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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 2:02 am 
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Bram wrote:

I stand corrected. My analysis was just plain wrong. Battle Gear does not, in fact, jumble.

That leaves the Split Hex as the simplest example of a jumbling puzzle which only has rational angles.

Battle Gear has been removed from the list. Also, what is the degree of rotation on Pascal's Jumble?

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 7:45 am 
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Bram wrote:
That leaves the Split Hex as the simplest example of a jumbling puzzle which only has rational angles.


Split Hex... simple!? I still have yet to get my head around this one. Is there a video of it somewhere?

If you merge a 24-Cube and a 2x2x2 you get a puzzle which jumbles at 90 degrees along the 24-Cube cuts and at 45 degrees along the 2x2x2 cuts. And it still has the acos(1/3) jumbling too. And if I recall correctly it may have had one or two other jumbling moves... I think we've talked about this one before.

This may not be a simplier puzzle to make but the geometry is a bit easier to get one's mind around.

Carl

P.S. I didn't recall correctly. This is the thread I was thinking of:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=15603
And there we had added a Skewb to the 24-Cube and 2x2x2 mix. It's that puzzle that has 5 different jumbling moves.

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 7:55 am 
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PuzzleMaster6262 wrote:
This is like the battle gear if it could jumble. One wheel is rotated blocking the other. Because the only move is to continue rotating the first wheel, should it even be considered bandaged?


Yes, because not all the states were turns are allowed are identical. It's not a doctrinaire puzzle.

PuzzleMaster6262 wrote:
This ties back to the top of my post. I would say the mixup cube cannot jumble because no rotations are blocked. The cuts can be extended to form a jumbling puzzle just like with my 3x3x3 example. However, because after rotating the top 45 degrees only that axis can continue rotating, I believe it is not bandaged or jumbled in any way.


This I agree with. The Mixup Cube is not bandaged. It can't jumble. And it IS a doctrinaire puzzle. The issue Matt raised and I agree with is that its not a pure twisty puzzle. I'd simply put it in a class that I'd call a Twisty/Slidey puzzle.

Carl

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 10:48 am 
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The Split Hex is different from the 2x2x2 plus 24 cube because after sliding 30 degrees, the Split Hex can jumble. The 2x2x2 plus little chop jumbles at 45 degrees, 90 degrees, and around 70 degrees. Once I finish making mine, it will belong in the regular and irragular jumbling subgroup :wink:

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 1:53 pm 
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So the types of movements so far are twists, slides, and jumbles. All movements can be bandaged and jumbling can be rational and irrational. A puzzle can exist with only twists, slides, or jumbles. An example of a twisting only puzzle is the Rubik's cube. Sliding only could be the 15 puzzle. Jumbling only could be the Deep Cut 3x3.

I think this is a good grouping idea :D

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 2:17 pm 
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PuzzleMaster6262 wrote:
Jumbling only could be the Deep Cut 3x3.


Deep Cut 3x3? You mean a 2x2x2? If not could you post a link as you just lost me. A 2x2x2 doesn't jumble. More Madness by OSKAR I think may be a puzzle that only Jumbles. Its so hard to understand the geometry of these odd creations by only looking at pictures.

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 2:39 pm 
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Deep cut 3x3 was a design I posted a couple days ago of a puzzle that looks like a 3x3x3 on the top and bottom but an X on the sides. It also is harder to scramble then solve because after jumbling, all moves get blocked. The more madness I think would also only have jumbling moves.

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 3:27 pm 
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Warning: EXTREMELY LONG post :roll:

Monopoly wrote:
All vertex-turning pentagonal dodecahedra where 2 non-adjacent corners on each face-pentagon overlap can also jumble. I used Jaap's sphere applet, and it appeared to work for all vertex-turning dodecahedra...can anyone please explain exactly WHY this is? (it does explain why the Polaris would jumble though....) sorry for the very improper usage of grammar and math, but that's about as far mathematically as my brain can go with this ._.


YES! 8-)
First off, you should realize that vertex-turning dodecahedra puzzles will have icosahedral cores. Nothing about a purely dodecahedral cored puzzle will ever jumble. However Icosahedral-cored puzzles WILL jumble IF they are cut deep enough. Here's why:

On all twisty puzzles rotations are made about an axis perpendicular to one face on the core of the puzzle. Additional twists are possible iff the cut due to one face before the move lines up with the cut from a different face after the move. Since nearly every puzzle, including the icosahedron, has all faces equidistant from the center of the puzzle, the only thing we need for the next twist to be available is the angles between the faces in question to match. Specifcally, the dihedral angle between the rotating face and another face must match the dihedral angle between the rotating face and the face about which the next move will be made. Typically, this only happens when every face on the core lines up, which means every move is available. i.e. 3x3x3, pyraminx, megaminx, and 99% of all twisty puzzles. However it could be the case where partway through a move, some faces line up while others don't. This is called jumbling :)

I know that was too confusing to make sense, so here's some pictures:
Attachment:
WhyIcosahedronsJumble1.png
WhyIcosahedronsJumble1.png [ 35.84 KiB | Viewed 5360 times ]

