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TBTTyler

Post subject: Jumbling Puzzles Posted: Tue Oct 07, 2008 10:00 pm 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

This is a continuation of the thread viewtopic.php?t=10854Danny Devitt wrote: The spheres are actually the only jumblable puzzles on GB. While you can't use the scramble feature to scramble them in a nonjumblable way, you can scramble it yourself using only 90˚ turns (and solve the same way) and it will stay nonjumbled. But this also takes all the fun out of it because that turns it into an edgesonly 3x3. I'm not sure that I would qualify the spheres as jumblable. I think that they more resemble bandaged puzzles because their rest state contains incomplete circles. But then that begs the question, what about a jumblable bandaged puzzle. What constitutes the difference?


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the.drizzle

Post subject: Re: Jumbling Puzzles Posted: Tue Oct 07, 2008 10:24 pm 

Joined: Thu Apr 05, 2007 8:05 pm Location: Seednee

This whole discussion clearly illustrates the problem of using jargon as a substitute for plain language How about this: The puzzles based off a rhombic dodecahedron have the rather interesting feature that the slices line up for subsequent moves at increments of both $\pi$ and $\acos\left(\frac{1}{3}\right)$. What makes this interesting is that if subsequent moves are made on the slice that "appears" as a non$\pi$ option, then subsequent twists of the puzzle cause further moves to be blocked. The behaviour is easily seen in the Battle Gear puzzle (thanks Bram for pointing that out), but I still need to do some math to fully explain the move sequences. One of these days... Thus, we see that the spherical puzzles mentioned in the previous puzzles do, in fact, share this property but in a different manner. That is, the non$\pi/2$ slices line up as there are bandaged cuts on the sphere allowing the moves to exist, as opposed to it being a property of the geometry of the underlying mechanism. Which one counts as "jumblable"? Who knows!
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Danny Devitt

Post subject: Re: Jumbling Puzzles Posted: Tue Oct 07, 2008 10:32 pm 

Joined: Tue Mar 25, 2008 2:51 am Location: Malibu, California

the.drizzle wrote: This whole discussion clearly illustrates the problem of using jargon as a substitute for plain language Quote: $\pi$ and $\acos\left(\frac{1}{3}\right)$. This is clearly not jargon lol
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kastellorizo

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 2:19 am 

Joined: Fri Nov 04, 2005 12:31 am Location: Greece, Australia, Thailand, India, Singapore.

TBTTyler wrote: This is a continuation of the thread viewtopic.php?t=10854 Nice one! This topic indeed deserves its own thread. the.drizzle wrote: if subsequent moves are made on the slice that "appears" as a non$\pi$ option, then subsequent twists of the puzzle cause further moves to be blocked. That is a brilliant way to describe it in a general way! Anyone agrees/disagrees? Danny Devitt wrote: the.drizzle wrote: This whole discussion clearly illustrates the problem of using jargon as a substitute for plain language Quote: $\pi$ and $\acos\left(\frac{1}{3}\right)$. This is clearly not jargon lol LaTeX, although misunderstood by many (who mainly use other software) is wonderful to use, especially when it comes to making maths documents. It is no wonder that there are two main choices for IEEE papers, one of them being LaTeX. Pantazis
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 9:27 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

TBTTyler wrote: This is a continuation of the thread viewtopic.php?t=10854Thanks for starting this thread. In hind sight it is something I should have done instead of taking the.drizzle's thread off topic. TBTTyler wrote: I'm not sure that I would qualify the spheres as jumblable. I think that they more resemble bandaged puzzles because their rest state contains incomplete circles. But then that begs the question, what about a jumblable bandaged puzzle. What constitutes the difference? Ahhh... very interesting point. If those "incomplete circles" were completed in the "rest state" then the puzzle wouldn't be jumblable so you are saying this puzzle is just a bandaged version of the nonjumblable puzzle. Very very interesting... Pulling up this quote of yours from the other thread... TBTTyler wrote: I see jumbling moves as those that are allowed when the spherical structure of a puzzle is not at rest.
The spherical structure of a puzzle is like from Jaap's sphere applet (specifically to get shape out of the argument) Rest is when all circles resemble a solved state. (I can't think of any nonbandaged puzzles that deviate from this, but I could easily be wrong)
This of course is a shaky definition, but I think it's on the right track. So this gives me an idea which I'm having a very hard time visualizing. Let's say with take the 24Cube. Where the.drizzle says... the.drizzle wrote: What makes this interesting is that if subsequent moves are made on the slice that "appears" as a non$\pi$ option, then subsequent twists of the puzzle cause further moves to be blocked. This tells me we have "incomplete circles" on the spherical structure (think Jaap's sphere applet version) of the 24Cube when the puzzle is in this what I assume we can call nonrest state. Can we then complete these circles to form a new puzzle? I think so but I'm having a very hard time wrapping my mind around what the puzzle would look like at the moment. If that is true then is it fair to say the 24Cube is a bandaged version of this other puzzle? If so I think we can go back to TBTTyler's question regarding what constitutes the difference between bandaged and jumblable and say there is non. If that makes sense... what would the unbandaged version of the 24Cube look like? And is it possible that it too could have nonrest states where twists are allowed? Carl
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 9:49 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

kastellorizo wrote: the.drizzle wrote: if subsequent moves are made on the slice that "appears" as a non$\pi$ option, then subsequent twists of the puzzle cause further moves to be blocked. That is a brilliant way to describe it in a general way! Anyone agrees/disagrees? I agree it works well for the 24Cube but I'm not sure it serves as a general definition of what jumblable means. the.drizzle wrote: $\pi$ and $\acos\left(\frac{1}{3}\right)$. kastellorizo wrote: LaTeX, although misunderstood by many (who mainly use other software) is wonderful to use, especially when it comes to making maths documents. It is no wonder that there are two main choices for IEEE papers, one of them being LaTeX. Back in the mid 90's I took a brief stab at trying to learn TeX which LaTeX is based on and was real impressed with what it could do but that was a long time ago and I must confess I don't have a clue how to read $\pi$ and $\acos\left(\frac{1}{3}\right)$ now. I'm pretty sure $\pi$ is just pi but is the other just acos(1/3)? Does the forum here support LaTeX and for some reason I'm the only one seeing a bunch of dollar signs, back slashes, etc? Don't get me wrong... I love the output from LaTeX. I'm not not seeing it here. Carl
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Mike G

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 10:41 am 

Joined: Wed Oct 31, 2001 4:19 am Location: Manchester, UK

wwwmwww wrote: I love the output from LaTeX. I'm not not seeing it here. Even as a TeX/LaTeX user, I have to agree that "acos(1/3)" would have been much, much clearer, especially for such a simple expression.


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kastellorizo

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 10:52 am 

Joined: Fri Nov 04, 2005 12:31 am Location: Greece, Australia, Thailand, India, Singapore.

wwwmwww wrote: kastellorizo wrote: the.drizzle wrote: if subsequent moves are made on the slice that "appears" as a non$\pi$ option, then subsequent twists of the puzzle cause further moves to be blocked. That is a brilliant way to describe it in a general way! Anyone agrees/disagrees? I agree it works well for the 24Cube but I'm not sure it serves as a general definition of what jumblable means. It also works for the Bevel Cube (as far as I remember from last year's IPP27). Can anyone verify this? In general, jumblability can be restrictive for future moves. And I am not sure if that is a good or a bad thing...! wwwmwww wrote: Back in the mid 90's I took a brief stab at trying to learn TeX which LaTeX is based on and was real impressed with what it could do but that was a long time ago and I must confess I don't have a clue how to read $\pi$ and $\acos\left(\frac{1}{3}\right)$ now. I'm pretty sure $\pi$ is just pi but is the other just acos(1/3)? Does the forum here support LaTeX and for some reason I'm the only one seeing a bunch of dollar signs, back slashes, etc? Don't get me wrong... I love the output from LaTeX. I'm not not seeing it here.
Carl Everyone here only sees the input jargon. But I am pretty sure me and the.drizzle are not the only ones here who are familiar with this (jumblable?) code. I used LaTeX for both my PhD and Masters Thesis, IEEE papers use it as a standard, while wikipedia also uses it extensively when editing a mathematical document. It is a language which takes some time to get used to, especially when there are so many different class files and templates, but just like you said, the end result is very rewarding. Pantazis
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 11:28 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

kastellorizo wrote: It also works for the Bevel Cube (as far as I remember from last year's IPP27). Can anyone verify this? In general, jumblability can be restrictive for future moves. And I am not sure if that is a good or a bad thing...! I think it may work for all rhombic dodecahedron based puzzles but are only rhombic dodecahedron based puzzles jumblable? And if jumblable and bandaged are the same (still an open question at least in my mind) then jumblability MUST be restrictive for future moves. Here is what the spherical version of the 24Cube looks like in the "rest state": Note: all lines are parts of complete circles. Can someone show me what the "nonrest state" looks like where jumblable moves are possible? If all lines are completed to form full circles at this point what does it look like? Now return it to the "rest state"... are all the 'new' circles still complete? And if so is it possible to see this 'new' puzzle in Jaap's sphere applet? If not... why not? This is all so hard to picture in my head without some pretty pictures. Carl
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TBTTyler

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 1:20 pm 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

I think we need a better definition of rest. The Golden Cube just popped into mind, and the solved state is equivalent to 1 click on a skewb. Also, I would describe a bandaged puzzle as one where the arcs are unable to all be part of complete circles. Perhaps then we can define a rest state as one where all arcs are part of complete circles. (Could a puzzle be constructed with multiple nonsimilar rest states using this definition?)
Carl, instead of using the 24 cube to visualize jumbling, try using the helicopter cube. The smaller circles should make it easier to see.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 2:26 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

