That build can be done using the shells approach. On the very inside you have two axes, at a 90 degree angle to each other, opposite where the green slices are. Then there are layers building up until they get deeper than the origin. A novel, if straightforward, application of basic techniques, and probably a very difficult and interesting puzzle.

When I made a comprehensive list of rubik's cube variants, I intentionally didn't include any deeper than the origin cuts, because that would have way more than doubled the already large number of possibilities. There are a bunch of interesting twists, like whether slice moves make the centers move with the center slice or the outer slices. The build described above is probably the simplest of the zany deeper than origin things to do, with making one of the two axes half turn only up next.

Unless I'm missing something the shells much doesn't work. Because the slice moves only affect edges and centers the corners can't stop the pieces from sliding.

With my design the 6 armed core has square prism arms instead of cylinders. This makes the locking pieces only able to move up and down but not rotate. When the locking pieces are pushed down the split between the center piece and core is above the locking piece. When the locking piece isn't pushed down the split is hidden within the locking piece making rotations impossible.

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Jul 25, 2012 5:51 pm 
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Just spotted this topic. I need to think a bit more before I comment but lots of great questions and a very interesting puzzle idea. Not sure I understand the mech drawings that have been posted so I need to study those a bit more as well.

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Jul 25, 2012 9:27 pm 
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Hi All
This idea was published a year ago my friend pytlivyj_1: http://twistypuzzles.com/forum/viewtopi ... 02#p261902
I and my friend Aleh few months working on this puzzle. We found several different mechanisms (2-axis deepcut shell mech, 6-axis with tricky latches), now I have done a good mechanism (similar to the idea of ​​Mike PuzzleMaster6262).
I named it: "Bingo Cube" :) you will soon see a workable puzzle (as well as several others with this mechanism)!

this is a very difficult puzzle for those who want to solve it, Pytlivyj_1 found the algorithm, the shortest of his solution in 1400 steps

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Jul 25, 2012 10:56 pm 
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Hmm, yes, there is some goofy subtlety here due to the layers being exactly parallel. I'm not too fond of mechanisms with locking doohickies in them - they tend to not work all that well. I suspect it's possible to make a nice mechanism using gearing though, where you have floating centers and the core has some gears in it arranged so that when an edge opposite the bandaged piece moves it rotates a gear in the core which in turn forces the floating centers next to it to move in lock step.


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 Post subject: Deeper-than-Origin Puzzles
PostPosted: Thu Jul 26, 2012 1:25 pm 
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Hi All.
Recently I have found a way of solve Bingo Cube from any situation no more than for 1000 turns!
My new method:
1) Permutation of the corner pieces;
2) Orientation of the corner pieces;
3) Permutation of the centers;
4) Permutation of the edges;
5) Orientation of the edges.
:D


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 Post subject: Deeper-than-Origin Puzzles
PostPosted: Sun Aug 12, 2012 2:46 am 
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How to solve a Bingo Siam Cube 3x3x3 (or RwUw-Cube).


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Mar 25, 2013 1:49 pm 
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Hi TP-definition debaters,

Is Mixup Cube a deeper-than-origin puzzle?

My argument to say "yes" comes from studying its cut curve, see below. The cut curve of Mixup Cube has the classic zig-zag shape, like any shells puzzle. However, study that cut carefully, and follow it from the screw to the point where it surfaces. As the cut passes the origin depth (90 degrees), the mechanism would classify as a deeper-than-origin mechanism, similar to Dave Pitchers award-winning Rotoprism.
Attachment:
Mixup Cube v5 - cut curve - view 1.jpg
Mixup Cube v5 - cut curve - view 1.jpg [ 28.44 KiB | Viewed 5375 times ]


Naysayer may object that "deeper-than-origin cut" is a property of the puzzle, not of the mechanism. However, the only way to deduce the deeper-than-origin cuts is hidden bandaging, which blocks moves that seem possible from looking at the surface cuts. By that argument, Enabler Cube would also qualify as a deeper-than-origin puzzles, despite its shallow-cut mechanism. By that argument, Rotoprism would suddenly become a shallow-cut puzzle, if Dave found a way to unbandage his brilliant puzzle (which some frustrated solvers at IPP would highly appreciate, hint, hint, ...). By that argument, deeper-than-origin puzzles would be a subset of twisty-puzzles-with-hidden-bandaging. Correct?

I believe that deeper-than-origin is a property of a mechanism. Rotoprism has a deeper-than-origin mechanism. Mixup Cube has too. And Enabler Cube doesn't. Hidden bandaging is a property of a twisty puzzle, which could be achieved with a deeper-than-origin-cut mechanism or by other means.

Here is another question to the naysayers. What if I bandage one of the visible edge pieces of Mixup Cube to one of the 24 hidden moving pieces that keep the visible centers and edges in place? Then we have a twisty puzzle with hidden bandaging, combined with a deeper-than-origin mechanism. Surely that hypothetical puzzle would qualify as deeper-than-origin?

Oskar

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Last edited by Oskar on Tue Mar 26, 2013 5:34 am, edited 1 time in total.

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Mar 25, 2013 2:32 pm 
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Oskar wrote:
Here is another question to the naysayers.
Ok... I'm the "naysayer" that pointed Oskar to think thread. So forgive the bump...

Anyway via email here is most of what I had to say:

I take deeper-then-deep-cut and deeper-then-origin to mean the same thing. And I would not consider the Mixup Cube to be either. To get to what I understand the meaning of deeper-then-origin to be I think one must define a few other terms:

Order = The number of independent layers per axis on the puzzle.

Holding point = A point with remains stationary under a set of operations which is general enough to represent the entire puzzle. For example a 3x3x3 can have these 6 operations: U, D, L, R, F, and B which all represent clockwise quarter turns of a face. With this set of operations the core of the 3x3x3 is the holding point.

I believe for a puzzle to be deeper-then-origin it must be Order = 1, it must have a holding point, and I think there is usually a piece (can be hidden inside the mechanism) that turns with every operation from the holding point definition. Just as the holding point doesn't turn with any of these operations.

Terms like deeper-then-origin, deep-cut, order, etc... I think apply to puzzles. Not necessarily mechanisms.

The Handi Cube is a great example. The puzzle built with the Handi Cube mechanism is still a 3x3x3. It doesn't change the nature of the properties associated with the 3x3x3. You analyze the 3x3x3 as if it were 27 cubes that just turn under a given set of operations. The details of the mechanism don't come into play when determining if its deep-cut or deeper-then-deep-cut or if its Order = 2, etc.

Yes. The Enabler Cube is deeper-then-deep. Here is why:

(1) This puzzle is Order = 1. On any one axis you have an independent layer and a dependent layer. Which is which depends on your definitions of the operations.