This is the core of, as you called it, a vertex-turning dodecahedra puzzle, aka an icosahedral cored puzzle. We are going to rotate around the yellow ray, aka rotate the blue face. This core will not move, but the pieces above it (not shown) will rotate. Note that the marked dihedral angles between the blue face and each green face are exactly equal. This means that the cuts through the pieces caused by one green face will line up with the other green face when we rotate it...
Attachment:
WhyIcosahedronsJumble2.png
WhyIcosahedronsJumble2.png [ 37.82 KiB | Viewed 5360 times ]

...about 44.5 degrees. So if we rotate the pieces above the blue face (again, not shown) 44.5 degrees clockwise, the pieces that could have been moved by rotating the right green face before the turn can now be rotated by the left green face. At this point many faces, including the right green face will be blocked. This example of jumbling is unusual because the blue face and each of the green faces are NOT adjacent, whereas nearly every other jumbling puzzle only exploits jumbling across adjacent faces (More Madness technically has jumbling across both adjacent and non-adjacent core faces)

I hope either the explanations or the pictures or both helped cleared up a little. The same angle also causes some faces even farther away to jumble, but that would require a deeper cut icosahedron than currently exists.



On to bigger and better things! :P

wwwmwww wrote:
Bram wrote:
In my book, any puzzle which is primarily permutation based is a twisty puzzle, including every puzzle mentioned in this thread
I tend to agree and I did say that I thought non-twisty puzzles wasn't the the best name choice for this group/sub-group.

Fair enough. :)
wwwmwww wrote:
Going back to Matt's terminology maybe we could call his two sets twisty puzzles with stable cores and twisty puzzles without stable cores.

If you'd like. But thank you for recognizing the difference here.
wwwmwww wrote:
When I look at a 3x3x3 I see a 3D twisty puzzle. To me the Mixup Cube is an impure twisty puzzle because it has slidey elements to it. It physically is impossible to construct as a pure twisty puzzle. 2D slidey puzzles need a void for the pieces to be able to move but that isn't the case in 3D as loops can be made. Look at the Chromoball for example.

This is an excellent analysis and is in fact how the Mixup Cube and Hex Cube family were constructed: a chromoball type design combined with a 2x2x2.
wwwmwww wrote:
P.S. I still haven't fully understood the Hex Cube, Split Hex, and Mixup 2x4x4. Maybe we have Jumble Twists AND Jumble Slides.

They're just like the Mixup Cube, which you seem to now be comfortable with. Take a Mixup Cube (Chromoball combined with 2x2x2) remove the chromoball ring of pieces from the xy-plane, and add an extra ring on both the xz and yz planes. Then bandage/shape-mod appropriately to get each of these puzzles. The 2x2x2 pieces (the triangles on the hexes, the corners on the 2x4x4) still behave exactly the same on the inside and take up the same space mechanically, the visible surface has just been shoved closer to the xy plane.
PuzzleMaster6262 wrote:
The mixup cube being a slide puzzle is a good idea but it COULD be made without sliding.
Not like a standard twisty puzzle... No matter how you make this, either R or L will move the core, and that is NOT normal on standard twisty puzzles with an odd number of layers...
wwwmwww wrote:
Again if I'm following Matt's reasoning correctly you'd need an infinite number of axes of rotation. Ok... make the core a sphere. Now if your cut planes (2 per axis) contain any volume the infinite number of pieces in this puzzle don't have any volume to them. However there should be 18 zero volume pieces in the mix that DO behave as your Mixup Cube pieces.

So if (BIG IF as I haven't proven it) I'm correct the Mixup Cube could be considered a subset of this infinite axes of rotation pure Twisty Puzzle or it could be considered a Twisty/Slidey Puzzle

:lol: it's funny to hear you say it that way but yes that's exactly right. And using the core you showed here you can indeed create a puzzle that looks exactly like the Mixup 3x3x3 that I would consider to be a standard, perfectly normal pure twisty puzzle with a stable core. It would even behave exactly like a Mixup 3x3x3... for a FEW turns. But it would be quite simple to get to a point where a move would be available on Oskar's Mixup 3x3x3 but not be available on the puzzle produced from this core. :wink:

PuzzleMaster6262 wrote:
I agree to disagree. The internal mechanics of any puzzle should not be involved when deciding how to classify it. The core of a puzzle is not important. Only the visible pieces involved in solving a puzzle matter. If the mixup cube was created with hundreds of parts and a stable core, it is still a mixup cube.