TBTTyler wrote: I think we need a better definition of rest. The Golden Cube just popped into mind, and the solved state is equivalent to 1 click on a skewb. I'd agree the Golden Cube isn't in its rest state in the solved position. TBTTyler wrote: Also, I would describe a bandaged puzzle as one where the arcs are unable to all be part of complete circles. So you would say the 24Cube is NOT bandaged? Hmmm... I'm still not sure. A part of me still thinks the 24Cube may be a bandaged version of some nonjumblable puzzle. TBTTyler wrote: Perhaps then we can define a rest state as one where all arcs are part of complete circles. (Could a puzzle be constructed with multiple nonsimilar rest states using this definition?) I don't see how... but I certainly can't prove the answer is "no" at the moment. But I may have a proof that will work for deep cut puzzles like the 24Cube. Say you have a deep cut puzzle. Any move rotates half of the sphere. In any rest state the the arcs on the half not being rotated still have to be complete when the puzzle stops at a "rest state". The only cut that doesn't appear on the half not being rotated is the one the rotation is along and that circle is never broken. Maybe all that proves is if a puzzle does have 2 nonsimilar rest states you can't get from one to the other with a rotation along a single axis and even if that is true this proof only works for puzzles with only deep cuts. TBTTyler wrote: Carl, instead of using the 24 cube to visualize jumbling, try using the helicopter cube. The smaller circles should make it easier to see. Well the Helicopter cube has twice as many circles too and I don't have either a 24Cube or a Helicopter cube at hand so I'm trying this all in my head at the moment. I'm thinking about taking a shot a making a spherical 24Cube in POVRay that I can play with and hopefully make some pictures to help me see what is going on. If there is such a thing as an unbandaged 24cube I'd really like to see what it looks like... I'm still not sure there is such a beast so I'm sure it will be fun trying to make the thing in POVRay. Carl
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Puzzlemaster42

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 2:45 pm 

Joined: Wed Aug 17, 2005 8:26 pm Location: Saratoga Springs, NY


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TBTTyler

Post subject: Re: Jumbling Puzzles Posted: Wed Oct 08, 2008 2:48 pm 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

wwwmwww wrote: Well the Helicopter cube has twice as many circles too and I don't have either a 24Cube or a Helicopter cube at hand so I'm trying this all in my head at the moment. I'm thinking about taking a shot a making a spherical 24Cube in POVRay that I can play with and hopefully make some pictures to help me see what is going on. If there is such a thing as an unbandaged 24cube I'd really like to see what it looks like... I'm still not sure there is such a beast so I'm sure it will be fun trying to make the thing in POVRay. It wasn't the number of circles, but the size that I was interested in. With the 24 cube on Jaap's applet, it becomes difficult to see because the cuts are all so far apart. With the helicopter cube, it's much easier to see. Look at the circle in front and center. Rotate it clockwise and you can easily see that the arc from the upper left circle will connect with the upper right circle. Analogous twists are possible with the deep cut (24Cube) Attachment:
pscr.jpg [ 26.89 KiB  Viewed 13796 times ]
[edit] See in the upper right corner of the individual posts where it says "posted" and there's a little sheet of paper. The sheet of paper contains a direct link


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Thu Oct 09, 2008 5:11 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Ok... I did some tinkering in POVRay last night. Here is a 24Sphere with the top portion turned acos(1/3) degrees. So if the 24Cube is a bandaged puzzle I should be able to unbandage it... correct? So if I take the above (and it's mirror image which is also possible if I had turned the top acos(1/3) degrees in the opposite direction) and complete all the arcs I get this: Looks good. So I just need to do the above for the other 5 planes that bisect the puzzle... correct? Wrong! I'm not done with this plane yet as I can now rotate the top by acos(1/3) degrees again and I get this: And at this point I realize I'm fighting a lossing battle. I'll never be able to complete all the arcs as I can always come back and rotate another acos(1/3) degrees and as that isn't a rational number of degrees the pattern never overlaps and I'll end up with a sphere bisected by an infinate number of planes at the end of the day. So unless we are willing to talk about puzzles with an infinate number of cuts (I'm not) the 24Cube is NOT a bandaged version of some more general puzzle. The end result of all this is then that: jumblable ≠ bandagedSo what about this puzzle that was mentioned earlier: http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/sphere_f5.htmWell that is a bandaged version of this: But that puzzle has a portion that is based off of rhombic dodecahedron symetry so the parent puzzle is jumblable. So do we say the bandaged version is bandaged AND jumblable? And am I now just getting to the point the.drizzle was at in the second post above. Interesting stuff. So are all Rhombic Dodecahedron and Rhombic Triacontahedron based puzzles jumblable? Are ONLY Rhombic Dodecahedron and Rhombic Triacontahedron based puzzles jumblable? Carl
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Last edited by wwwmwww on Thu Oct 09, 2008 7:22 pm, edited 1 time in total.


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TBTTyler

Post subject: Re: Jumbling Puzzles Posted: Thu Oct 09, 2008 5:33 pm 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

Are only Rhombic**hedron puzzles jumblable? No. Counterexample? Polaris Are all? I would venture to say yes as long as the circles are equal and large enough.
I would only call the f5 sphere both jumblable and bandaged if it was possible to perform moves analogous to the jumbling moves on the sphere applet. Otherwise I would say the bandaging prohibits jumbling.
Because you have shown that bandaging != jumbling, I see no reason that a puzzle can have both properties. Bandage a 24 cube, and it's still possible to jumble.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Fri Oct 17, 2008 5:03 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

TBTTyler wrote: Are only Rhombic**hedron puzzles jumblable? No. Counterexample? Polaris Is this the current Polaris thread? http://twistypuzzles.com/forum/viewtopic.php?f=9&t=10863I'm eager to learn more about this one. Can you make a spherical Polaris using Jaap's sphere applet? TBTTyler wrote: Are all? I would venture to say yes as long as the circles are equal and large enough. Agreed. TBTTyler wrote: Because you have shown that bandaging != jumbling, I see no reason that a puzzle can have both properties. Bandage a 24 cube, and it's still possible to jumble. I agree for the most part but I now see a hole in my proof that bandaging != jumbling. That proof is really only valid for the 24Cube. I have an idea for a counter example that I'll post soon. Carl
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Fri Oct 17, 2008 5:39 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Ok, let's take a look at the 24Cube + the 2x2x2 or what I'll call the 48Cube (unless there is a better name out there). I'll make the puzzle a sphere so its easier to see what is going on. And as with the 24Cube this puzzle is jumblable with turns of acos(1/3) degrees along the 24Cube planes. Those I won't look at too closely. The states I'm interested in are those seen when the puzzles is turned by 45 degrees along one of the 2x2x2 planes: and when it is turned by 90 degrees along one of the 24Cube planes: Are the turns allowed here jumblable? I suspect these might be states that could be unbandaged as its easy to see these turns are half the standard nonjumblable turns so if we unbandaged the puzzle with respect to any one given plane its easy to see the pattern repeats and we don't end up needing to bisected the puzzle with an infinate number of planes. However this is the case when we just look at one plane at a time. Let's try unbandaging just the 2x2x2 planes. If we do that we get this puzzle: And you note that if I rotate the top of this by 45 degrees there will still be broken arcs. So that fails. Let's try unbandaging just the 24Cube planes for the 90 degree turns. If we do that we get this puzzle: And it just so happens the cuts we got when trying to unbandage the 2x2x2 planes are a subset of these. So we can look at this as an attempt to unbandage the 2x2x2 planes to 45 degree turns AND to unbandage the 24Cube planes to the 90 degree turns. Both fail. And I'm sure I've created so many new jumblable moves in this process that even if it had worked I'd really have a hard time seeing it as a success. A few questions... (1) If I repeated the above process a finite number of times is it possible to unbandage this puzzle to 45 degree turns along the 2x2x2 planes and the 90 degree turns along the 24Cube planes? I suspect no but I'm not sure I understand why not. (2) Is there any polyhedron that would be symetric (i.e. not change shape) after these 2 types of turns? I tried to create one and failed and I suspect the reason that I failed is similiar to the reason the above attempt at unbandaging these two types of turns failed. Carl
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Fri Oct 17, 2008 6:07 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Ok... let's look at a much simpler puzzle. And look at this statement again... TBTTyler wrote: Also, I would describe a bandaged puzzle as one where the arcs are unable to all be part of complete circles. The above puzzle is cut by 3 planes and all arcs are complete circles. So does that mean this is an unbandaged puzzle? Let's turn the top 90 degrees clockwise. We get this: You now see you can turn the front of this puzzle but other arcs are blocked so the puzzle is not in a rest state. Does that make this a jumblable puzzle? I'd say no. I'd called this a bandaged puzzle. Can you see why? So how do our definitions of rest state, bandaged, and jumblable look now? Could it be said that for this puzzle jumblable = bandaged? Carl
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TBTTyler

Post subject: Re: Jumbling Puzzles Posted: Fri Oct 17, 2008 9:29 pm 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

I can see why you would say your second example is bandaged and not jumblable. But using a similar (though not exactly the same) argument You could contend that a 3x3 is just a bandaged 5x5. Though it does help to think of it like that when solving. So, here's a question that your example poses. Can we call removing an entire slice "bandaging"?