(2) If we consider the unique corner to be in the D,L,B position we can define our operations as U, R, and T which each turn a face layer and the slice layer next to it by a quarter clockwise turn. This is a general representation of the Enabler Cube. All other operations that can be performed on the Enabler Cube can be thought of as a liner combinations of these 3 plus global rotations of the entire puzzle. And with this picture it is clear that our holding point is the DLB corner.

(3) Note the core of the Enabler cube (the center hidden cubie if the puzzle is visualized as 27 perfect cubes) moves with each of these 3 operations.

Its these details which put the Enabler Cube into the deeper-then-deep class. In short you should be able to analyze a puzzle without needing to know how to make a mechanism which allows the puzzle to function. A 3x3x3 puzzle on Gelatinbrain for example is still the same puzzle as your Handi Cube and it doesn't have a physical mechanism.

Oskar via email wrote:
I hope you see the humor in me classifying you as a “naysaying TP-definition debater” :-)
No problem at all. In the twisty puzzle community being called just about ANYTHING by Oskar is an honor... ;)
Oskar via email wrote:
I do not follow your point of “order=1”. A Rubik’s Cube has six axes, each of order=1?
The way I view the 3x3x3 is as a puzzle with 3 axes (the x-axis, y-axis, and z-axis). Each axis has 3 turnable layers on it, 2 face layers and a slice layer. Only 2 of these layers are independant and you are free to pick any 2 you like but once you do the 3rd layer becomes a dependant layer. Granted this isn't the only definition of order in use here at the forums and there are certainly others that consider the Rubik's cube an order=1 puzzle. This is why I included my definition above.
Oskar via email wrote:
As for “holding point”, my post proposes an internally bandaged Mixup Cube with a single internally bandaged piece. Would that qualify as holding point?
I doubt it but I'll need to think about it a bit more before I'm certain. You also appear to be getting into very muddy water with the use of the word bandaging. You view your mechanism for the Enabler Cube as "hidden bandaging" however by the definition I proposed above the Enabler cube is doctrinaire. So the Enabler Cube doesn't behave as a banadged puzzle. I agree the definitions get rather messy here. I can also fuze pieces of a 3x3x3 together (i.e. bandage them) to form a Fuzed Cube which is another example of a doctrinaire puzzle so you could also say that puzzle isn't bandaged.

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Mar 25, 2013 2:58 pm 
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It just dawned on me that I can also show an application of the above check on a Rubik's Cube that can "almost" prove that its deeper-then-origin.

Let's create a different picture of the Rubik's Cube then the typical one:

Let's define these 6 operations:

R = a quarter clockwise turn of the R face AND the adjacent slice layer.
L = a quarter clockwise turn of the L face AND the adjacent slice layer.
U = a quarter clockwise turn of the U face AND the adjacent slice layer.
D = a quarter clockwise turn of the D face AND the adjacent slice layer.
F = a quarter clockwise turn of the F face AND the adjacent slice layer.
B = a quarter clockwise turn of the B face AND the adjacent slice layer.

Take a minute or two but you should be able to convince yourself that these 6 operations completely define all the operations available on a Rubik's Cube. Any other operation you think of can be viewed as a linear combination of these 6 plus maybe a global rotation of the entire puzzle.

Using "another" definition of order you may be able to get away with calling this an Order = 1 puzzle. Check if you are willing to take a bit of hand waving.
The puzzle even contains a piece which rotates with ALL 6 of these operations, again its the center of the 27 perfect cubies. Check.
But here is where this fails. The puzzle as defined this way doesn't have a holding point. So I don't think you can view it as deeper-then-origin.

Just thought as I had provided a positive check above I should also provide a negative to show how I use this "test" for lack of a better word.

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Mar 26, 2013 1:49 pm 
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Carl,

Thank you for accepting this TP-definition debate. Such debates are important to refine our definitions and classifications. Let me respond to some of your points.
wwwmwww wrote:
Order = The number of independent layers per axis on the puzzle.
That is a vague definition, as the word "independent" is rather debatable. It makes the Rubik's Cube a degenerate case. Counting layers from the origin outward is unambiguous, which is why that is the classic definition.
wwwmwww wrote:
Holding point = A point with remains stationary under a set of operations which is general enough to represent the entire puzzle.
That is a vague definition too. If I take the "set of operations" as all possible ways to scramble a puzzle, then any of its visible pieces would qualify as "holding point". Paraphrasing Einstein's theory of relativity: "all holding points are equal".
wwwmwww wrote:
... Enabler cube is doctrinaire ...
Please do not mix the terms "doctrinaire" and "having a vertex-transitive graph". Many (all?) single-bandaged Rubik's Cube puzzles have a vertex-transitive graph, but you would make Bram and Andreas cry by calling those doctrinaire.
wwwmwww wrote:
So the Enabler Cube doesn't behave as a bandaged puzzle.
Hand the puzzle to any puzzle person (or layperson) and he/she will discover some turns that are blocked, which is the very definition of bandaged behaviour.
wwwmwww wrote:
I agree the definitions get rather messy here.
That happens when you try to apply "deeper-than-origin" to classify a puzzle. Classifying a mechanism "deeper-than-origin" is easily tested by taking the mechanism apart and checking the cut line(s). You would not need to change the definitions of "doctrinaire", "bandaged" or "order". Nor would you need to define monsters like "holding point".

The consequence of accepting that "deeper-than-origin" is (only) a property of a mechanism, is that Enabler Cube is no longer classified as "deeper-than-origin". Like Pluto is no longer classified as a planet. Another consequence is that Mixup Cube would be classified as "deeper-than-origin". Just check its cut line.

Please refute.

Oskar

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Mar 26, 2013 10:59 pm 
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On the enabler cube/definition debate. Should the definition be strict to the point where any possible view of the puzzle that would classify it as not deeper than origin cut disallow it from being classified as deeper than origin cut? in other words: I think there is a strong argument to be made for the enabler cube to be classified as a puzzle that is a regular 3x3x3 with moving bandaging. Should this mean that it is not deeper than deeper cut?

I don't think this only applies to the Enabler cube either. I think you can extend this sort of argument to David pitchers puzzles. The common theme among puzzle that are considered to be deeper than origin cut is that an axis of rotation does not coincide with an axis at its antipodal location. This may be a mere coincidence, but I think it may help provide a deeper insight into this conundrum we are currently in. This relates back to Oskar's Enabler cube because he had to impose a very unique and innovative mechanism to employ (what I consider to be) moving bandaging.


just my two cents!