I dunno if you've changed your mind since here, but at this point it sounds to me like you think we're debating over something that doesn't matter for your purposes. There is no point in looking into the possibilities of a 3x3x3 as a bandaged 6x6x6 because as far as the analysis is concerned, it is a 3x3x3 regardless of what it's made of. I think everyone would agree with that. We have seen identical puzzles made in completely different ways, for example the Master Skewb has been made with an octahedral core by Okamoto and Tony Fisher and a tetrahedral core by Drewseph. These two designs actually have a different number of moving parts, and yet the outside result is the same. It is not incorrect to state a Master Skewb can be constructed from a tetrahedral core, but at the same time, it is generally considered an octahedral cored puzzle. It MAY be considered either octahedral or tetrahedral. The case is NOT the same here. The Mixup 3x3x3 was constructed with "slidey" pieces because it MUST be. Or at least it must NOT be constucted with a stable core in order for the puzzle to work. (CAPS are for emphasis only, not frustration :) )


PuzzleMaster6262 wrote:
Also the puzzle list has been moved into subgroups based on rational/irrational jumbling. If possible I would like to include the angle of rotation for each puzzle that allows jumbling moves. If you know the angle, please post it :D

This I can do, but it's actually alot more complicated than that...
There is not just one jumbleable angle for every jumble puzzle. For example take the Meteor Madness Puzzle. Look at one face while the puzzle is in its solved shape, oriented however you like. Imagine drawing a line from the center of the face straight up on the surface of the face. Now when you rotate this face, keeping the orientation of the rest of the puzzle fixed, this line will rotate like the hand of a clock. The twelve o'clock position is where you started. This diagram shows where other moves become available:
Attachment:
MeteorMadnessAngles.png
MeteorMadnessAngles.png [ 45.13 KiB | Viewed 5360 times ]

So what's the "jumbling" angle? My intuition tells me we should classify this on its functionality. Start at the top and follow the circle clockwise. After the 83 degree turn one of three potential cutting planes on the current face is able to move again, so that subgroup of pieces can move on either side of the 83 degree turn. After the additional 56 degree turn, the other 2 cutting planes line up, so there are no complete cuts that line up on both sides of the 56 degree angle, however if you mash the 83 and 56 together and look at that angle of 139 degrees, then the other 2 cutting planes line up on both sides of the 139 degree angle. So basically we have two things happening here:
Attachment:
MeteorMadnessAngleBreakdown1.png
MeteorMadnessAngleBreakdown1.png [ 34.17 KiB | Viewed 5360 times ]
Attachment:
MeteorMadnessAngleBreakdown2.png
MeteorMadnessAngleBreakdown2.png [ 37.74 KiB | Viewed 5360 times ]

Which in my opinion should be the angles considered for this particular puzzle. The exact measurements are arccos(1/8) and arccos(-3/4), respectfully. 8-) After this weekend I'll try to come up with the angles for some of these more obscure puzzles. Oh, by the way, EVERY rhombic dodechaedron cored puzzle has the same jumbling angle which has been correctly identified as arccos(1/3) or approx. 70.53 degrees. That includes: Rua, Face-Turning Rhombic Dodecahedron and Toru. On the Shapeways site, Oskar mentions that the Distorted Cube's faces are set at 92 degrees, I'm guessing that means the jumbling angle would be 92 degrees as well but that's just a guess.

I think that's enough for now :lol:

Peace,
Matt Galla

PS It still bugs me to say the hex family jumbles. It's just not the same thing as normal puzzles. I don't know...
What I can tell you is that without a doubt, the Mixup 2x4x4 and the Hex Cube are both just bandaged versions of the Split Hex, so whatever the Split Hex classifies as, the other two will follow. Then again, which is more natural: the Split Hex or the Hex Cube? Because in a sense, the Hex Cube isn't bandaged... Going from the Hex Cube to the Split Hex is just like going from a 3x3x3, cutting it down the middle on all three axes and calling it an unproportional 4x4x4... Sounds to me like we're all gonna have to agree on a definition of jumbling for these non-stable core puzzles...


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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 4:59 pm 
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PuzzleMaster6262 wrote:
So the types of movements so far are twists, slides, and jumbles.

It seems like you could have jumbling slide moves too if the tile sizes were irrational. This would lead to shapeshifting and multiple holes. Is this getting too far from being a "twisty"/"slidey" puzzle? Here's a couple examples of sliding puzzles that are bandaged and have large holes.
Rush Hour
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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 5:44 pm 
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Allagem wrote:
Sounds to me like we're all gonna have to agree on a definition of jumbling for these non-stable core puzzles...


Nice post!!! I just got time to reply to this last part at the moment.

I like Oskar and Bram's notion of doctrinaire puzzles.

Quote:
The term “doctrinaire” refers to a (classic) twisty puzzle that returns to the same three-dimensional shape after each turn.


Actually I would have worded things not in reference to the puzzles shape but its cut positions as you could have a Helicopter Sphere that doesn't change shape when it jumbles. I.e. a doctrinaire puzzle is one that has all the same cuts in all the same positions before and after a turn. Maybe it would be even better to say that all moves available before a move are also available after a move. The use of the word turn implies twists where I use moves to include twists and slides. This comes onto play below.

Lets assume all twisty/slidey puzzles can be turned into spheres (or at least some shape that is invariant to all turns) to remove the shape changing issue. We can also remove the issue of bandaging as by definition all baddaged puzzles can be unbandaged into their basis puzzle. So this leaves us with the following two types of turns for doctrinaire puzzles.

Twists and Slides

Jumbling comes into play when one talks about non-doctrinaire puzzles and I'm inclined to say non-doctrinaire puzzles can also have two types of moves.