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Bram

Post subject: Re: Jumbling Puzzles Posted: Fri Oct 17, 2008 10:14 pm 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

Carl's graphical explanation between the difference between bandaging and jumbling is quite good. Bandaging is when there's a finite extension which is a clean group, jumbling is when there isn't. The simplest proof that a particular motion is jumbling is to show that the angle of rotation is irrational, although surprisingly enough even though practically all trigonometric functions on rationals are irrational an most specific ones are conjecture to be irrational, proving so is in many cases an open problem, except in a few exceptional cases when the continued fraction representation is known or something similar to that.
Tyler, a 5x5x5 can be bandaged to form a 3x3x3, but that doesn't mean that a 3x3x3 is a bandage puzzle  its permutations are a completely closed group, hence its not bandaged.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sun Oct 19, 2008 1:43 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Bram wrote: The simplest proof that a particular motion is jumbling is to show that the angle of rotation is irrational Yes, but I think I proved above that the 48Cube has two types of turns with rational angles of rotation (45 degrees and 90 degrees) that are jumbling. So if that is the simplest proof it doesn't appear to be the only proof. What is the angle of rotation of the jumbling turns on a Polaris? Carl
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TBTTyler

Post subject: Re: Jumbling Puzzles Posted: Mon Oct 20, 2008 3:30 pm 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

Here's a picture of the jumbling circles outlined. The polaris is a bit deeper cut, but this is a lot easier to see.
I'm not sure the angle. I'll try and figure it out.
Attachments: 
jumbler.png [ 34.04 KiB  Viewed 13809 times ]



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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Thu Jul 23, 2009 6:41 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Bram wrote: Carl's graphical explanation between the difference between bandaging and jumbling is quite good. Bandaging is when there's a finite extension which is a clean group, jumbling is when there isn't. Hello Bram, Just noticed you were here too. Anyways... this is not meant to be a bump. This is a continuation of the discussion that is going on here: http://twistypuzzles.com/forum/viewtopic.php?f=15&t=14189And is more relevant to this topic. Looks like my "graphical explanation" wasn't quite good enough as it seems to have failed me. Your Mixup Cube can't be unbandaged with a finite number of cuts yet it isn't jumbleable.... I see that now as only turns are allowed when the puzzle is in its "rest state". Can you give me a definition of "clean group" or "closed group" as I'm still not sure we have a nice definition of what a jumbleable move is in the general case? Or maybe we do and I just don't understand it yet. Thanks, Carl
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Bram

Post subject: Re: Jumbling Puzzles Posted: Thu Jul 23, 2009 7:12 pm 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

Carl, I think I see where you're getting confused now (actually, you explained it, but maybe didn't quite realize all the details). The Mixup Cube has two types of moves: Rotate a face 90 degrees, and rotate a center band 45 degrees. It sort of looks like there's a rotate a face 45 degrees move, because if you move two opposite faces that way you can continue to move the puzzle, but that's very misleading  what you've really done is rotate the center band 45 degrees one way and then the puzzle as a whole 45 degrees the other way.
'Closed group' is a mathematical term  the 'closed' means that none of the group operations result in a state outside the group.
Uncanny Cube and Meteor Madness are both bandage puzzles although at first blush it isn't obvious that they are. That's because they've been 'unbandaged' to a plateau of maneuverability where an unusually high proportion of moves are allowed. There are still some moves which are blocked though, and if slices start getting added to allow those moves then the puzzle quickly becomes a thing which tends to get into a total dead end, which is generally how jumble puzzle behave. The helicopter cube/24 cube family is a bit of an exception, because it has moves which are two jumble moves and then a 180 degree rotation of a now symmetric but somewhat irregularly shaped slice, and without those moves there's a very limited set of states it can get into, so they're very important.


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Allagem

Post subject: Re: Jumbling Puzzles Posted: Fri Jul 24, 2009 4:47 pm 

Joined: Sun Oct 08, 2006 1:47 pm Location: Houston/San Antonio, Texas

Bram's got it. Or certainly at least he's on the right track.
How many moves are available on a 2x2x2? Some will say there 18 different moves you can make: 90, 180, or 270 rotations about any of the 6 faces. Others will (more accurately?) say there are only 9 different types of moves noting that opposite rotations of opposite faces actually accomplish the same thing. It depends on how you look at it.
Anyway my point is that the mixup cube has the same problem. On the inside, the actual mechanism is merely a 2x2x2 with the mosaikalike tiles sliding around. Now USUALLY, 3layer puzzles have stable, nonmoving centers and cores. So we determine all mathematics from that fixed reference
Carl is trying to impose a mathematically stable pattern onto the mixup cube. Bram is defending the actual mechanism within the mixup cube  which of course doesn't follow Carl's pattern because the centers  the fixed point of reference upon which Carl's whole system is based  DO move.
So really, you are both correct. The Mixup Cube as is, in my opinion at least, can't be classified as jumbleable or not jumbleable (or bandaged or not) because it introduces the sliding of center pieces not around a fixed rotating single piece.
BUT, if the Mixup Cube was made in a much more complicated way, then Carl's pattern does apply and the Mixup cube, made that way, IS a jumbleable puzzle, and one that is bandaged in such a way that it restricts movement to only a certain subgroup of the peices, resulting in the mixup cube.
So this whole thing depends on how you look at it.
Peace, Matt Galla


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Fri Jul 24, 2009 5:34 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Bram wrote: Carl, I think I see where you're getting confused now (actually, you explained it, but maybe didn't quite realize all the details). The Mixup Cube has two types of moves: Rotate a face 90 degrees, and rotate a center band 45 degrees. It sort of looks like there's a rotate a face 45 degrees move, because if you move two opposite faces that way you can continue to move the puzzle, but that's very misleading  what you've really done is rotate the center band 45 degrees one way and then the puzzle as a whole 45 degrees the other way. Agreed... Bram wrote: 'Closed group' is a mathematical term  the 'closed' means that none of the group operations result in a state outside the group. Let's use the 24Cube as an example. A state in its group would be any state when the cube is in its "rest state". Correct? However does a group's operations have to be defined as a rotation about a single axis? If so I see how the 24Cube isn't a closed group. However could an operation be created that takes the 24Cube through a jumbleable move and back to a "rest state" through several rotation? If yes... the next question is (and its hard for me to answer this question without a 24Cube I can play with) does the 24Cube have a finite or an infinite number of such operations? If infinite... then I again see how its not a closed group. However if it is finite then wouldn't the 24Cube be considered a closed group despite being jumbleable. This is what I mean by the "rest state" of the 24Cube: http://www.jaapsch.net/puzzles/sphere.htm?blue=0&sym=4&angle=330,105,355Its any state where this is how the cut planes are positioned. I'm just questioning the statement that a jumbleable puzzle is one that cannot be represented by a closed group if that is the definition we should be using... Bram wrote: Uncanny Cube and Meteor Madness are both bandage puzzles although at first blush it isn't obvious that they are. Do you consider your Mixup Cube bandaged? If not how would you refer to the cut planes that don't propagate into the face layers? Bram wrote: That's because they've been 'unbandaged' to a plateau of maneuverability where an unusually high proportion of moves are allowed. There are still some moves which are blocked though, and if slices start getting added to allow those moves then the puzzle quickly becomes a thing which tends to get into a total dead end, which is generally how jumble puzzle behave. Interesting... I just watched those two videos. The Uncanny Cube appears to allow turns when the top and bottom triangle face centers are out of align so you are making turns from states other then the rest state. Do you consider this puzzle jumbleable? I'm not as sure with the Meteor Madness puzzle but looking at the shape of the face centers it looks like only 360 degree turns would return the puzzle to its rest state so I assume this one would be considered jumbleable too? Bram wrote: The helicopter cube/24 cube family is a bit of an exception, because it has moves which are two jumble moves and then a 180 degree rotation of a now symmetric but somewhat irregularly shaped slice, and without those moves there's a very limited set of states it can get into, so they're very important. Hmmm... not sure if you are saying something about the nature of jumbleable puzzles here or just making an observation about the family of puzzles with rhombic symmetry. Let me test my understanding... Is it fair to say your Mixup Cube is a bandaged version of this puzzle? In a similiar fasion we can bandage this related puzzle: To make this puzzle: http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/sphere_f5.htmCorrect? This puzzle, sphere_f5, unlike yours DOES allow rotation when the puzzle isn't in its rest state. For example rotate one of the faces by 45 degress and observe the state of the puzzle. It is NOT in its rest state. And by playing with that applet I can easy see how many... many... turns can be made without the puzzle ever returning to its rest state. This would appear to NOT be a closed group as there are probably an infinite number of operations. Or if you define operations as single turns then several such operations that result in a state outside the group. As such this puzzle is jumbleable.... correct? How all this applies to the Uncanny Cube and the Meteor Madness puzzle I'm still not sure. Do you consider these closed groups? If so why? Carl P.S. I still need to dig out my dino cube and look at those 60 degree rotations you mentioned.
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TBTTyler