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Mar 27, 2013 12:05 pm 
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Oskar wrote:
Thank you for accepting this TP-definition debate. Such debates are important to refine our definitions and classifications. Let me respond to some of your points.
You are welcome. I like a good debate. But its certainly possible to develope seperate internally consistant pictures and at that point I think things just reply on personal preferences.
Oskar wrote:
That is a vague definition, as the word "independent" is rather debatable.
How so? Any operation which can be expressed as a linear combination of the already existing operations plus a global rotation of the entire puzzle I view as "dependent". If that isn't the case its "independent".
Oskar wrote:
It makes the Rubik's Cube a degenerate case. Counting layers from the origin outward is unambiguous, which is why that is the classic definition.
I disagree. I see nothing degenerate with this definition. In fact its only when I use your proposed definition that I start to see degeneracies. For example, if we just count layers out from the Origin you say the Rubik's cube is the Order=1 puzzle based on the orthogonal 3 axis geometry. Correct? If so what does that make the 2x2x2? What differentiates the 3x3x3 and the 2x2x2 is precisely the number of independent layers per axis.
Oskar wrote:
That is a vague definition too. If I take the "set of operations" as all possible ways to scramble a puzzle, then any of its visible pieces would qualify as "holding point". Paraphrasing Einstein's theory of relativity: "all holding points are equal".
Totally disagree with the first sentence. But I'm in 100% agreement with the second. My degree is in Physics so yes I'm aware that all inertial frame of reference are equal. And in physics when one sets up a problem and solves it he picks one. You don't typically solve the problem in multiple different reference frames at once, though that can be done too. This is the PERFECT analogy for a puzzles holding point. On a 3x3x3 all 27 cubes can serve as the holding point. But you pick just one when you define the independent operations you are going to allow on the cube. Feel free to try. For any of the 27 cubes you'd pick I can give you the 6 operations that would define a 3x3x3 with that choice of holding point. And just as its possible to solve problems using multiple inertial frame of reference in physics that is EXACTLY what I did when I went to define the Complex NxNxN puzzles for N=even. Did you see Brandon's quote in my sig? ;)
Oskar wrote:
Please do not mix the terms "doctrinaire" and "having a vertex-transitive graph". Many (all?) single-bandaged Rubik's Cube puzzles have a vertex-transitive graph, but you would make Bram and Andreas cry by calling those doctrinaire.
Ok... my degree is in Physics and not Math so I can't say I know for sure what vertex-transitive graph means. However are you telling me Bram and Andreas would say the Enabler Cube is NOT doctrinaire? I think you are wrong. Bram himself has said the Fused Cube is doctrinaire. See his post here. Its a puzzle defined with just 3 operations just as the Enabler Cube is.
Oskar wrote:
Hand the puzzle to any puzzle person (or layperson) and he/she will discover some turns that are blocked, which is the very definition of bandaged behaviour.
Its A definition of bandaged behavior. That is part of the problem here. In these forums I think I can find about 10 different ways the word bandage has been used. More to the point of your statement... I would counter with the Enabler Cube is defined with only 3 valid operations. The layperson is going to assume the puzzle has 6 valid operations based on their familiarity with the Rubik's Cube. So yes, it would be reasonable to expect they would think these operations are bandaged. When the truth is they simply aren't valid operations.

Let's approach this a different way... let's assume the Mixup Cube was produced before the Rubik's Cube. This same layperson would now view the Rubik's Cube as a bandaged Mixup Cube. In fact the Mixup cube could be viewed as a bandaged version of the puzzle which allowed 45 degree face turns too. And so on... If you accept this definition of Bandaged then ALL puzzles are Bandaged puzzles. And I just don't view that as a very useful definition. What your layperson is calling bandaged... I would call stored cuts.
Oskar wrote:
That happens when you try to apply "deeper-than-origin" to classify a puzzle. Classifying a mechanism "deeper-than-origin" is easily tested by taking the mechanism apart and checking the cut line(s). You would not need to change the definitions of "doctrinaire", "bandaged" or "order". Nor would you need to define monsters like "holding point".
You are ALREADY using a different definition of bandaged then Bram is using... and I'm not sure of your definition of doctrinaire as you seem to think the Enabler Cube isn't doctrinaire. And there is no agreed definition of order... I'm just using mine. And doing away with holding point is akin to physics doing away with inertial reference frames. I think its needed.
Oskar wrote:
The consequence of accepting that "deeper-than-origin" is (only) a property of a mechanism, is that Enabler Cube is no longer classified as "deeper-than-origin".
Well the Enabler Cube is a Puzzle... so if you make "deeper-than-origin" a mechanism only property then doesn't that mean that NO puzzles are "deeper-than-origin"?
Oskar wrote:
Like Pluto is no longer classified as a planet. Another consequence is that Mixup Cube would be classified as "deeper-than-origin". Just check its cut line.
Using that definition I guess you could call that a "deeper-than-origin" cut line but as the Mixup Cube is a puzzle this term no longer applies. If someone hands me a Mixup Cube on the street I cann't see that cut line. All the pieces could be held to a sphere core via magnets as far as I know. And if it was... it would still be a Mixup Cube... would it not?.
Oskar wrote:
Please refute.
I tried... I doubt I make any headway... but I tried.

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Mar 27, 2013 1:11 pm 
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Since I was mentioned I feel forced to answer here.
Oskar wrote:
wwwmwww wrote:
Holding point = A point with remains stationary under a set of operations which is general enough to represent the entire puzzle.
That is a vague definition too. If I take the "set of operations" as all possible ways to scramble a puzzle, then any of its visible pieces would qualify as "holding point". Paraphrasing Einstein's theory of relativity: "all holding points are equal".
I want to add some examples to Carls answer in this detail topic.
<U D L R F B> represents the 3x3x3. Holding Point in this case is the core.
<U Us L Ls F Fs> represents the 3x3x3. Holding Point in this case is the DRB-corner. Fs stand for a slice move.
You are right in one point. Every (?) physically implemented puzzle I can think of consists only of pieces which can serve as holding point. The concept of the Holding point piece was only introduced because we discovered the Complex puzzles.
Carl: Please give an example to a physically implemented non-holding-point pieces if you know any.
Please do not ask me about a proper definition of Holding Point. Above sentences illustrate my intuitive concept. I assume Carl has the same thoughts.
Oskar wrote:
Please do not mix the terms "doctrinaire" and "having a vertex-transitive graph". Many (all?) single-bandaged Rubik's Cube puzzles have a vertex-transitive graph, but you would make Bram and Andreas cry by calling those doctrinaire.
Carl already pointed out that Bram considers the Fused Cube as doctrinaire. I do so too.
Coming back to the examples above the Fused Cube could be defined as <U, L, F>.
The Siamese Cube could be defined as <U Us L Ls>
The Enabler Cube could be defined as <Uw Lw Fw> or <UUs LLs FFs>
(Uw=UUs => U and Us are forced to move together)
Therefore I consider the Enabler-Cube doctrinaire.
I consider Gear Cube (in its rhombic dodecahedral shape) as doctrinaire too. <UD LR FB> or <DsD2 RsR2 BsB2>
The same with Gear Cube Extreme: <U D RsR2 BsB2>
And with the "Cube with a single gear": <UD Rs R Bs B>

To distinguish between the Fused Cube and the 3x3x3 I use the term "restricted" as opposed to "truly bandaged" (for things like Square-1 or Mefferts bandaged cube). I know that with considering the Fused Cube as doctrinaire the terms become even more complicated.
There is very human reason to consider all the puzzles I mentioned as doctrinaire: It is very easy to define those restricted 3x3x3s I mentioned when you already have defined the 3x3x3. I have never defined a truly bandaged 3x3x3 for a reason.