Jumbling Twists and Jumbling Slides (make that 4 as they can also have normal Twists and Slides)

If one goes this way then one could say the 15-Puzzle is a Jumbling puzzle. It's not doctrinaire as the moves available depend on where the void is and that moves with each "turn". It's just a Jumbling Slidey puzzle and not a Twisty puzzle.

And you could even call the Polo Cube by ALEX and OSKAR a Twisty/Slidey puzzle with a Jumbling Slide move.

You could probably break the slide moves into two further sub-groups. There are slide moves that make use of a void as the 15-Puzzle and the Polo Cube do. In this group you could add the PantaCube, Twist & Slide, Bram's Black Hole, Bram's Rocket, RotaCubes, Alex Black Hole, and PantaBram to name a few other puzzles (all from Oskar's Shapeways Shop).

And there are slide moves that don't made use of a void. In this group you have SphereXYZ, the Mixup Cube, the Hex Cube, the Split Hex, the Mixup 2x4x4, Chromoball, and Dave's Elemental puzzles.

Carl

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 7:09 pm 
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GuiltyBystander wrote:
PuzzleMaster6262 wrote:
So the types of movements so far are twists, slides, and jumbles.

It seems like you could have jumbling slide moves too if the tile sizes were irrational. This would lead to shapeshifting and multiple holes. Is this getting too far from being a "twisty"/"slidey" puzzle? Here's a couple examples of sliding puzzles that are bandaged and have large holes.
Rush Hour
Klotski

Twists, slides, and jumbles are just the ways of moving the puzzle. They could be intermixed with eachother.

The 15 puzzle does not jumble and I believe no 2d sliding puzzles can. A jumbling puzzle according to Bram cannot be unbandaged. This definition of jumbling can be used for non stable core puzzles.

Also with the mixup cube, I was not meaning a 6x6x6 being bandaged into a 3x3x3. I meant internal pieces with many layers that would allow a stable core.

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 7:24 pm 
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GuiltyBystander wrote:
It seems like you could have jumbling slide moves too if the tile sizes were irrational. This would lead to shapeshifting and multiple holes. Is this getting too far from being a "twisty"/"slidey" puzzle? Here's a couple examples of sliding puzzles that are bandaged and have large holes.
Rush Hour
Klotski


Yes, you could consider these non-doctrinaire slidey puzzles and therefore they have jumbling slide moves.

This got me thinking can one puzzle have both jumbling twists and jumbling slides? The Mixup Cube, Mixup 2x4x4, Hex Cube, and Split Hex are all built up by adding sliding pieces to the equator of a 2x2x2. So lets apply that method to a base puzzle that has jumbling twists. If you take a 24-Cube and add pieces that now slide along the 6 equators between the pieces what do you have? I'm not sure what shape the puzzle should be... could anything other then a sphere with the sliding peieces being balls that roll in grooves around the equators be made to work? If you try to make it a cube the geometry gets rather messy at the corners.

Carl

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 7:49 pm 
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PuzzleMaster6262 wrote:
The 15 puzzle does not jumble and I believe no 2d sliding puzzles can. A jumbling puzzle according to Bram cannot be unbandaged. This definition of jumbling can be used for non stable core puzzles.


Hmmm... maybe the term "Jumbling Slide" move should be limited to Slidey puzzles that don't make use of a void. I tend to agree that trying to unbandage a Jumbling puzzle should result in that puzzle being cut to dust. But it doesn't make much sense to talk of trying to unbandage the 15-puzzle. Similarly it doesn't make much sense to talk of trying to unbandage the Polo Cube by ALEX and OSKAR.

Then again maybe Matt's resistance to calling any of these sliding moves jumbling should be looked at closer. Is the slide move on a Mixup Cube, a doctrinaire puzzle, fundamentally different then the slide move on a Mixup 2x4x4? Arg... I could go either way.

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 Post subject: Re: Puzzles that jumble
PostPosted: Fri Jul 02, 2010 9:57 pm 
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Just like jumbling has an exact definition, I think sliding needs one as well. A slide could be moving any pieces along a path, not around a center point.

Yes this would mean the mixup cube can't slide, that is if my definition works :D

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 12:33 am 
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PuzzleMaster6262 wrote:
The 15 puzzle does not jumble and I believe no 2d sliding puzzles can. A jumbling puzzle according to Bram cannot be unbandaged. This definition of jumbling can be used for non stable core puzzles.

What about this simple 3 piece sliding puzzle based on the golden ratio. Looks to me like I can keep cutting it forever. Horizontal arrows are making cuts, diagonal ones are moving the free space in a CCW fashion around the edge of the puzzle.
Attachment:
GoldenTiles.png
GoldenTiles.png [ 7.1 KiB | Viewed 5282 times ]



PuzzleMaster6262 wrote:
Just like jumbling has an exact definition, I think sliding needs one as well. A slide could be moving any pieces along a path, not around a center point.

Yes this would mean the mixup cube can't slide, that is if my definition works :D

It sounds like you're making up definitions to suite your needs. The pieces on the mixup cube look like they're on a path to me. Who said paths have to be straight?