Post subject: Re: Jumbling Puzzles Posted: Fri Jul 24, 2009 5:57 pm 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

wwwmwww wrote: I'm just questioning the statement that a jumbleable puzzle is one that cannot be represented by a closed group if that is the definition we should be using... If I understand correctly: If we define the basic operation as a 180 degree twist around an axis, then it is a closed group. However, if we define the basic operation as either a 180 degree twist or an acos(1/3) twist, then those twists would have to be possible after every single basic operation. As this would continue to infinity, I would say it's a jumblable puzzle. Perhaps there's another layer of complexity which we can add? A jumbling depth. If we cut off the jumbling (acos(1/3)) moves after x distance from rest (or perhaps x "completions of broken arcs"), we still form a closed group. However, x COULD continue to infinity so I would still call this a jumbling move.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Fri Jul 24, 2009 6:06 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Allagem wrote: So this whole thing depends on how you look at it. I don't think its that complicated. I think its just a problem of multiple definitions being used that don't necessarily all agree with each other. A puzzle should either be jumbleable or not and how it is made shouldn't make a difference. I wish I knew more group theory then I do but a Mixup Cube is defined by the same set of operations regardless of how it is made. And if there are operations that result in state that is outside the group those operations relate to jumbleable moves on the puzzle. In the case of the Mixup Cube any state that allows rotations to be made has the exact same geomety of cuts regardless of the puzzles outward appearance so there are no jumbleable moves that take it to a different state. With puzzles like Uncanny Cube and Meteor Madness I'm not entirely sure if the geomety of cuts alone defines the states that are "inside the group" or not. And I'm also not 100% sure of the definition of bandaged. In my mind any puzzle that has cut planes that don't completely penetrate the puzzle in a state where some rotations are allowed is bandaged. Is this a fair definition? Till I came across the Mixup Cube I had made the incorrect assumption that any bandaged puzzle could be unbandaged with a finite number of cuts if it wasn't also jumbleable. The Mixup Cube appears to be a bandaged puzzle that can't be unbandaged and isn't jumbleable. I still feel ALL jumbleable puzzles are also bandaged. For example, look at the 24Cube. In its rest state all the cuts completely penetrate the puzzle however there are other states that DO allow rotation and in these states there are cuts that DON'T completely penetrate the puzzle. It is those cuts that are bandaged. Having said that... is it possible for a puzzle to have a state outside its rest state where all the cuts are unbandaged? I can't think of an example but I suspose it may be possible to have a jumbleable puzzles that isn't also bandaged. Carl
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Fri Jul 24, 2009 6:32 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

TBTTyler wrote: If I understand correctly: If we define the basic operation as a 180 degree twist around an axis, then it is a closed group. If we are talking about the 24Cube, yes the 180 degree twists form a closed group. However this group doesn't repesent all the moves that are possible on a 24Cube. TBTTyler wrote: However, if we define the basic operation as either a 180 degree twist or an acos(1/3) twist, then those twists would have to be possible after every single basic operation. As this would continue to infinity, I would say it's a jumblable puzzle.
Perhaps there's another layer of complexity which we can add? A jumbling depth. If we cut off the jumbling (acos(1/3)) moves after x distance from rest (or perhaps x "completions of broken arcs"), we still form a closed group. However, x COULD continue to infinity so I would still call this a jumbling move. Not having a 24Cube in hand I'm a little lost. How many acos(1/3) twists can be made in a row without returning to the "rest state" or repeating yourself? I think you are saying its infinite. Correct? I just can't picture a series of those turns in my head. Carl
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TBTTyler

Post subject: Re: Jumbling Puzzles Posted: Fri Jul 24, 2009 7:10 pm 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

wwwmwww wrote: Not having a 24Cube in hand I'm a little lost. How many acos(1/3) twists can be made in a row without returning to the "rest state" or repeating yourself? I think you are saying its infinite. Correct? I just can't picture a series of those turns in my head. Because acos(1/3) is irrational, then without having to picture anything I can say "infinite". You showed it yourself HERE. After each defined twist *we resolve all incomplete cuts before we make another twist*. If this goes on forever, it's jumblable. If it repeats a finite number of times, we have bandaging. If we never have to resolve any incomplete cuts, we've got a base puzzle Side question: So would a "rest state" just be any state in the full closed group of the puzzle (the largest closed group that is not part of a subgroup)? I only ask because we've taken to putting "quotes" around it every time we say it.


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Bram

Post subject: Re: Jumbling Puzzles Posted: Fri Jul 24, 2009 8:56 pm 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

wwwmwww wrote: Bram wrote: 'Closed group' is a mathematical term  the 'closed' means that none of the group operations result in a state outside the group. Let's use the 24Cube as an example. A state in its group would be any state when the cube is in its "rest state". Correct? However does a group's operations have to be defined as a rotation about a single axis? If so I see how the 24Cube isn't a closed group. However could an operation be created that takes the 24Cube through a jumbleable move and back to a "rest state" through several rotation? If yes... the next question is (and its hard for me to answer this question without a 24Cube I can play with) does the 24Cube have a finite or an infinite number of such operations? If infinite... then I again see how its not a closed group. However if it is finite then wouldn't the 24Cube be considered a closed group despite being jumbleable. If you even once find that a natural operation is blocked, then a puzzle is at a minimum bandaged, and possibly jumbleable. All jumble puzzles are bandaged. wwwmwww wrote: Do you consider your Mixup Cube bandaged? If not how would you refer to the cut planes that don't propagate into the face layers?
They... uh... don't exist. The Mixup Cube doesn't have any cut planes which the regular Rubik's Cube doesn't have, it's just that it has more positions that the center band motions (not the face motions) can stop in. Maybe the source of confusion has to do with what's a 'natural' move. You seem to be fixating on the fact that from the point of view of individual pieces in the Mixup Cube it looks like there are extra cut planes which could be added, but the puzzle doesn't have to feel guilty for not having those. For example, consider the Sphere Xyz. It has only a single slice, and even though the pieces suggest a LOT of other slices it could have, it still clearly isn't a jumble or even bandage puzzle. wwwmwww wrote: Bram wrote: That's because they've been 'unbandaged' to a plateau of maneuverability where an unusually high proportion of moves are allowed. There are still some moves which are blocked though, and if slices start getting added to allow those moves then the puzzle quickly becomes a thing which tends to get into a total dead end, which is generally how jumble puzzle behave. Interesting... I just watched those two videos. The Uncanny Cube appears to allow turns when the top and bottom triangle face centers are out of align so you are making turns from states other then the rest state. Do you consider this puzzle jumbleable? I'm not as sure with the Meteor Madness puzzle but looking at the shape of the face centers it looks like only 360 degree turns would return the puzzle to its rest state so I assume this one would be considered jumbleable too? Yes, the Uncanny Cube is jumbleable. The easiest way to see this is to note that it feels bandaged in places, and think through what would happen if you try to 'unbandage' it. wwwmwww wrote: Bram wrote: The helicopter cube/24 cube family is a bit of an exception, because it has moves which are two jumble moves and then a 180 degree rotation of a now symmetric but somewhat irregularly shaped slice, and without those moves there's a very limited set of states it can get into, so they're very important. Hmmm... not sure if you are saying something about the nature of jumbleable puzzles here or just making an observation about the family of puzzles with rhombic symmetry. I was making a comment about jumble puzzles's tendency to wind up in a lot of dead ends, nothing to do with the definition of jumbling. Sorry for the confusion.


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Allagem

Post subject: Re: Jumbling Puzzles Posted: Sat Jul 25, 2009 2:34 am 

Joined: Sun Oct 08, 2006 1:47 pm Location: Houston/San Antonio, Texas

Ok..... if you are insisting that these types of definitions should remain unchanged no matter how a puzzle is built (a fair requirement btw), then the mixup cube is a bandaged version of a jumbleable puzzle, just as Carl wrote. This is NOT how it was made, it's just how the logic explains it. I personally think there is a certain setup of jumbleable puzzles that can be considered nonbandaged. I'll take a shot at a definition. A nonbandaged, jumbleable puzzle is a jumbleable puzzle in which all moves are possible from the rest state, and any two adjacent pieces can be seperated in one move from rest state (in other words all cuts between any two pieces are accesible from rest state). Examples of Jumbleable, non bandaged Helicopter Cube Rua Toru Little Chop (24Cube) Jumble Prism (I know I'm disagreeing with the given definition in the YouTube vid, but I believe that is wrong)What's weird about jumbleable puzzles is that even these "non bandaged" ones can have jumbling moves that block other moves  but I don't think bandaged puzzles are uniquely defined by the fact that some moves are blocked. I think bandaged puzzles are those where at least two mathematically diferent pieces on the puzzle are fused together. What you CAN do on a jumbleable puzzle is split the pieces in the spots where these "natural" move blockages occur and thereby generating new pieces. As everyone here knows this process can be repeated and will continue to generate more pieces, but will never end. However, you could for example, "split" the "natural" bandages once and then stop, or twice and then stop. Examples of jumbleable puzzles where the natural piece formations have been split Meteor Madness Uncanny Cube Fairly Twisted (this one's complicated, don't get me started..... but the red face at least has a split piece as well as all around the tiny square yellow face)FINALLY, just as with other puzzles you can bandage jumbleable puzzles and cause many brilliant minds to argue about what is what on a puzzling forum Since Jumbleable puzzles already have so many different possible moves, there is more freedom to bandage pieces in a way that somewhat change the symmetry/function of the puzzle. This is very uncommon as of now, and the only example I can think of (other than the mixup cube) is the sphere on gelatinbrain's applet: http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/sphere_f5.htm as Carl also pointed out. Now even though the mechanics of the mixup cube is much, much, simpler than the mech needed to demonstrate this point, the mixup cube IS a bandaged, jumbleable puzzle, which again Carl provided. The gelatinbrain applet sphere's have an interesting property though (Well two actually) 1. (less important) the shallow cut spheres obviously only have 6 faces that can rotate (the other 12 have been bandaged out of existence), but the deepcut spheres are actually not fully functional. The 100% mathematically natural state of that puzzle would allow rhombic dodecahedron based twists as well. 2.(more important) Let me start by saying that the fact that the sphere's can always rotate is a planned coincidence. Now let me modify the word coincidence by explaining why it works. The sphere's nonjumbling moves are 90 degree turns about the "cubic" faces (there would also be 180 degree turns about the "rhombic dodecahedron" faces, were these moves allowed). Play with one of Gelatinbrain's sphere puzzles for awhile and you'll notice the moves are usually 90 degrees, following the same basic symmetries as a 2x2x2 cube. A nonjumbling move on a jumble puzzle is always possible provided all affected slices of the move (planes between the moving pieces) were clear on the previous move < a rather obvious, but necessary comment to make Since all the faces of a rhombic dodecahedron are offset from a cube's faces by a 45 dgree rotation about a face, gelatinbrain's sphere can take advantage of thatby allowing 45 degree twists in several states. This virtually swaps a potential set of 4 2x2x2 with another set created by combing half of each piece with the half of the adjacent pieces (swapping the cubical symmetry with a subset of the rhombic dodecahedron symmetry), but only when such a rotation is available (this can be viewed as the only jumbling move available). So by the previous two statements it is guaranteed that the puzzle can always behave as a 2x2x2. So in response to Carl's thoughts, it is correct that the gelatinbrain sphere can move infinitely whilst not in rest state, but that's only because the applet will force you to make the 90 degree turn (the nonjumbling move) when the 45 degree turn (the jumbling move) is unavailable. The 90 degree turn is always available as I said above because the cubical and only available subset of the rhombic dodecahedron share the same cubical symmetries. Remember, the explanations are found after someone finds something that needs explaining, not the other way around. I know I give long explanations that probably aren't as clear as they could be but I'm fairly sure that I've finally figured this whole jumbleable thing out. Let me know your thoughts guys! Peace, Matt Galla PS If I didn't make it clear, some jumbleable puzzles have symmetries that allow for nonjumbling moves. This included practically every jumbleable puzzle we knew about before Bram came along The JumblePrism and Meteor Madness puzzles have 0 non jumbling moves, while the Uncanny Cube and Fairly Twisted each have some faces that are always jumbling and others that have at least one nonjumbling move.
Attachments: 
File comment: If this helps at all the core shape for both the gelatinbrain spheres and the nonexistant "mathematically correct" Mixup Cube would be this shape:
45Degree Jumbleable Core.jpg [ 12.02 KiB  Viewed 13080 times ]