BTW:
One of my serieses finished in the past contains all the physically implementable subgroups of the 3x3x3 which could be defined like demonstrates above. Parts of that series can be found here:
viewtopic.php?f=14&t=15016
viewtopic.php?f=15&t=24080
viewtopic.php?f=15&t=24753


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Mar 27, 2013 2:54 pm 
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Andreas Nortmann wrote:
Carl already pointed out that Bram considers the Fused Cube as doctrinaire. I do so too.
In that case Bram and you should consider all single-bandaged Rubik's Cube as doctrinaire too. Argument: take the single bandaged piece as "holding point" to define the turning operations. You can easily convince yourself that all turning operations map the puzzle onto itself.

Following this line of reasoning, we can stop use the term "doctrinaire", as mathematics does already have a proper term, namely vertex-transitive.

Oskar

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Mar 27, 2013 4:10 pm 
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Oskar wrote:
In that case Bram and you should consider all single-bandaged Rubik's Cube as doctrinaire too. Argument: take the single bandaged piece as "holding point" to define the turning operations. You can easily convince yourself that all turning operations map the puzzle onto itself.
That sounds correct to me.
Oskar wrote:
Following this line of reasoning, we can stop use the term "doctrinaire", as mathematics does already have a proper term, namely vertex-transitive.
That may very well be true as well. Not being a mathematician, I'm personally more comfortable with Bram's definition of doctrinaire as I'm not certain I understand all the ramifications of that wikipedia definition of vertex-transitive. Looking at that wikipedia page makes me feel like the layperson though I get that same feeling looking at most legal documents too. ;)
Andreas Nortmann wrote:
Carl: Please give an example to a physically implemented non-holding-point pieces if you know any.
Well that's tricky. You can define the 3x3x3 with the operation set I did a few posts above and say the 3x3x3 doesn't contain a holding point. But I know that's not what you are after. Maybe the best example would be the puzzle I call the The Augmented Skewb. This doesn't exist as a physical puzzle yet but it easily could and is on my to-do-list. And now looking at this closer this also doesn't answer the question you ask as actually both cores CAN act as a holding point. So I guess the answer to your question is no... I'm not aware of any puzzles which currently implement what you call "non-holding-point" pieces. I think I typically call them "imaginary" pieces and yes, Andreas is correct, these come into play on the Complex Puzzles.

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Mar 27, 2013 6:27 pm 
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wwwmwww wrote:
Andreas Nortmann wrote:
Carl: Please give an example to a physically implemented non-holding-point pieces if you know any.
... Maybe the best example would be the puzzle I call the The Augmented Skewb ...
A hidden-bandaged version of Dino Pent may come close to what Carl is referring to. Dino pent is a pentagon split in two, with ten dino-style-turning corners. The core consists of two pivot-connected hemispheres, each of which has five hidden rotating "centers" under each corner. When you glue some of those hidden rotating centers to its hemisphere, the visible puzzle is still fully functional, albeit with hidden bandaging. I suspect that it is pretty hard to define an unambiguous "holding point" for that puzzle.
wwwmwww wrote:
Oskar wrote:
Following this line of reasoning, we can stop use the term "doctrinaire", as mathematics does already have a proper term, namely vertex-transitive.
That may very well be true as well. Not being a mathematician, I'm personally more comfortable with Bram's definition of doctrinaire as I'm not certain I understand all the ramifications of that wikipedia definition of vertex-transitive.
I have no objections to having a layman's explanation associated with a formal definition. After all, most laws and technical standards have that. Still, we need to agree which definition takes precedence for those cases where Bram's definition of doctrinaire is inconclusive or contradictory with the formal mathematical definition. Therefore I would like to hear Andreas' and Bram's view about calling all single-bandaged Rubik's Cube doctrinaire.

Personally, I prefer a more narrow definition of doctrinaire. For example a combination of vertex-transitive and no stored cuts (inactive cuts?) in any canonical state of the puzzle, with proper definition of "inactive/stored cut" and "canonical state".

Oskar

P.S. Are there any objections against saying that Mixup Cube has a deeper-than-origin mechanism?

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Mar 27, 2013 7:26 pm 
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Oskar wrote:
A hidden-bandaged version of Dino Pent may come close to what Carl is referring to. Dino pent is a pentagon split in two, with ten dino-style-turning corners. The core consists of two pivot-connected hemispheres, each of which has five hidden rotating "centers" under each corner. When you glue some of those hidden rotating centers to its hemisphere, the visible puzzle is still fully functional, albeit with hidden bandaging. I suspect that it is pretty hard to define an unambiguous "holding point" for that puzzle.
Unless I'm mistaken the Dino Pent jumbles. And I think the way I've defined holding point it may only apply to doctrinaire puzzles.
Oskar wrote:
I have no objections to having a layman's explanation associated with a formal definition. After all, most laws and technical standards have that. Still, we need to agree which definition takes precedence for those cases where Bram's definition of doctrinaire is inconclusive or contradictory with the formal mathematical definition. Therefore I would like to hear Andreas' and Bram's view about calling all single-bandaged Rubik's Cube doctrinaire.
Since I'm not sure I understand the formal mathematical definition can you point out a case where Bram's definition is inconclusive or contradictory?
Oskar wrote:
Personally, I prefer a more narrow definition of doctrinaire. For example a combination of vertex-transitive and no stored cuts (inactive cuts?) in any canonical state of the puzzle, with proper definition of "inactive/stored cut" and "canonical state".
There are MANY puzzles which require stored cuts. For example take my Bubbloid122 puzzle. Do you want to consider this a bandaged puzzle?
Oskar wrote:
P.S. Are there any objections against saying that Mixup Cube has a deeper-than-origin mechanism?
Before I answer that let me ask a question... can't I get the exact same puzzle with this cut surface?
Attachment:
Cut.png
Cut.png [ 19.27 KiB | Viewed 5008 times ]
And if I did, would you consider this a deeper then origin cut?

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Mar 27, 2013 7:52 pm 
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Oskar wrote:
What if I bandage one of the visible edge pieces of Mixup Cube to one of the 24 hidden moving pieces that keep the visible centers and edges in place? Then we have a twisty puzzle with hidden bandaging, combined with a deeper-than-origin mechanism.
Hmmm... I've been thinking about this question more and it has made me aware of an issue with the definition of holding point. In the post just above I thought that term could be applied to just about any piece on any real doctrinaire puzzle. But I don't think that is the case with the Mixup Cube. On the Mixup Cube I think only the corners can serve as valid holding points. If I try to make a face center a holding point then once end up in a position where that face center is in an edge location the available operations are different. I think the holding point must also be fixed with respect to the axis system of the puzzle when in a doctrinaire position. Does this make a Mixup Cube slice layer pieces and example of physically implemented non-holding-point pieces? They aren't imaginary pieces so I still don't think they are what Andreas is after. But it does point out a "weakness" of the holding point definition.