I think you're trying to put too much of a distinction between sliding and twisting. You're trying to make a hard distinction between the two and I'm not sure one exists. What's important is that the pieces are being permuted. I'm not sure I would call the Topsy Turvy a sliding puzzle but it is clearly a permutation based puzzle and the pieces do "slide." Do you want to make up a new classification just for this puzzle because it doesn't fit nicely into the two buckets you have labeled "slide" and "twist?" What about the next puzzle that doesn't fit in that bucket?

Which brings me to my final question, do puzzle classifications jumble?


*edit* Could Jason or Bhearn show us their face-turning icosahedra jumbling? I think we'd all love to see that.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 1:27 am 
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GuiltyBystander wrote:
What about this simple 3 piece sliding puzzle based on the golden ratio. Looks to me like I can keep cutting it forever. Horizontal arrows are making cuts, diagonal ones are moving the free space in a CCW fashion around the edge of the puzzle.


Now there's an interesting idea. Someone should build an actual puzzle based on that one :-)


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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 1:39 am 
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I never said a path was straight. The Topsy Turvy would slide going with my definition. No more groups will ever need to be made for any new puzzles. If the pieces rotate around a central point, it's a twist. If they move along a path without a central point, they slide. Because this covers any way of moving pieces it covers all puzzles. Ofcourse someday a 4d puzzle might become reality and a new type of movement will be needed to describe it but that will be a while.

With the mixup cube, I have no need of it being classified as just a normal puzzle. I would like it to be classified as one because that's what I think it is but if you change my mind I'm all for it :D

I understand what you are trying to say with the golden ratio but from the pictures, it's hard to tell what's going on.

Puzzle classifications jumbling, I like that idea :mrgreen:
Now we need to start over and group our puzzle classifications :lol:

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 1:42 am 
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I have to agree with Bram. That could make a great puzzle.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 1:48 am 
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I reread this older definition of Bram:
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.
My point is that according to this definition the FusedCube is doctrinaire. I think this stays true under the differently worded definition in the upcoming CFF-article.
That was my reason for distinguishing between "bandaged" and "restricted" further up.
Is the FusedCube doctrinaire or isn't it?

EDIT: The best example for the Fused cube is:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=266


Last edited by Andreas Nortmann on Sat Jul 03, 2010 2:59 am, edited 1 time in total.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 1:55 am 
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Andreas Nortmann wrote:
I reread this older definition of Bram:
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.
My point is that according to this definition the FusedCube is doctrinaire. I think this stays true under the differently worded definition in the upcoming CFF-article.
That was my reason for distinguishing between "bandaged" and "restricted" further up.
Is the FusedCube doctrinaire or isn't it?

Do you have a link for the fused cube? I'm sorry but I've never heard of it :shock:

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 2:27 am 
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@Allagem, thanks for clearing up the problem, at least I now SOMEWHAT get it...xD Trying to comprehend your post nearly made my brain implode though..
Another kind-of off-task thing relating to the building of jumbling icosahedra- If you were to actually try to build one off of a spherical Dino Dodecahedron, it would not be able to jumble; you would need a slightly deeper-cut icosahedron for the core:
Image

this would allow the core to use the jumbling moves...in its natural form, the pieces that form a 'flower' will probably be very tiny and unnecesary on the core (but would probably make a very nice puzzle in itself). so the solution would be to: remove those pieces completely. Then you would have a icosahedral-based shallow-cut core puzzle that is essentially a spherical Dino-Dodecahedron with gaps that allow it to make 'jumbling' moves while another face is out of perfect alignment. I also used Sphere to draw up Jason's (?) FTI:
ImageNow that I look at it, I see it _should_ be about to jumble. Now my question is: if the FTI were to get its (I'm assuming) Dino Dodecahedron core replaced with a core of this shape that allows it to jumble, would it be able to jumble? or would the moves be blocked due to the shape of the pieces?

sorry for 'hijacking' the thread.....should I start a new one?

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 2:46 am 
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It's fine :D This thread is about putting puzzles in a list and yet we have been talking about everything to do with puzzles.

Both the first and second puzzles would be able to jumble so I don't understand what you are asking.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 2:59 am 
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I think this is where we que for some puzzle animations for easier understanding... only wwwmwww can save us...

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 3:09 am 
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I think elijah is right...

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 9:39 am 
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Bram wrote:
GuiltyBystander wrote:
What about this simple 3 piece sliding puzzle based on the golden ratio. Looks to me like I can keep cutting it forever. Horizontal arrows are making cuts, diagonal ones are moving the free space in a CCW fashion around the edge of the puzzle.


Now there's an interesting idea. Someone should build an actual puzzle based on that one :-)

I'm not sure how practical that would be. In my simple 3 tile starting example, every move you would like to make has to be unbandaged first. This leave you to have very little choice on where to move the pieces unless they had been unbandanged before. The number of possible states would be close to a linear function (maybe squared) of the number of pieces. This is different from the 15-puzzle where the number of states is exponential compared to the number of pieces.
I'm not saying it's impossible to make an irrational sliding puzzle, but you would have to start with something a little more interesting than the 3 pieces I showed.