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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sat Jul 25, 2009 9:18 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

TBTTyler wrote: Because acos(1/3) is irrational, then without having to picture anything I can say "infinite". You showed it yourself HERE. After each defined twist *we resolve all incomplete cuts before we make another twist*. If this goes on forever, it's jumblable. If it repeats a finite number of times, we have bandaging. If we never have to resolve any incomplete cuts, we've got a base puzzle. Yes, but on the actual 24Cube after you have made one acos(1/3) turn many of the other turns available to you are bandaged and can't be resolved on the physical object in your hands. Mathematically yes they can be but then you are opening up more states that aren't available on the actual 24Cube. I'm just curious, if I picked up a 24Cube and started to jumble it are there a finite number of twists I could make before I either started repeating myself or the only option available was to return to the rest state. This may seem obvious after you've handled one but its a different question then how many cuts are needed to completely unbandage a 24cube which is what I showed above. Maybe the two questions are linked in a way I don't see at the moment. TBTTyler wrote: Side question: So would a "rest state" just be any state in the full closed group of the puzzle (the largest closed group that is not part of a subgroup)? I only ask because we've taken to putting "quotes" around it every time we say it. I think I started that and I was just doing that as I was using the term without putting out a solid general definition. If I understand you correctly, yes. In the case of a 24Cube the rest state is any state where all the cuts line up like this: http://www.jaapsch.net/puzzles/sphere.htm?blue=0&sym=4&angle=330,105,355Is the 24Cube ALWAYS a cube in this state? I think so... but I'm not 100% sure. Again this may be obvious to someone that has actually played with one. Taking a more simple case... A Rubik's cube is ALWAYS in a rest state when its in the form of a cube. If you rotated the top layer 45 degrees you could argue you still have rotations available to you in that position. For example you are free to rotate the bottom face and you can continue to rotate the top face but I don't consider this a rest state. How to say this is different then the case of making an acos(1/3) turn on a 24Cube in proper group theory terms I'm not sure. Is the term rest state defined in group theory? Carl
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sat Jul 25, 2009 9:52 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Bram wrote: They... uh... don't exist. The Mixup Cube doesn't have any cut planes which the regular Rubik's Cube doesn't have, it's just that it has more positions that the center band motions (not the face motions) can stop in. I think I agree with Allagem. Physically those cut planes aren't obvious on the innards of your puzzle due to the way it is made. I think I'd argue mathematically they are there. Take this image and bandage it and I have a Mixup Cube: I agree that is not how you made yours but it is the same puzzle... the same mathematical group. If someone DID make a Mixup Cube in this fashion should they call their's a bandaged Mixup Cube and yours a nonbandaged Mixup Cube when they have the exact same functionality? Bram wrote: Maybe the source of confusion has to do with what's a 'natural' move. You seem to be fixating on the fact that from the point of view of individual pieces in the Mixup Cube it looks like there are extra cut planes which could be added, but the puzzle doesn't have to feel guilty for not having those. For example, consider the Sphere Xyz. It has only a single slice, and even though the pieces suggest a LOT of other slices it could have, it still clearly isn't a jumble or even bandage puzzle. I'm NOT trying to make the puzzle or its creator feel guilty about anything. I LOVE the Mixup Cube and think its perfect just as is. It's the "term" bandaged in this case that should feel guilty. I'm using it one way, you are using it another, and I see Allagem is using it in yet a third way. The part of Allagem definition that I agree with is this: Quote: any two adjacent pieces can be seperated in one move from rest state (in other words all cuts between any two pieces are accesible from rest state). This is NOT true for the Sphere Xyz puzzle so I would say its bandaged. Where Allagem and I differ is in regards to jumbleable puzzles. I agree a 24Cube is NOT bandaged when its in its rest state but I would say it IS bandaged after an acos(1/3) turn. As such I would call it a bandaged puzzle. I believe its fair to sort jumbleable puzzles the way he has but I'd add the qualifier unbandaged in the rest state. Bram wrote: Yes, the Uncanny Cube is jumbleable. The easiest way to see this is to note that it feels bandaged in places, and think through what would happen if you try to 'unbandage' it. Do you agree Meteor Madness is jumbleable too? Carl
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TBTTyler