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Wed Mar 27, 2013 11:56 pm 
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A singly bandaged Rubik's Cube (glueing just one edge to just one corner) is doctrinaire. Whether it's deeper than deep cut hinges on a sillier question: Is the slice-only cube deeper than deep cut?

I think it makes sense to talk separately about deeper than origin mechanisms and deeper than origin puzzles. Both concepts are meaningful, and refer to slightly different things.

All deeper than origin puzzles must either have stored cuts or hidden bandaging, otherwise there's no way to differentiate them from the shallow cut puzzle with axes going opposite.

Whether a puzzle 'seems' bandaged to most lay people who hold it isn't really relevant to the properties of the puzzle. To most lay people (and even most specialists, including me) Rotoprism 2 will at first seem like a trivial non-puzzle with three things which can rotate and nothing else possible. That doesn't mean we should conclude that it isn't really a scrambleable puzzle.

I think it makes sense to expand the definition of doctrinaire to include everything vertex transitive, not just based on strict permutations. That makes it even more clearly defined, and also makes it clearly include Tracker Ball and Spider Gear, which seem like they should qualify. I still prefer the term 'doctrinaire' though, it's less clinical and conveys more of the gestalt, especially when it comes to the solving experience, where the techniques in doctrinaire and non-doctrinaire puzzles are wildly different.

Deeper Madness is an interesting case. If you flip the vertices of a triangular prism to the opposite point from the origin, you wind up with a triangular prism in a different orientation, so because it's almost exactly deep cut the two halves should be isomorphic to each other and the puzzle could be viewed as either side being the one which is actually moving, as long as you're completely consistent with all later slices about which side it is which is moving (although keeping that straight sounds extremely difficult in practice). The pictures don't appear to have two isomorphic halves on either side of each slice though, so I'm not sure what to make of that.


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 12:05 am 
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Oskar wrote:
Therefore I would like to hear Andreas' and Bram's view about calling all single-bandaged Rubik's Cube doctrinaire.
I am neither Andreas or Bram and am not yet ready to weigh in on the full "deeper-than-origin" question, but this to me was worth engaging.

I believe perhaps our best solution for this narrow issue is to refine Bram's definition to match what I believe was his intent. Part of what I feel he is getting at with his definition does involve some symmetry.
Bram wrote:
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.
From a frame of reference point you can argue that holding the single bandage piece means the rest will look identical for all positions. But I think we need to expand the definition to require this "exactly the same" property for all frames of reference: And clearly if you choose any of the non-bandaged pieces as the "hold" piece it is immediately not "exactly the same" once that bandaged piece is permuted.

So while by the strict definition a singly bandaged cube might squeak by as doctrinaire but only with that single frame of reference caveat. I think we lose nothing of the original intent by requiring all frames of reference to be identical, and it certainly avoids this otherwise confusing outcome.

Dave

P.S. I think Carl gets close to my intuitive feeling of deeper-than-origin being questionable by mentioning the difference between "puzzle" and "puzzle mechanism". I think there is value in talking about properties of abstract puzzles without considering their mechanism.

P.P.S. It seems Bram posted in the time I was composing this...

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 9:43 am 
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Why are we using the term "deeper than origin cut puzzles", anyway? I think the term "moving origin puzzles" might be better...


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 10:19 am 
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Bram wrote:
Whether it's deeper than deep cut hinges on a sillier question: Is the slice-only cube deeper than deep cut?
Maybe silly but that is a very interesting question. My gut wanted to say "no" but it does pass my test. It is order=1. The set of 8 corners can be picked as the holding point. And if you do, the core or central cubie moves with all 3 operations. So I guess at least in my picture at the moment it is deeper-then-deep cut. I'm not sure that feels right.

However I'm not sure how the singly bandaged Rubik's Cube (glueing just one edge to just one corner) being deeper-then-deep cut hinges on this? I would say that isn't an order=1 puzzle. Two of its axes have 2 independent layers.

Hmmm...
Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 12:09 pm 
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wwwmwww wrote:
I'm not aware of any puzzles which currently implement what you call "non-holding-point" pieces. I think I typically call them "imaginary" pieces and yes, Andreas is correct, these come into play on the Complex Puzzles.
You call them "imaginary", right. You might remember that I do not like this term. "non-holding-point" is more descriptive.

On the Dino Pent: Here I have been a coward in the past: I haven't dared to tackle jumbling puzzles. This answer does not help but at least true.
Oskar wrote:
Therefore I would like to hear Andreas' and Bram's view about calling all single-bandaged Rubik's Cube doctrinaire.
In my earlier posting I forgot to write that I consider all puzzles as doctrinaire which can be redefined by a notation like the examples given there. Therefore I consider every puzzle bandaged by a single block (can have more than one bond, like the Fused Cube) as doctrinaire. The example behind my links demonstrates that there are also some multiply bandaged 3x3x3s (more than one block) which are doctrinaire under Brams definition.
Sometimes I want to distinguish between the puzzles of Oskars approach and the ones which fall under Brams more open definition. Therefore I tried to introduce the term "restricted".

The slice-move-only cube Bram mentioned is an interesting case. Like with the 3x3x3 ALL of its pieces can be used as holding points. Therefore it qualifies for the narrower definition of Dave. And it has already been implemented in plastic under the name B12C111.


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 12:52 pm 
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Andreas Nortmann wrote:
You call them "imaginary", right. You might remember that I do not like this term. "non-holding-point" is more descriptive.
Yes, I remember. As I recall the mapping was:

non-holding-point = imaginary
zero-volume-holding-point = virtual

I still like the terms imaginary and virtual and they help support the additional naming conventions I like to use:

real+imaginary = complex
real+virtual = augmented

But now that I look at things closer I may see another issue with your naming. Look at the Mixup Cube. I believe the only valid holding points are the corners. Yet I don't consider the edges or face centers (which are actually the same piece type) as either imaginary or virtual. Do you have a method that allows you to treat these as holding points? I can't think of a clean way to define the operations if you do.

Aother example would be the Bubbloid122 puzzle. On that puzzle I believe the core is the only valid holding point. As with the Mixup Cube if you try to use any other piece as a holding point the valid operations end up moving around in space.

And these 2 examples are both what I consider doctrinaire puzzles. I have no idea how apply these classifications to a jumbling puzzle.

Looking at Dave's stricter definition I'm curious if he would view either the Mixup Cube or Bubbloid122 as doctrinaire.