As a practical problem, as you cut up the pieces, you'll have several tiles that are smaller than the free space. These pieces could end up rotating inside the free space and generally rotating pieces in a sliding puzzle are illegal moves. Hmm... I wonder if this feature could be used to make the puzzle more interesting.



Here's a list of face turning icosahedra that have been made. They look like they could jumble but don't know if they actually can.
Bob Hearn's Face-Turning Icosahedron
Jason's Radiolarian
Cornered Radiolarian
Jason's Circo-Radiolarian / Radiolarian II
Jason's Radiolarian 3

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 10:42 am 
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Andreas Nortmann wrote:
IMy point is that according to this definition the FusedCube is doctrinaire. I think this stays true under the differently worded definition in the upcoming CFF-article.


The FusedCube is doctrinaire. So is the half turns only cube. Puzzles do have subgroups. you know.

By the way, the term 'subgroup' has a specific mathematical meaning. The set of all transformations which can be done on a puzzle form a group, and the a subgroup corresponds to a subset of positions where if you combine the transforms to get from any one of them to any other one of them then you reach another position still in the subgroup.


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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 2:20 pm 
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Bram wrote:
The FusedCube is doctrinaire. So is the half turns only cube. Puzzles do have subgroups. you know.


I agree. The Fused Cube just looks odd as all it's cut planes don't extend all the way through the puzzle. To me its in a similiar boat as the Mixup Cube. It too has cut planes that don't extend all the way through the puzzle but there you just don't see them on the surface. Both fit the definition of doctrinaire.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 2:51 pm 
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I agree about the mixup cube and fused cube being the same. Both puzzles could have additional cuts but on their own they are unbandaged.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 4:10 pm 
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PuzzleMaster6262 wrote:
I think elijah is right...

Help us wwwmwww, you're our only hope...


Ok. I'll take a quick shot at this.

Allagem wrote:
:lol: it's funny to hear you say it that way but yes that's exactly right. And using the core you showed here you can indeed create a puzzle that looks exactly like the Mixup 3x3x3 that I would consider to be a standard, perfectly normal pure twisty puzzle with a stable core.


Let's start here. First I see we are both wrong about the core. It's not a typical Truncated rhombic dodecahedron. At least its not one where the faces are hexagons and squares. All faces need to be the same distance from the center so I need to move the square faces further in. This removes the 60 degree symmetry of the hexagons and leaves them with only 180 degree symmetry. Here is the core.

Attachment:
Core.png
Core.png [ 10.04 KiB | Viewed 5171 times ]


We can now build this puzzle up till it is the size of a Mixup 3x3x3. It now looks like this:

Attachment:
Mixup.png
Mixup.png [ 63.8 KiB | Viewed 5171 times ]


Here, I'll take the top 3x3x3 layer off to show you the interior cuts and a surface of the core inside.

Attachment:
Mixup2.png
Mixup2.png [ 56.96 KiB | Viewed 5171 times ]


Now we can bandage up just the pieces on the very surface that keep this puzzle from looking like a Mixup Cube. We know what they look like. And I think we've made the puzzle that Matt is talking about.

Allagem wrote:
It would even behave exactly like a Mixup 3x3x3... for a FEW turns. But it would be quite simple to get to a point where a move would be available on Oskar's Mixup 3x3x3 but not be available on the puzzle produced from this core. :wink:


Arg... here is where I get stuck. I don't have the time to animate my model but I'd be happy to give my POV-Ray code to anyone that wants to play with it. But I think as long as I just make 45 degree slice turns this will behave exactly as the Mixup 3x3x3 does. So I think the issue is with the 90 degree face turns but I must confess I don't see it at the moment. Even if I were to animate this puzzle (something that would take a few days I don't have) I think the problem will happen with a few of the interior pieces which wouldn't be seen from the surface. So even if I could clearly see the problem in my head (which I can't at the moment) I'm not sure of the best way to show it to you.

Let me think about this one a bit more,
Carl

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 4:18 pm 
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That is a great job. That puzzle would jumble but if it was only rotated like the mixup cube, I believe it would be exactly the same as the mixup cube.

So this puzzle could be bandaged into the Mixup cube, but is the mixup cube a bandaged version of this jumbling puzzle? I think it is not for the same reasons the fused cube is not bandaged.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 4:35 pm 
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wwwmwww wrote:
Arg... here is where I get stuck. I don't have the time to animate my model but I'd be happy to give my POV-Ray code to anyone that wants to play with it.

If it's all set to go and you just need someone to burn some CPU cycles, I would like to help. I've got an i7 I don't "use" much besides doing some OCR stuff for my thesis.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 5:17 pm 
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Ok... I have it figured out now and I DO see the problem. Let's take this pure twisty puzzle again.

Attachment:
Mixup.png
Mixup.png [ 63.8 KiB | Viewed 5158 times ]


Before I cut off just the top 3x3x3 layer. Now lets cut off ALL the 3x3x3 layers and see what the middle cubie of this 3x3x3 looks like.

Attachment:
Core1.png
Core1.png [ 22.14 KiB | Viewed 5158 times ]


You see its a shallow cut helicopter cube. The red faces are the true core again. Now let's take a vertical slice of the puzzle (first picture above) and rotate it 45 degrees. This slice contains the whole of the central shallow cut helicopter cube so after that slice rotation it now looks like this.