Post subject: Re: Jumbling Puzzles Posted: Sat Jul 25, 2009 10:41 am 

Joined: Thu Jan 06, 2005 8:53 pm Location: Los Angeles

This was the question I was answering: wwwmwww wrote: I'm just questioning the statement that a jumbleable puzzle is one that cannot be represented by a closed group if that is the definition we should be using... So then using the cut resolutions, the definition of jumbling should hold. wwwmwww wrote: I'm just curious, if I picked up a 24Cube and started to jumble it are there a finite number of twists I could make before I either started repeating myself or the only option available was to return to the rest state. Since we're dealing with a finite number of pieces I don't see how we could have an infinite number of states. So I would venture to say that, yes, you would eventually repeat yourself or have to return to rest. Are you asking: How many jumbling moves are possible without repeating a state? What is the most jumbled position on a 24 cube? That I don't know, and I can see where you're stuck.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sat Jul 25, 2009 12:15 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Allagem wrote: Ok..... if you are insisting that these types of definitions should remain unchanged no matter how a puzzle is built (a fair requirement btw), then the mixup cube is a bandaged version of a jumbleable puzzle, just as Carl wrote. This is NOT how it was made, it's just how the logic explains it. Agreed... I'm just after a set of definitions for terms like jumbleable, badgaged, rest state, etc. that I think should be independant of the actual mechanics of the puzzle. Allagem wrote: I personally think there is a certain setup of jumbleable puzzles that can be considered nonbandaged. I'll take a shot at a definition. A nonbandaged, jumbleable puzzle is a jumbleable puzzle in which all moves are possible from the rest state, and any two adjacent pieces can be seperated in one move from rest state (in other words all cuts between any two pieces are accesible from rest state). The bit in ()'s required. If one were to just define a nonbandaged puzzle as one where any two adjacent pieces can be seperated in one move from the rest state then one would called the Mixup Cube nondandaged. The bit about ALL cuts between any two pieces be accesible is critical. Allagem wrote: Examples of Jumbleable, non bandaged Helicopter Cube Rua Toru Little Chop (24Cube) Jumble Prism (I know I'm disagreeing with the given definition in the YouTube vid, but I believe that is wrong) I haven't got my mind around the Jumble Prism yet but I more or less agree here. I'd just categorize these as jumbleable but nonbandaged in their rest state. Allagem wrote: What's weird about jumbleable puzzles is that even these "non bandaged" ones can have jumbling moves that block other moves  but I don't think bandaged puzzles are uniquely defined by the fact that some moves are blocked. I think bandaged puzzles are those where at least two two mathematically diferent on the puzzle are fused together. If one had this puzzle... You would be fusing mathematically different pieces together to get a 24Cube. Allagem wrote: What you CAN do on a jumbleable puzzle is split the pieces in the spots where these "natural" move blockages occur and thereby generating new pieces. As everyone here knows this process can be repeated and will continue to generate more pieces, but will never end. However, you could for example, "split" the "natural" bandages once and then stop, or twice and then stop. Agreed... but do this process once and you'll see how the 24Cube is a bandaged version of that puzzle. Allagem wrote: Examples of jumbleable puzzles where the natural piece formations have been split Meteor Madness Uncanny Cube Fairly Twisted (this one's complicated, don't get me started..... but the red face at least has a split piece as well as all around the tiny square yellow face) Interesting... can you unsplit these puzzles to get their root jumbleable puzzles? I'd love to see what they looked like. Regarding the Fairly Twisted puzzle... what is meant by lefthanded and righthanded faced? Can the faces just be turned in one direction? (Just rewatched the video... nope that's not it.) Allagem wrote: FINALLY, just as with other puzzles you can bandage jumbleable puzzles and cause many brilliant minds to argue about what is what on a puzzling forum Since Jumbleable puzzles already have so many different possible moves, there is more freedom to bandage pieces in a way that somewhat change the symmetry/function of the puzzle. This is very uncommon as of now, and the only example I can think of (other than the mixup cube) is the sphere on gelatinbrain's applet: http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/sphere_f5.htm as Carl also pointed out. Now even though the mechanics of the mixup cube is much, much, simpler than the mech needed to demonstrate this point, the mixup cube IS a bandaged, jumbleable puzzle, which again Carl provided. I'd say the Mixup Cube can be made by bandaging a jumbleable puzzle, but I wouldn't call the Mixup Cube itself jumbleable. I guess you could say its been bandaged to disallow the jumbleable moves. Allagem wrote: The gelatinbrain applet sphere's have an interesting property though (Well two actually) 1. (less important) the shallow cut spheres obviously only have 6 faces that can rotate (the other 12 have been bandaged out of existence), but the deepcut spheres are actually not fully functional. The 100% mathematically natural state of that puzzle would allow rhombic dodecahedron based twists as well. I wouldn't call the cuts spheres. They form circles on the sphere but are best thought of as planes. And in this case I believe all the cuts are at the same depth. You are calling the cubic cuts shallow and the rhombic dodecahedron based cuts deep but I don't think that is the case. Allagem wrote: 2.(more important) Let me start by saying that the fact that the sphere's can always rotate is a planned coincidence. Now let me modify the word coincidence by explaining why it works. The sphere's nonjumbling moves are 90 degree turns about the "cubic" faces (there would also be 180 degree turns about the "rhombic dodecahedron" faces, were these moves allowed). Play with one of Gelatinbrain's sphere puzzles for awhile and you'll notice the moves are usually 90 degrees, following the same basic symmetries as a 2x2x2 cube. A nonjumbling move on a jumble puzzle is always possible provided all affected slices of the move (planes between the moving pieces) were clear on the previous move < a rather obvious, but necessary comment to make Since all the faces of a rhombic dodecahedron are offset from a cube's faces by a 45 dgree rotation about a face, gelatinbrain's sphere can take advantage of thatby allowing 45 degree twists in several states. This virtually swaps a potential set of 4 2x2x2 with another set created by combing half of each piece with the half of the adjacent pieces (swapping the cubical symmetry with a subset of the rhombic dodecahedron symmetry), but only when such a rotation is available (this can be viewed as the only jumbling move available). So by the previous two statements it is guaranteed that the puzzle can always behave as a 2x2x2. So in response to Carl's thoughts, it is correct that the gelatinbrain sphere can move infinitely whilst not in rest state, but that's only because the applet will force you to make the 90 degree turn (the nonjumbling move) when the 45 degree turn (the jumbling move) is unavailable. The 90 degree turn is always available as I said above because the cubical and only available subset of the rhombic dodecahedron share the same cubical symmetries. I think I followed all that. Allagem wrote: Remember, the explanations are found after someone finds something that needs explaining, not the other way around. Is that your way of saying that despite what set of definitions we might agree to now Bram will just go off and make a puzzle that breaks or bends a few of them to the point we are back here debating this again? I'd take that as a good thing. Allagem wrote: PS If I didn't make it clear, some jumbleable puzzles have symmetries that allow for nonjumbling moves. This included practically every jumbleable puzzle we knew about before Bram came along The JumblePrism and Meteor Madness puzzles have 0 non jumbling moves, while the Uncanny Cube and Fairly Twisted each have some faces that are always jumbling and others that have at least one nonjumbling move. I think I agree... I just wish there was an affordable way I could get one of each of these into my hands so I'd have a better understanding of what is actually going on with a few of them. For example, how does one approach the Jumble Prism with group theory? Does that puzzle have a rest state? What are its operations? I'm guessing it just can't be treated as a closed group. How do you deal with open groups in group theory? Carl
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Last edited by wwwmwww on Sat Jul 25, 2009 2:18 pm, edited 2 times in total.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sat Jul 25, 2009 1:07 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

TBTTyler wrote: This was the question I was answering: wwwmwww wrote: I'm just questioning the statement that a jumbleable puzzle is one that cannot be represented by a closed group if that is the definition we should be using... So then using the cut resolutions, the definition of jumbling should hold. wwwmwww wrote: I'm just curious, if I picked up a 24Cube and started to jumble it are there a finite number of twists I could make before I either started repeating myself or the only option available was to return to the rest state. Since we're dealing with a finite number of pieces I don't see how we could have an infinite number of states. So I would venture to say that, yes, you would eventually repeat yourself or have to return to rest. Are you asking: How many jumbling moves are possible without repeating a state? What is the most jumbled position on a 24 cube? That I don't know, and I can see where you're stuck. I agree the 24Cube is jumbleable. What I'm less sure of is the statement that it cannot be represented by a closed group. If we allowed the 180 degree twists as basic operations and didn't allow the acos(1/3) twists could we get away with forming a list of all possible combinations of moves that could be perfomed before returning to a rest state after an acos(1/3) twist was perfomed and call these a list of basic operations? I can see what could easily happen here. You could end up with a list of operations that would take the puzzle from any state to any other state in a single operation. I'd consider this akin to solving a Rubik's cube by taking it apart and putting it back together again. While being a closed group it wouldn't really be intereating or of much help in actually solving a 24Cube. What got me thinking along these lines is that I saw this video of someone solving a Bevel Cube (aka Helicopter Cube) http://www.youtube.com/watch?v=uti5OEE1E3Uand it gives the appearence that just about all jumbleable moves lead back to the rest state very quickly. Granted that is probably the solver just not want to go off in that direction but for someone posting solve times for a Bevel Cube I also noticed he started with the cube in a rest state. Shouldn't one start with a cube that is as scrambled AND as jumbled as possible? Getting to the rest state MAY be half the fun... I just don't know. It's videos like this that make me feel the jumbleable space of these puzzles may be rather small and thus maybe a useful closed group could be created that accurately represents these puzzles without the number of operations approacting the number of states of the puzzle itself. Carl
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Allagem

Post subject: Re: Jumbling Puzzles Posted: Sat Jul 25, 2009 2:25 pm 

Joined: Sun Oct 08, 2006 1:47 pm Location: Houston/San Antonio, Texas

wwwmwww wrote: I wouldn't call the cuts spheres. They form circles on the sphere but are best thought of as planes. And in this case I believe all the cuts are at the same depth. You are calling the cubic cuts shallow and the rhombic dodecahedron based cuts deep but I don't think that is the case.
Ah sorry, you misunderstood me here. I guess I wasn't too clear. When I am refering to spheres I mean the whole puzzle. There are two spheres on Gelatinbrain: the deep cut one and the shallower cut one. I did mean spheres I agree that the cuts are best thought of as planes and you are correct all cuts within either puzzle are the same depth. Otherwise the 45 degree moves wouldn't line up. Sorry for that confusion. wwwmwww wrote: The bit in ()'s required. If one were to just define a nonbandaged puzzle as one where any two adjacent pieces can be seperated in one move from the rest state then one would called the Mixup Cube nondandaged. The bit about ALL cuts between any two pieces be accesible is critical. Good point, but to get really picky, the two definitions should always be complimentary. If one is true the other is true as well. Remember, we are looking at the Mixup Cube if it was built up from a jumbling core (the one I attached in my previous post) A Mixup Cube created this way would have a ton of little mathematical pieces on the inside. ALL of these are included in my definitions. On this Mixup Cube there are pieces within the puzzle that could not move until the top and bottom were rotated 45 degrees. wwwmwww wrote: You would be fusing mathematically different pieces together to get a 24Cube. Your picture isn't loading for me, but I'm assuming you have made a puzzle that has many more cuts than a 24Cube, but shares all the same cuts. This is the same as saying a 3x3x3 is the same thing as a bandaged 6x6x6. While it is true, I would say that only a subset of the moves are available  and these are identical to a higher symmetry, so we might as well determine things from that higher symmetry. Afterall, EVERY puzzle is a bandaged form of a sphere with an infinite number of cuts right? wwwmwww wrote: Interesting... can you unsplit these puzzles to get their root jumbleable puzzles? I'd love to see what they looked like. Yes! Unsplitting the pieces of the Meteor Madness gives you a shape mod of the Jumple Prism It's actually fairly simple to visualize while just looking at the puzzle. Unsplit any two pieces that can't be separated on the first move by fusing them together like a bandage and you'll have the nonsplit form. Their "root" jumbleable puzzle as you say (I like that! ) It sounds like the only real difference between us is that you are looking at a jumble puzzle jumbled one move and are saying that it has become bandaged even if it was unbandaged the move before. Although I see why you say this, this mindset will fall back into the old pattern have having infinitely bandaged puzzles. Perhaps I can aid my argument by explaining how the Jumble Prism works: It's core shape is a triangular dipyramid  that is two triangular pyramids with their bases fused together. The angle between each pair of adjacent faces around one pyramid matches the angle between the two pyramids. This narrows the core down to only one possible shape. I'll attach a picture of this. (BTW it's the matching angles that allow the jumbling). Quick analogy to a Rubik's cube. The core is a cube, and the pieces that attach directly to the core, the centers, rotate on the axis that is normal to the face to which they are attached. Subsequent moves are available whenever the edges of the center piece line up with another face on the core. This clearly happens every 90 degrees, because the face of the core that the piece is attached to is square and while rotating lines up with itself every 90 degrees. On the Jumble Prism, there are similar pieces attached to the trinagular faces of the core. Since this shape is an isosceles triangle, it does not have rotational symmetry and only lines back up with itself after 360 degrees. BUT certain edges line up with different edges  never all at the same time  and since all angles between two faces on the core are equal, the cutting planes line up at these certain angles  hence jumbling happens. But notice that not all the edges line up at the same time. If they did, the core face would have rotational symmetry and we would have a non jumbling puzzle  or at least a nonjumbling move. So ofcourse there will always be bandaging on a jumble puzzle. What appears to be a jumble puzzle that never has blocked moves (mixup cube, gelatinbrain's spheres) is really an illusion created by the fact that entire FACES are blocked. Each of these puzzles mathematically have cores with 18 faces and therefore 18 potential axes to rotate around (the core in my previous post's attachment shows these 18 faces) Gelatinbrain's spheres always block the 12 faces that aren't the x y and z axes and the mixup cube always trades out 4 cubical faces for 4 rhombic dodecahedron faces. My point is that ALL jumbleable puzzles will have blocked moves, so I think their rest state  ex. when the pieces that attach to the Jumble Prism's core all line up with the faces of the core, should determine whether or not the puzzle is bandaged (or split however many times). Besides don't you find it odd that a characteristic such as bandaged or not can change as the puzzle moves? The mathematical pieces are either glued together or they're not. Make up your mind! Peace, Matt Galla PS: wwwmwww wrote: Regarding the Fairly Twisted puzzle... what is meant by lefthanded and righthanded faced? Can the faces just be turned in one direction? (Just rewatched the video... nope that's not it.) Did you just add this later? Honestly, I think this is just Oskar's interpretation of the asymmetrical shape of the puzzle, and has no real mathematical meaning. The shape of this thing is tricky, but I'm 90% sure it IS symmetrical  by mirror symmetry. At one point in the video Oskar sets down the puzzle with the small yellow square side facing the camera. At this point imagine a vertical mirror going straight through the puzzle. The exception (of course there is an exception!! ) would be what would be the back face at that orientation. It has one piece sticking off to one side that creates the awkward teal face. Perhaps this left and right thing has to do with which side of the puzzle the face is on compared to what Oskar is assuming to be the front.
Attachments: 
File comment: Here is the core for the jumble prism (and meteor madness). The outline of the Jumble Prism itself is also included, but my program cut off the very top. Sorry about that!
Jumble Prism Core.jpg [ 20.55 KiB  Viewed 12903 times ]