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 1:46 pm 
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Bram wrote:
I think it makes sense to expand the definition of doctrinaire to include everything vertex transitive, not just based on strict permutations. That makes it even more clearly defined, and also makes it clearly include Tracker Ball and Spider Gear, which seem like they should qualify. I still prefer the term 'doctrinaire' though, it's less clinical and conveys more of the gestalt, especially when it comes to the solving experience, where the techniques in doctrinaire and non-doctrinaire puzzles are wildly different.

Deeper Madness is an interesting case.
Stating that adding vertex transitive to doctrinaire EXPANDS its definition implies to me the two aren't exactly the same. Someone want to spell out the differences for the layman? I stopped to look up each of these puzzles hoping that would help:

Tracker Ball
Spider Gear
Deeper Madness

And it looks like only Tracker ball has been physically made at the moment and I can't say I understand any of them very well and I also wasn't sure if Bram was Deeper Madness was a doctrinaire puzzle or just interesting. I think (thought I'm far from sure) that it jumbles so maybe its not doctrinaire.

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 3:41 pm 
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Well, since I started this whole mess (which I think is actually a wonderful debate), I figure I should throw in my 2 cents as well.

I still see the definition for deeper-than-origin which I proposed at the beginning to be the easiest to understand. I think of it as applying to puzzles and not to mechanisms, as the mechanical implementation of a puzzle geometry may take many forms. Simply said, I see a puzzle as deeper-than-origin turning if, for the axis system of the puzzle to remain stationary in space, a slice containing the origin point of that axis system must be turned, and the smaller slice held stationary.

For RotoPrism 2, if 90 degree turns were not allowed, it would become a shallow-cut puzzle (no matter that the mechanism extends beyond the origin). It is the 90 degree turns that demand that the larger slice (including the axis system origin point) be turned in order to prevent the axis system itself from flipping by 90 degrees.

For Trapentrix, it is easier to see what is happening since there are no "special" moves such as RotoPrism 2's 90 degree turns, and it is easy to see where the axes are located. The larger part of the puzzle (origin point included) must be rotated, or else the other axis will be flipped into a new position. Thus it is deeper-than-origin turning.

I think the same logic does apply to the two cubic puzzle concepts (incredibly, both of which have been physically realized in the short time since they were proposed). Where these two get potentially muddy is that since the axis system is orthogonal, one could argue that the axis system really hasn't changed if the shallower slices are turned, since the axes land in the same line after such a turn. Perhaps this is what leads to the thought that these two puzzles are merely bandaged, and not deeper-than-origin turning. However, I think it is most obvious in the two axis cube that it is really deeper than origin turning since you can readily see that the large edge moves around the puzzle if the shallow slices are turned rather than the origin-containing slices.

Deeper Madness is an interesting case, in that if viewed as a face-turning puzzle it is indeed sliced deeper than the origin of the axis system. However, the actual turning centers are hidden in the mechanism, and to truly determine if the puzzle is deeper-than-origin (at least according to my definition), you need to see how these pieces move with each turn of the puzzle. If they stay in position no matter which portion of the puzzle is held while the other one is turned, then I think the puzzle should be viewed as edge-turning (despite the fact that the slice planes are not parallel to the edges). In this case it would not be a deeper-than-origin turning puzzle, as there is no consequence (rearrangement of the axis system in space) of turning the shallow portion of the puzzle. However, if the axes are relocated in space no matter which side of the slice plane is held stationary, then we have a different beast entirely.

Regarding the Mixup Cube, I think this one is also shallow cut, despite the fact that the best known mechanism may have a slice pattern that extends beyond the origin (I say may because Carl's redrawing of Oskar's slice pattern might indeed be just as viable). With this puzzle, the hidden centers that actually define the axis system must be considered, and not the visible face centers. If you look at the puzzle this way, I believe it is clear that neither shallow turns, nor slice turns reorient the axis system in space. Therefore it does not meet my definition of a deeper-than-origin turning puzzle.

No matter what conclusions are eventually arrived at, I think it would be great to have one central place (the Twistipedia?) where such terminology and definitions can reside for easy reference and editing as our knowledge expands.

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 5:56 pm 
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Jared wrote:
Why are we using the term "deeper than origin cut puzzles", anyway? I think the term "moving origin puzzles" might be better...


Can I get a serious answer on why this would or wouldn't be good?


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 6:57 pm 
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Jared wrote:
Why are we using the term "deeper than origin cut puzzles", anyway? I think the term "moving origin puzzles" might be better...
There are a few reasons I think the term "deeper than origin cut" (or some such variant) is appropriate.

First, it follows in the established lexicon of shallow-cut and deep-cut puzzles.

Second (and possibly more importantly), the origin actually doesn't move in these puzzles unless they are turned as if they are shallow-cut. Particularly with the RotoPrism 2, this is something to be avoided, as the solver would quickly lose their frame of reference, and have to reorient the puzzle in order to proceed. Therefore the description of "deeper-than-origin-turning" accurately describes how the puzzle is best played.

Last, "moving origin" implies to me that the origin point is actually relocating in space. In reality, if the puzzles are turned holding the larger slice still (as if they were shallow cut), the origin only rotates, it does not relocate. One might make an argument for "rotating origin", but I still think "deeper-than-origin-turning" is more appropriate for the above two reasons.

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Last edited by David Pitcher on Thu Mar 28, 2013 7:02 pm, edited 1 time in total.

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 7:02 pm 
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David Pitcher wrote:
No matter what conclusions are eventually arrived at, I think it would be great to have one central place (the Twistipedia?) where such terminology and definitions can reside for easy reference and editing as our knowledge expands.
I like this idea and I know it has been proposed before. The biggest problem as I see it is we rarely reach a conclusion everyone agrees with. There are at least 3 definitions of "order" here and just as many definitions of "deep cut". Which come to think of it wasn't there supposed to be a new thread on that topic here shortly? If I missed it PLEASE feel free to point me to it. Here is a thread where the meaning on Complex NxNxN was debated for even N and I was actually able to convince Brandon that my definition was better then his. That is actually a very rare thing here as everyone (myself included) seems to get stuck in their own mindset and really resists change. It leads to alot of fun (though sometimes hair pulling) debating but I'm not sure about conclusions.
Jared wrote:
Why are we using the term "deeper than origin cut puzzles", anyway? I think the term "moving origin puzzles" might be better...
Sorry... not meaning to ignore you. And wecome to the debate. As for your question... If you want to consider deeper-then-origin, deeper-then-deep-cut, and moving-origin to all mean the same thing I'm perfectly ok with that. We can call it GEORGE if we want to. I'm less concerned about the name then the I am about the definition of what this property means. Personally I think I ever so slightly prefer deeper-then-deep-cut as origin is a term that hasn't really been talked about much in this context. But seeing as there is still debate about "deep-cut" it certainly has its own problems too. If we want to talk about moving origin... what about puzzles that have 2 or more points where axes of rotation intersect. The examples I can think of off the top of my head still contain all these points in a single puzzle piece (can anyone thing of an exception?) so maybe this piece can be refered to as the origin piece... which I think most would simply call the core. So maybe the term becomes "moving-core". But if you take this too literally, i.e. the name is the definition, then I guess you could say the 3x3x3 is a "moving-core" puzzle as its only with the choice of the core as a holding point that it doesn't move. There are 26 other potential holding points which could be used to define the operations of a 3x3x3 and any of those allow the core to move.