Attachment:
Core2.png
Core2.png [ 25.07 KiB | Viewed 5158 times ]


Here is the full puzzle in this state.

Attachment:
Puzzle2.png
Puzzle2.png [ 102.4 KiB | Viewed 5158 times ]


Now lets rotate the horizontal slice of the puzzle by 45 degrees. The central shallow cut helicopter cube now looks like this.

Attachment:
Core3.png
Core3.png [ 29.06 KiB | Viewed 5158 times ]


And the full puzzle is in this state.

Attachment:
Puzzle3.png
Puzzle3.png [ 114.32 KiB | Viewed 5158 times ]


Now here is the problem. The Mixup Cube could go on and move the first vertical slice again by 45 degrees. This puzzle can't make that move due to the state of the central shallow cut helicopter cube. It blocks that rotation.

See?

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 5:22 pm 
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isn't that a shallow cut helicopter cube?

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 5:31 pm 
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PuzzleMaster6262 wrote:
That is a great job. That puzzle would jumble but if it was only rotated like the mixup cube, I believe it would be exactly the same as the mixup cube.


No its not. I can see that now.

PuzzleMaster6262 wrote:
So this puzzle could be bandaged into the Mixup cube, but is the mixup cube a bandaged version of this jumbling puzzle?


No. No level of bandaging will turn this into a Mixup Cube. You can try to unbandage this jumbling puzzle to turn it into a Mixup Cube but then you end up cutting the core to dust. The exact issue Matt mentions in his first post. And I've got to hand it to Matt for being able to see that without the pictures.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 5:32 pm 
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elijah wrote:
isn't that a shallow cut helicopter cube?


Arg!!!! YES!!! Brain fart on my part. I'll correct that now.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 5:33 pm 
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wwwmwww wrote:
elijah wrote:
isn't that a shallow cut helicopter cube?


Arg!!!! YES!!! Brain fart on my part. I'll correct that now.

Carl

Does that effect the puzzle in any way?

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 5:36 pm 
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PuzzleMaster6262 wrote:
Does that effect the puzzle in any way?


No... just a naming issue. I was thinking helicopter puzzle but for some reason typing dino. My hands weren't following good direction. All the pics above are correct and the naming is now corrected too.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 5:39 pm 
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he just explained how it affects the puzzle, because the mixup cube and helicopter cube have different "jumbling angles",(this probably isn;t the correct vocab to use at this point, but hopefully you get what I mean) the interior puzzle bandages the exterior puzzle...

But this new puzzle is in reality a pure face turning truncated rhombic dodecahedron shape modded into a cube, correct? So now we have 2 almost identical puzzles based on the same geometry that are in reality completely different?

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 5:45 pm 
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How though could this puzzle be different then the Mixup Cube? If it is the same puzzle with all of the cuts extended, how would it get blocked moves that the mixup cube doesn't have?

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 6:02 pm 
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elijah wrote:
he just explained how it affects the puzzle, because the mixup cube and helicopter cube have different "jumbling angles",(this probably isn;t the correct vocab to use at this point, but hopefully you get what I mean) the interior puzzle bandages the exterior puzzle...


The Mixup Cube doesn't jumble and the helicopter cube doesn't support 45 degree rotation except when its being rotated as a whole.

elijah wrote:
But this new puzzle is in reality a pure face turning truncated rhombic dodecahedron shape modded into a cube, correct? So now we have 2 almost identical puzzles based on the same geometry that are in reality completely different?


Yes, the pure twisty puzzle is a face turning truncated rhombic dodecahedron shape modded into a cube. It jumbles at 45 degrees and arccos(1/3) and normal twists at 90 degrees and 180 degrees.

The Mixup Cube isn't a pure twisty puzzle and isn't really that strongly tied to the truncated rhombic dodecahedron shape. I look at it as a 2x2x2 with slidey parts added around the 3 equators. It's doctrinaire shape is the rhombicuboctahedron.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 6:17 pm 
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PuzzleMaster6262 wrote:
How though could this puzzle be different then the Mixup Cube? If it is the same puzzle with all of the cuts extended, how would it get blocked moves that the mixup cube doesn't have?


This puzzle is the Mixup Cube with its cuts extended. But the issue is this puzzle with the extended cuts isn't doctrinaire any more. After a 45 degree turn its NOT back in the same state again. The Mixup Cube is. So we haven't added ALL the cuts needed to make it act like a Mixup Cube. We could rotate a slice layer of this puzzle by 45 degrees and there would be more cuts to extend. As this is a jumbling puzzle this goes on forever and the puzzle is turned to dust TRYING to make it behave as a Mixup Cube.

If the pictures don't show you the problem I'm running out of things to say.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 6:21 pm 
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I thing the truncated rhombic dodecahedron jumbling puzzle should be made, it's has so many different moves that can be made, more than I've ever seen on a twistypuzzle. :D

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 6:25 pm 
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I know this puzzle jumbles and would end up having moves blocked, but if it was rotated EXACTLY like the mixup cube, the only cuts that would get blocked are the the ones the mixup cube doesn't have. If this puzzle could have its pieces glued together to form a mixup cube, just like a 5x5x5 could be glued together into a 3x3x3, why would it be blocked in rotation when the mixup cube isn't?