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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sat Jul 25, 2009 5:05 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Allagem wrote: Ah sorry, you misunderstood me here. I guess I wasn't too clear. When I am refering to spheres I mean the whole puzzle. There are two spheres on Gelatinbrain: the deep cut one and the shallower cut one. I did mean spheres I agree that the cuts are best thought of as planes and you are correct all cuts within either puzzle are the same depth. Otherwise the 45 degree moves wouldn't line up. Sorry for that confusion. Ahhh... Isn't that deep cut puzzle just a 2x2x2 mixed with a 24Cube? The applet seems to only let me make 45 or 90 degree turns along the 2x2x2 cuts. Is there a way to make turns along the 24Cube cuts? I'm guessing no. Allagem wrote: Good point, but to get really picky, the two definitions should always be complimentary. If one is true the other is true as well. Remember, we are looking at the Mixup Cube if it was built up from a jumbling core (the one I attached in my previous post) A Mixup Cube created this way would have a ton of little mathematical pieces on the inside. ALL of these are included in my definitions. On this Mixup Cube there are pieces within the puzzle that could not move until the top and bottom were rotated 45 degrees. I see. I was only thinking about surface pieces. When all pieces are considered I believe you are correct. Allagem wrote: Your picture isn't loading for me, but I'm assuming you have made a puzzle that has many more cuts than a 24Cube, but shares all the same cuts. This is the same as saying a 3x3x3 is the same thing as a bandaged 6x6x6. While it is true, I would say that only a subset of the moves are available  and these are identical to a higher symmetry, so we might as well determine things from that higher symmetry. Afterall, EVERY puzzle is a bandaged form of a sphere with an infinite number of cuts right? In my mind the 3x3x3 isn't bandaged so there is no need to try and unbandage it. The 24Cube appears to have bandaged cut planes after an acos(1/3) twist so that is the reason to consider it a bandaged puzzle. If its bandaged I believe its fair to consider what happens when you try to unbandage it. Yes, a 3x3x3 can be made by bandaging a 6x6x6 but the 3x3x3 isn't a bandaged puzzle. I think this is similiar to saying a Mixup Cube can be made from bandaging a jumpleable puzzle but the MixUp Cube isn't jumbleable. [added later] Here is a similiar idea... you can take a Mixup Cube (which we consider bandaged) and make it such that it has same sized cubies. This would in effect bandage the 45 degree turns and turn it into a normal 3x3x3. So you've taken a bandaged puzzle, bandaged it some more, and made an unbandaged puzzle. Interesting... I wonder if there would be any advantage to making a 3x3x3 in this fashion? Allagem wrote: Yes! Unsplitting the pieces of the Meteor Madness gives you a shape mod of the Jumple Prism It's actually fairly simple to visualize while just looking at the puzzle. Unsplit any two pieces that can't be separated on the first move by fusing them together like a bandage and you'll have the nonsplit form. Their "root" jumbleable puzzle as you say (I like that! ) Interesting... Yes, I can almost see that. Allagem wrote: It sounds like the only real difference between us is that you are looking at a jumble puzzle jumbled one move and are saying that it has become bandaged even if it was unbandaged the move before. Although I see why you say this, this mindset will fall back into the old pattern have having infinitely bandaged puzzles. Perhaps I can aid my argument by explaining how the Jumble Prism works: It's core shape is a triangular dipyramid  that is two triangular pyramids with their bases fused together. The angle between each pair of adjacent faces around one pyramid matches the angle between the two pyramids. This narrows the core down to only one possible shape. I'll attach a picture of this. (BTW it's the matching angles that allow the jumbling). Quick analogy to a Rubik's cube. The core is a cube, and the pieces that attach directly to the core, the centers, rotate on the axis that is normal to the face to which they are attached. Subsequent moves are available whenever the edges of the center piece line up with another face on the core. This clearly happens every 90 degrees, because the face of the core that the piece is attached to is square and while rotating lines up with itself every 90 degrees. On the Jumble Prism, there are similar pieces attached to the trinagular faces of the core. Since this shape is an isosceles triangle, it does not have rotational symmetry and only lines back up with itself after 360 degrees. BUT certain edges line up with different edges  never all at the same time  and since all angles between two faces on the core are equal, the cutting planes line up at these certain angles  hence jumbling happens. But notice that not all the edges line up at the same time. If they did, the core face would have rotational symmetry and we would have a non jumbling puzzle  or at least a nonjumbling move. So ofcourse there will always be bandaging on a jumble puzzle. What appears to be a jumble puzzle that never has blocked moves (mixup cube, gelatinbrain's spheres) is really an illusion created by the fact that entire FACES are blocked. Each of these puzzles mathematically have cores with 18 faces and therefore 18 potential axes to rotate around (the core in my previous post's attachment shows these 18 faces) Gelatinbrain's spheres always block the 12 faces that aren't the x y and z axes and the mixup cube always trades out 4 cubical faces for 4 rhombic dodecahedron faces. My point is that ALL jumbleable puzzles will have blocked moves, so I think their rest state  ex. when the pieces that attach to the Jumble Prism's core all line up with the faces of the core, should determine whether or not the puzzle is bandaged (or split however many times). Besides don't you find it odd that a characteristic such as bandaged or not can change as the puzzle moves? The mathematical pieces are either glued together or they're not. Make up your mind! The difference between us here is that I was considering a puzzle that had any bandaged moves to be a bandaged puzzle. True the 24Cube doesn't have any bandaged moves in its rest state but I was still considering it a bandaged puzzle. Your statement above "So of course there will always be bandaging on a jumble puzzle." is more or less saying that itself. However I agree... your definition of bandaged puzzle may be more useful in the long run. I certainly see the merits of your position. But to acknowlege bandaging on a nonbandaged puzzle sounds like it might have some problems of its own too. By the way, did you figure all that out about the Jumble Prism and the Meteor Madness by just looking at the pictures and watching the videos? If so you're better at this then I am. Allagem wrote: PS: Did you just add this later? Honestly, I think this is just Oskar's interpretation of the asymmetrical shape of the puzzle, and has no real mathematical meaning. The shape of this thing is tricky, but I'm 90% sure it IS symmetrical  by mirror symmetry. At one point in the video Oskar sets down the puzzle with the small yellow square side facing the camera. At this point imagine a vertical mirror going straight through the puzzle. The exception (of course there is an exception!! ) would be what would be the back face at that orientation. It has one piece sticking off to one side that creates the awkward teal face. Perhaps this left and right thing has to do with which side of the puzzle the face is on compared to what Oskar is assuming to be the front. Yes, I added that latter. Actually it was there the first time I tried to post but after typing up 90% of my post I was pulled away from the PC and when I went to submit the post the system had logged me off and asked me to log in again so my whole post was lost and I had to type it again. I forgot to add that comment the second time and when I remembered it I came back and added it later. Interesting... I think I see the mirror symmetry in that yellow piece he holds up too. It looks like a pentagon with two tetrahedrons on it and a square based pyramid on its side between the two tetrahedrons. The mirror symmetry would cut the square based pyramid and the pentagon in half. Carl
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Last edited by wwwmwww on Sun Jul 26, 2009 9:12 am, edited 1 time in total.