Here let me try to make my definition more formal:

If Puzzle "X" can be defined by operations "O1", "O2", "O3", ... and "On" and the "signature" of some (maybe not all) of the pieces in that puzzle can be expressed as a list of operations which move that piece, THEN Puzzle "X" is GEORGE if and only if Puzzle "X" contains the pieces with signatures:

{O1, O2, O3, ... On} i.e. a piece which is moved with ALL operations
and
{} i.e. a piece which is moved by NONE of the operations

For a list of operations to be able to completely define a puzzle I believe that means the puzzle must be doctrinaire or vertex transitive or maybe I should say SALLY and that these operations are all available from the doctrinaire state and each operation restores the puzzle to the doctrinaire state.

Given this definition and nothing more its easy to see why the puzzle must be Order=1 if its not a Complex puzzle which I believe covers all current physical puzzles. This definition however I guess does mean that the Complex 3x3x3 would be GEORGE as well even though I consider it Order=2.

Note: Order=1 basically boils down to 1 operation per axis. I.e. the turning of the independent layer on that axis. If your list above has more operations then the puzzle has axes then that means there are 2 independent layers on at least one axis. If there exists a piece which is in BOTH of these layers at the same time then it is imaginary and you have a complex puzzle.

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 7:25 pm 
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David Pitcher wrote:
Last, "moving origin" implies to me that the origin point is actually relocating in space. In reality, if the puzzles are turned holding the larger slice still (as if they were shallow cut), the origin only rotates, it does not relocate. One might make an argument for "rotating origin", but I still think "deeper-than-origin-turning" is more appropriate for the above two reasons.
I generally agree... though I think I'm willing to accept rotation as a form of movement. If you ask your child to go stand in the corner and not to move and he goes to the corner and spins in circles I guess he could say "See Dad I'm not moving I'm just rotating". LOL!!!

Carl

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 9:17 pm 
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wwwmwww wrote:
Stating that adding vertex transitive to doctrinaire EXPANDS its definition implies to me the two aren't exactly the same. Someone want to spell out the differences for the layman? I stopped to look up each of these puzzles hoping that would help:

Tracker Ball
Spider Gear
Deeper Madness

And it looks like only Tracker ball has been physically made at the moment and I can't say I understand any of them very well and I also wasn't sure if Bram was Deeper Madness was a doctrinaire puzzle or just interesting.


Deeper Madness is a jumbling only puzzle (please don't start a discussion about what 'jumbling only' means) and is most definitely not doctrinaire. My musing on it had to do with whether it (or maybe a slightly deeper cut version of it) qualifies as a deeper than origin cut puzzle.

For the Tracker Ball and Spider Gear the movement is that there are several different axes which the central ball can be rotated on, but the interesting this is that on each axis it can move back and forth between two different states but not keep going (the other axes are changed in the process, which is why the puzzle can be scrambled). Technically that makes moves not be a permutation, because the allowable moves of the central ball flip back and forth between two different states rather than always being the same, but the two different states are symmetric to each other, so it's still vertex transitive.

A video of the Spider Gear would probably clarify. Tracker Ball is fairly confusing even when you're holding it.


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Mar 28, 2013 10:44 pm 
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wwwmwww wrote:
I generally agree... though I think I'm willing to accept rotation as a form of movement. If you ask your child to go stand in the corner and not to move and he goes to the corner and spins in circles I guess he could say "See Dad I'm not moving I'm just rotating". LOL!!!

Carl


Yes, exactly. LOL!


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri Mar 29, 2013 5:55 am 
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I continue my objection against the term "deeper-than-origin puzzles". The fundamental problem is that it requires people to hypothesize a mechanism. Either you look at the puzzle as it is played, but then your have only the outside looks and feels to work with. Or you look at it as a mechanism, something to be taken apart and studied. This talk about deeper-than-origin puzzles is like having a cake and eat it. The issue about "holding points" is getting so muddy, as it is trying to model a puzzle's hypothetical mechanism while trying to desperately staying away from the real mechanism. Also, what if we start unbandaging these puzzles? How much bandaging should be eliminated until it no longer classifies as a "deeper-than-origin puzzle"? What if I remove just one of the hidden bandages of Enabler Cube? Or if I remove all except one? What if Dave Pitcher would manage to do the same with his RotaPrism 2?

In contrast, I do like the concept of "deeper-than-origin mechanisms". It is a fundamental class of puzzles, that can be easily identified by classifying a cut line. There are still degenerate cases like Mixup Cube, but those can be resolved.
    I would argue that only classic (non-degenerate) cut lines should be considered.
    Carl might argue that his cut line is as viable as mine.
    I would counter that his cut line would classify Mixup Cube as having a deep-cut mechanism, which feels strange next to the existing definition of deep-cut puzzle.
    Carl might agree or disagree.
    Bram would throw in the Slice-Only Cube, which we would all debate.
Depending on the outcome of the debate (which could be resolved by flipping coins, head butting, rock-paper-scissor-lizzard-spock, or voting like the IPCC or IAU do on climate models and astronomical classifications) we have a proper definition for "deeper-than-origin mechanisms", and perhaps some qualifiers like "degenerate", which we could include in Dave Pitcher's proposed Twistipedia.

Oskar

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri Mar 29, 2013 8:59 am 
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Oskar wrote:
I continue my objection against the term "deeper-than-origin puzzles". The fundamental problem is that it requires people to hypothesize a mechanism.


How about if we say that a puzzle is deeper than origin if the only way to view it as doctrinaire involves a slice where the half which moves contains more than half the pieces? That includes some goofy edge cases, for example if you take a Rubik's Cube, then take one of its center pieces and glue two opposite edges to it, then take the opposite center face and glue it to the core so it can't rotate in place, you wind up with a deeper than origin puzzle which doesn't really feel like a deeper than origin mechanism, but from a solving experience it clearly feels deeper than origin, so I think it's a useful insight to call it that.

I don't see the way that unbandaging can change a puzzle suddenly from very deep cut to very shallow cut as a problem. More like an interesting insight.


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri Mar 29, 2013 9:16 am 
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Bram wrote:
How about if we say that a puzzle is deeper than origin if the only way to view it as doctrinaire involves a slice where the half which moves contains more than half the pieces?
No go. A puzzle is doctrinaire, or it is not. Your goofy example remains doctrinaire (a.k.a. vertex-transitive), even if you play it while riding a roller-coaster. Also, your definition excludes RotoPrism 2, which happens to be the only non-controversial example of a puzzle with a deeper-than-origin mechanism.