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 6:27 pm 
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elijah wrote:
I thing the truncated rhombic dodecahedron jumbling puzzle should be made, it's has so many different moves that can be made, more than I've ever seen on a twistypuzzle. :D

I have been thinking the same thing for 4 years :lol: Now that I know about Shapeways, I can actually do something with the designs I made :mrgreen:

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 6:57 pm 
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PuzzleMaster6262 wrote:
I know this puzzle jumbles and would end up having moves blocked, but if it was rotated EXACTLY like the mixup cube, the only cuts that would get blocked are the the ones the mixup cube doesn't have.


On this puzzle the 45 degree slice turn IS a jumbling turn. On the Mixup Cube it isn't. One can infer from this that the Mixup Cube has MORE cuts that aren't obvious on its surface. In fact if you want to call it a pure twisty puzzle we've already seen it's core contains an infinite number of cuts. So its NOT an issue of the Mixup Cube not having all the cuts that are on this new truncated rhombic dodecahedron jumbling puzzle its the other way around. The new truncated rhombic dodecahedron jumbling puzzle only has a finite number of cuts and they get blocked when it tries to copy the moves of a Mixup Cube.

PuzzleMaster6262 wrote:
If this puzzle could have its pieces glued together to form a mixup cube,


It can't. Glue those pieces together and it looks like a Mixup Cube on the outside only. Rotate a slice layer by 45 degrees and the hidden cuts are NOT back in there starting positions. Therefore its still NOT a doctrinaire puzzle. To make it a doctrinaire puzzle on the inside requires its innards to be invariant under this 45 degree turn and that requires you ADD an infinite number of new cuts.

PuzzleMaster6262 wrote:
just like a 5x5x5 could be glued together into a 3x3x3, why would it be blocked in rotation when the mixup cube isn't?


When you glue the 5x5x5 up into a 3x3x3 its still a doctrinaire puzzle. The 5x5x5 has all the cuts that the 3x3x3 has and then some. That is NOT the case here.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 6:59 pm 
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PuzzleMaster6262 wrote:
I know this puzzle jumbles and would end up having moves blocked, but if it was rotated EXACTLY like the mixup cube, the only cuts that would get blocked are the the ones the mixup cube doesn't have. If this puzzle could have its pieces glued together to form a mixup cube, just like a 5x5x5 could be glued together into a 3x3x3, why would it be blocked in rotation when the mixup cube isn't?
The pictures show it fairly clearly. Imagine you take the vertical slice and turn it 45 degrees. Now take the horizontal slice and turn it 45 degrees. Now undo the first move. On the mixup cube, this isn't a problem. On a 'regular' twisty puzzle, this is impossible, unless you unbandage moves. And you'd have to unbandage infinitely.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 7:08 pm 
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theVDude wrote:
On a 'regular' twisty puzzle, this is impossible, unless you unbandage moves. And you'd have to unbandage infinitely.


Oskar could find a way to fudge it. ;)

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 7:14 pm 
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theVDude wrote:
PuzzleMaster6262 wrote:
I know this puzzle jumbles and would end up having moves blocked, but if it was rotated EXACTLY like the mixup cube, the only cuts that would get blocked are the the ones the mixup cube doesn't have. If this puzzle could have its pieces glued together to form a mixup cube, just like a 5x5x5 could be glued together into a 3x3x3, why would it be blocked in rotation when the mixup cube isn't?
The pictures show it fairly clearly. Imagine you take the vertical slice and turn it 45 degrees. Now take the horizontal slice and turn it 45 degrees. Now undo the first move. On the mixup cube, this isn't a problem. On a 'regular' twisty puzzle, this is impossible, unless you unbandage moves. And you'd have to unbandage infinitely.

The puzzle can have the first move undone. Unless we are looking at a different puzzle, there is no difference between the Mixup Cube and the extended cuts version of it.

I opened up a 3d model of the puzzle and it should be able to do the moves I think it can.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 7:42 pm 
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in the second to last picture, I would be wrong. But in the last picture, cuts exist that don't in the other that allow the vertical slice to rotate.

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 Post subject: Re: Puzzles that jumble
PostPosted: Sat Jul 03, 2010 7:54 pm 
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PuzzleMaster6262 wrote:
in the second to last picture, I would be wrong.


This picture?
Attachment:
Core3.png
Core3.png [ 29.06 KiB | Viewed 5100 times ]


PuzzleMaster6262 wrote:
But in the last picture, cuts exist that don't in the other that allow the vertical slice to rotate.


And just to make sure we are on the same page... this picture?
Attachment:
Puzzle3.png
Puzzle3.png [ 114.32 KiB | Viewed 5100 times ]


If we are on the same page... what you are looking at are the cuts as they appear on the surface of the puzzle. You can't tell from looking at just the surface if they go all the way through the puzzle. That's why I tried to show you what is on the inside. Hmmm that gives me an idea. Off to make another picture...

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