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Bram

Post subject: Re: Jumbling Puzzles Posted: Sun Jul 26, 2009 7:53 am 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

I think I've figured out how to explain what's jumbling in a welldefined manner. For the sake of simplicity, I'm going to be skipping a discussion of puzzles with gaps.
Let's define a 'doctrinaire' puzzle as one where if you were to remove all the coloration then every single position would look exactly the same. The Rubik's Cube is a doctrinaire puzzle, as is the Skewb and Megaminx. Also the Sphere Xyz, Chromo Ball, Puck puzzles, and a bunch of other puzzles which don't have slices like a Rubik's Cube but still have permutations.
A shape mod is a nondoctrinaire puzzle which can be shape modded to a doctrinaire puzzle. The Fisher Cube is a shape mod, as is the Mixup Cube.
A bandage puzzle is a nondoctrinaire one where by cutting the pieces into smaller parts it's possible to transform it into a doctrinaire puzzle.
A jumble puzzle is one which is nondoctrinaire but where it isn't possible to shape mod or unbandage it into a doctrinaire puzzle. Examples include the Helicopter Cube, 24cube, Jumbleprism, Uncanny Cube, and Battle Gears.
The 24cube is a somewhat confusing case because if one were to make an identicallooking puzzle which was physically blocked from doing anything but the 180 degree moves then it would be a doctrinaire puzzle, but as it is it's a jumble puzzle, and it's surprisingly difficult to figure out a way of keeping it from jumbling.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sun Jul 26, 2009 9:38 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Bram wrote: I think I've figured out how to explain what's jumbling in a welldefined manner. For the sake of simplicity, I'm going to be skipping a discussion of puzzles with gaps. By gaps, are you refering to puzzles like your Black Hole puzzle (in 3D) and the 15 Puzzle (in 2D)? I don't consider these "twisty puzzles" and I'm not sure how these terms would apply to those puzzles. But if you have a more complicated picture you avoided for simplicity I'd be interested in seeing it. 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. The Rubik's Cube is a doctrinaire puzzle, as is the Skewb and Megaminx. Also the Sphere Xyz, Chromo Ball, Puck puzzles, and a bunch of other puzzles which don't have slices like a Rubik's Cube but still have permutations.
A shape mod is a nondoctrinaire puzzle which can be shape modded to a doctrinaire puzzle. The Fisher Cube is a shape mod, as is the Mixup Cube. The doctrinaire Mixup Cube I guess you could call the Mixup Rhombicuboctahedron... interesting... Bram wrote: A bandage puzzle is a nondoctrinaire one where by cutting the pieces into smaller parts it's possible to transform it into a doctrinaire puzzle.
A jumble puzzle is one which is nondoctrinaire but where it isn't possible to shape mod or unbandage it into a doctrinaire puzzle. Examples include the Helicopter Cube, 24cube, Jumbleprism, Uncanny Cube, and Battle Gears.
The 24cube is a somewhat confusing case because if one were to make an identicallooking puzzle which was physically blocked from doing anything but the 180 degree moves then it would be a doctrinaire puzzle, but as it is it's a jumble puzzle, and it's surprisingly difficult to figure out a way of keeping it from jumbling. The Helicopter Cube is in that same "confusing case" as the 24Cube. Personally if someone were to take these puzzles and limit them to 180 degree moves I'd prefer they give them new names. Maybe something like the Limited Helicopter Cube and the Limited 24Cube. Thanks for these definitions and I see they are independant of the internal mechanism of the puzzle so I certainly consider them valid definitions. I don't think these are the only selfconsistent set of definitions that could be used but they may certainly be the simpliest and the best ones for this group to adopt. Carl
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sun Jul 26, 2009 11:22 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Bram wrote: There's a funny similar thing going on with the dino cube, which almost looks like it should form a closed group if you could rotate the faces 60 degrees instead of 120. Maybe there's an interesting jumble puzzle which allows some of that. I finally did get around to looking at this. Looks like its a combo of the Dino Cube and the 24Cube. The darker pieces would be interior pieces if the puzzle were in the shape of a cube. Considering the complexity of making a 24Cube I don't really expect this puzzle to be made any time soon but I'd LOVE to be proven wrong. Even without the acos(1/3) twist being allowed of the underlying 24Cube I believe this would be a jumbleable puzzle. The 60 degree twists appear jumbleable by themselves. Carl
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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sat Aug 01, 2009 10:25 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Hello Bram, wwwmwww wrote: Is that your way of saying that despite what set of definitions we might agree to now Bram will just go off and make a puzzle that breaks or bends a few of them to the point we are back here debating this again? I'd take that as a good thing. See... I saw this coming. Bram wrote: I think I've figured out how to explain what's jumbling in a welldefined manner. For the sake of simplicity, I'm going to be skipping a discussion of puzzles with gaps. I guess first I should ask if I interpreted your definition of "puzzles with gaps" correctly by assuming you ment puzzles like the Black Hole puzzle (in 3D) and the 15 Puzzle (in 2D)? 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. The Rubik's Cube is a doctrinaire puzzle, as is the Skewb and Megaminx. Also the Sphere Xyz, Chromo Ball, Puck puzzles, and a bunch of other puzzles which don't have slices like a Rubik's Cube but still have permutations.
A shape mod is a nondoctrinaire puzzle which can be shape modded to a doctrinaire puzzle. The Fisher Cube is a shape mod, as is the Mixup Cube.
A bandage puzzle is a nondoctrinaire one where by cutting the pieces into smaller parts it's possible to transform it into a doctrinaire puzzle.
A jumble puzzle is one which is nondoctrinaire but where it isn't possible to shape mod or unbandage it into a doctrinaire puzzle. Examples include the Helicopter Cube, 24cube, Jumbleprism, Uncanny Cube, and Battle Gears. Ok... I like these definitions but after seeing this thread about your (or is this one Oskar's idea?) Caution Cube: Quote: http://twistypuzzles.com/forum/viewtopic.php?f=15&t=14375 It got me thinking... how do these two puzzles fit into these definition? Caution CubeBram's CubeI'm guessing the Bram's Cube may be considered a doctrinaire puzzle as is but it appears the surface of the circles may not be in the same plane as the cube faces after a few turns though this may be just play that exists in this puzzle. And I'm also guessing the Caution Cube may be considered a shape mod and if you modded the edges to look like spheres you'd have a doctrinaire puzzle. Correct? Carl P.S. These puzzles though do fall into a class all their own of sorts. I got to thinking about this statement: Allagem wrote: Afterall, EVERY puzzle is a bandaged form of a sphere with an infinite number of cuts right? I think the Caution Cube and Bram's Cube fall outside of Allagem's definition of what I typically think of as a twisty puzzle. And maybe the Geary Cube should be included in this discussion. If so I then see how this use of gears might also just be considered a form of bandaging. Hmmm....
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Bram

Post subject: Re: Jumbling Puzzles Posted: Sat Aug 01, 2009 12:53 pm 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

The Caution Cube, Bram's Cube, and Geary Cube are all doctrinaire twisty puzzles. It isn't obvious in the videos, but the angling of the gears in the Caution Cube and Bram's Cube is really just a coloration thing  the teeth wind up in similar positions no matter what.
There are some interesting variants on the Geary Cube where only some of the gears are used. Removing just four gears all along the same plane results in an interesting doctrinaire puzzle, but leaving only two gears in results in something extremely bandaged and difficult.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sat Aug 01, 2009 2:23 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Bram wrote: The Caution Cube, Bram's Cube, and Geary Cube are all doctrinaire twisty puzzles. I can agree all the teeth may always return to their doctrinaire positions but the Caution Cube sure doesn't look like a cube any more after a 180 degree turn of one face. As such this one must be a shape mod... isn't it? I can see that it can easily be made into a doctrinaire puzzle by making the colored portion of each edge piece into the shape of a shere. Some other shapes may work too but I really can't tell at what angle the edge piece is rotated after one 180 degree face turn. Can the size of the gears be changed such that this does return to the cube shape after a 180 degree turn? If so I'd also call that a doctrinaire puzzle? Carl
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Bram

Post subject: Re: Jumbling Puzzles Posted: Sat Aug 01, 2009 3:51 pm 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

wwwmwww wrote: I can agree all the teeth may always return to their doctrinaire positions but the Caution Cube sure doesn't look like a cube any more after a 180 degree turn of one face. As such this one must be a shape mod... isn't it? Er, yeah, it's a shape mod, that's what I meant to say. You can't just change the number of teeth and have it work easily  there's a bunch of technical requirements about the teeth, both theoretical and physically practical, which make the particular numbers of teeth used very important.


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wwwmwww

Post subject: Re: Jumbling Puzzles Posted: Sat Aug 01, 2009 8:20 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Bram wrote: You can't just change the number of teeth and have it work easily  there's a bunch of technical requirements about the teeth, both theoretical and physically practical, which make the particular numbers of teeth used very important. Yes, I got to thinking about the number of teeth issue later and I realized that wouldn't work. I however did have another idea... Looking at the Caution Cube, I realized the reason one face has to be turned 180 degrees is because if you tried to stop after 90 degrees you'd be in a position where the center layer has only gone 45 degrees. And as we know... on a 3x3x3 you can't just turn the center layer 45 degrees. BUT on a Mixup Cube we know center rotations of 45 degrees are allowed. So I'm now curious... Is it possible to make a puzzle similiar to the Caution Cube (I guess you could call it a Mixup Caution Cube) where face rotations of 90 degrees ARE allowed? See what happens when you make GREAT puzzles! We expect even GREATER puzzles out of you down the road. Thanks, Carl
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Bram

Post subject: Re: Jumbling Puzzles Posted: Sat Aug 01, 2009 8:26 pm 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

wwwmwww wrote: Looking at the Caution Cube, I realized the reason one face has to be turned 180 degrees is because if you tried to stop after 90 degrees you'd be in a position where the center layer has only gone 45 degrees. And as we know... on a 3x3x3 you can't just turn the center layer 45 degrees. BUT on a Mixup Cube we know center rotations of 45 degrees are allowed. So I'm now curious... Is it possible to make a puzzle similiar to the Caution Cube (I guess you could call it a Mixup Caution Cube) where face rotations of 90 degrees ARE allowed? My guess is it's possible, but I don't know a mechanism


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