Oskar

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri Mar 29, 2013 10:59 am 
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Oskar wrote:
I continue my objection against the term "deeper-than-origin puzzles". The fundamental problem is that it requires people to hypothesize a mechanism. Either you look at the puzzle as it is played, but then your have only the outside looks and feels to work with. Or you look at it as a mechanism, something to be taken apart and studied. This talk about deeper-than-origin puzzles is like having a cake and eat it.
I think it's fine to distinguish between puzzles (which may have physical manifestations or not) and mechanisms. It's easy to come up with puzzle concepts that cannot be physically implemented, but that doesn't make them any less valid as puzzles. For example, here's a picture of an impossible 5x5x5 cube before and after a single turn operation. Note that the cube turns normally, except that a selection of center pieces not on the turning slice also turn with the slice:
Attachment:
5x5x5 demo.jpg
5x5x5 demo.jpg [ 114.44 KiB | Viewed 4584 times ]
Just because it may be impossible (at least until Oskar proves me wrong) to make this puzzle mechanically, it could still be played virtually. One could make virtual cubes (or other shapes) with any number of impossible to physically implement groupings. That doesn't mean that they aren't legitimate puzzles despite the lack of a mechanism. That said, I don't see a separation between the mechanical classifications of the puzzles we're talking about, and how they might be classified if we only viewed them from a "virtual" perspective.
Oskar wrote:
I would counter that his cut line would classify Mixup Cube as having a deep-cut mechanism, which feels strange next to the existing definition of deep-cut puzzle.
I think this might feel strange because I don't believe the cut line actually shows a deep-cut mechanism. I believe that the only reason Oskar's original cut line must extend past the origin is due to the limitations of Solidworks in only allowing a sketch used for the intended rotation operation to have a single end point. If multiple end points were allowed, Carl's sketch could be used. Further, even Carl's sketch does not imply a deep-cut mechanism because the reason the sketch touches the centerline of the puzzle is to create two separate parts. Without the cut line touching the center, the part that physically gets attached to the core would not be separate from the face centers and edge pieces that glide over the core rotators. The Mixup Cube is essentially a standard 3x3x3 mechanism at heart, with another 3x3x3 sliding over the core 3x3x3. Nothing deep cut about it.
Oskar wrote:
Also, what if we start unbandaging these puzzles? How much bandaging should be eliminated until it no longer classifies as a "deeper-than-origin puzzle"? What if I remove just one of the hidden bandages of Enabler Cube? Or if I remove all except one? What if Dave Pitcher would manage to do the same with his RotaPrism 2?
I don't see a problem here. Unbandaging any such puzzle creates a new puzzle, which may then fall into a different category. For example, if I were to create a new version of RotoPrism 2 that turned on 9 axes (eliminating the phantom axes and making them real), it would be a shallow-cut puzzle. If Oskar were to remove some of the hidden bandaging pieces in an Enabler Cube, that new puzzle may be deeper-than-origin turning on some axes, but not others (it becomes a hybrid, much like Cheese Block which is deep cut on some axes, and shallow cut on others). If enough bandages were removed it might become a shallow-cut puzzle. Either way, removing the bandaging fundamentally changes the nature of the puzzle and thus it's classification.
Oskar wrote:
Also, your definition excludes RotoPrism 2, which happens to be the only non-controversial example of a puzzle with a deeper-than-origin mechanism.
Do you consider the Trapentrix to be deeper-than-origin turning, and if not, why?

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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri Mar 29, 2013 2:18 pm 
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Oskar wrote:
Bram wrote:
How about if we say that a puzzle is deeper than origin if the only way to view it as doctrinaire involves a slice where the half which moves contains more than half the pieces?
No go. A puzzle is doctrinaire, or it is not. Your goofy example remains doctrinaire (a.k.a. vertex-transitive), even if you play it while riding a roller-coaster. Also, your definition excludes RotoPrism 2, which happens to be the only non-controversial example of a puzzle with a deeper-than-origin mechanism.


I keep forgetting that most of these deeper than origin cut puzzles are jumbling, which throws a wrench in the whole thing, because they're inherently fairly bandaged. regardless of which side is 'really' doing the turning.


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 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri Mar 29, 2013 2:35 pm 
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David Pitcher wrote:
The Mixup Cube is essentially a standard 3x3x3 mechanism at heart, with another 3x3x3 sliding over the core 3x3x3. Nothing deep cut about it.
Dave, we need a proper definition to support (or falsify) such a statement. You would (and should) not accept a statement from me like "The RotaPrism 2 is essentially a standard triangular prism at heart, with some floating triangle and edges over the core prism. Nothing deep cut about it." either. I claim that twisty puzzles are defined by their cut curve(s). One could chop up the cut curve in multiple pieces, flip and turn it (like Carl did), but I propose to require the (Solidworks-style?) non-degenerate continuous cut curve. I see no reason to make an exception for Mixup Cube, just because Mixup Cube hás to be classified as a classic twisty puzzle.

COMPARE: People have classified Pluto as planet for a long time. Then many Pluto-like Kuiper-belt objects were discovered, which made it necessary for the IAU to reclassify Pluto. Your RotoPrism 2 (and Trapentrix, I guess) is such Kuiper-belt object. And I argue that it may be needed to reclassify Mixup Cube.

David Pitcher wrote:
Oskar wrote:
Also, what if we start unbandaging these puzzles? How much bandaging should be eliminated until it no longer classifies as a "deeper-than-origin puzzle"? What if I remove just one of the hidden bandages of Enabler Cube? Or if I remove all except one? What if Dave Pitcher would manage to do the same with his RotaPrism 2?
I don't see a problem here. Unbandaging any such puzzle creates a new puzzle, which may then fall into a different category.
I object against sloppy definitions. My point is that the current definitions (plural) of "deeper-than-origin puzzle" in this thread are inconclusive and in several cases contradictory.
    -Bram's definition forgot to include RotoPrism 2 as deeper-than-deep-cut puzzle.
    -I cannot conclusively use any of the current definitions to tell whether or when a single-unbandaged deeper-than-origin puzzle is still a "deeper-than-origin puzzle".
Once we have a proper definition, we can check the result of the classification. And if there are multiple proper definitions, then we can debate which definition is the most elegant.

COMPARE: At one point in time, the IAU had multiple definitions of planet, including these.
    1) A planet is a body that is on the list "Mercury · Venus · Earth · Mars · Jupiter · Saturn · Uranus · Neptune · Pluto".
    2) A planet is a body that orbits the Sun, is massive enough for its own gravity to make it round, and has cleared its neighbourhood of smaller objects around its orbit.
The IAU held a heated debate, and then selected the most elegant definition, namely the latter. Of course there remain major controversies, and likely the definition of planet will be refined in the future. Until then, Pluto is a dwarf planet.

Oskar

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