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 Post subject: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 6:08 am 
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Hi!

This is my new puzzle - Constellation Six.

It's a vertex-turning trigonal dipyramid that has 3 complanar axes of rotation.
Each of the 6 faces has a star image on it and with only 3 possible moves the puzzle gets totally scrambled very easily.

This puzzle has some features that make it somewhat hard to classify.
If you consider Constellation Six as a vertex-turning puzzle, then it's a "normal" puzzle - it always has the same shape.
Only some locked (unturnable) cutting lines make you think that there's still something unusual. Yes, there is - if you consider it an edge turning puzzle, you can say that it constantly changes the shape, bandages some layers and unbandages others. Even the axes change their orientation in space, but the overall shape is "invariant". Can we say it shapeshifts or gets bandaged? You decide.

There are some other mentionable things about this puzzle:
1. Because the small face corner angle is not 36°, but 41.41°, the stars could not have rotational symmetry. And with 36° on the face 90°-turns would not be possible.
2. The ray pieces may play 3 different roles (head, arm, leg) and the small triangle pieces - 2 roles (on big egde or small egde).
3. The circle piece in the center could actually be 13-gon, because each 90° vertex turn turns it by 138.59°, and this repeated 13 times makes it turn by 1801.67°, which is only slightly different from 5 full turn-arounds (5x360° = 1800°).

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 6:10 am 
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Looks really good! Congratulations! :)

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 6:21 am 
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Wow that is crazy! I like it!

-Mark- :)

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 6:23 am 
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NICE! thats amazing =) will you offer it for sell on shapeways?

PS. I would like to see a bigger version tho.

Sharon

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 8:18 am 
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At first sight I just think: There'll be a gear-pyraminx? Well it's not. But looks beautiful. Great Work!


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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 8:20 am 
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Beautiful! I've been wondering what that was ever since I saw the mockup on your Shapeways page. I like the shape of the puzzle, and the way it looks. congratulations on a magnificent design!

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 10:04 am 
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Excellent puzzle, and with only 3 axes of rotation, it looks extremely complicated! A very interesting geometry.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 11:18 am 
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Amazing puzzle and great design! The fact that it only uses 3 axes just makes it better. Also is there any reason why the centers of the stars are circles and not pentagons?

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 11:26 am 
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That stuff at the start of your post is confusing, but the puzzle looks really cool with the stars on the faces. :D

Alex

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 11:49 am 
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Excellent puzzle. Absolutely beautiful. :D

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 12:27 pm 
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Just another challenge :cry:
I should give it up.
Nonetheless: Very cool!


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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 3:17 pm 
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Icredible Super PUZZLE!!!

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 3:22 pm 
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Amazing puzzle. You need to make a master version of this which would force it to be classified as edge and corner turning. Great puzzle!

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 3:26 pm 
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This is one of the most beautiful puzzles I've ever seen.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 3:30 pm 
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May I ask why the center pieces are circles? I know it's only a slight difference, but it would be slightly annoying if it was circlular without having a reason to be.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 3:35 pm 
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I agree with c1829, this is one of the most beautiful puzzles I have ever seen. Please continue your puzzle explorations in this direction!


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 Post subject: Re: Shim's Constellation Six
PostPosted: Mon Jan 17, 2011 3:40 pm 
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This is a very refreshing puzzle. It's hard to solve, isn't it?


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 Post subject: Re: Shim's Constellation Six
PostPosted: Tue Jan 18, 2011 2:34 am 
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SEBUVER wrote:
May I ask why the center pieces are circles? I know it's only a slight difference, but it would be slightly annoying if it was circlular without having a reason to be.

Unfortunalety it was impossible to make the center pieces pentagonal.
Normal pentagram that has a symmetrical pentagonal center has a corner angle of 36°. Such a pentagram would appear if the corner angle on the face were also 36°. But in a trigonal dipyramid that allows to rotate a corner by 90°, this angle is 41.41°.
It's obvious that with each 90°-turn a center piece turns by 180°-41.41°=138.59° - and you can not make a pentagon that replicates itself by 138.59°-turns (a 13-gon does, but that is almost the same as a circle).

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 Post subject: Re: Shim's Constellation Six
PostPosted: Tue Jan 18, 2011 6:50 am 
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Incredible. I needed that video to makes sense of it all, very complicated, but extremely attractive build. Thanks.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Tue Jan 18, 2011 10:28 am 
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This looks really cool to me - congratulations 8-)

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 Post subject: Re: Shim's Constellation Six
PostPosted: Tue Jan 18, 2011 2:21 pm 
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I love the uniqueness of this. All of your puzzles look deceptively simple but they're not at all! Hope to see this on Shapeways soon.


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 Post subject: Re: Shim's Constellation Six
PostPosted: Tue Jan 18, 2011 2:23 pm 
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I fell in love to this puzzle!
It is so beautiful!

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 Post subject: Re: Shim's Constellation Six
PostPosted: Tue Jan 18, 2011 11:25 pm 
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looks very cool but it's really hard to tell what it is or how it moves with the pictures
i dont want to sound picky but i'd appreciate if you put up a few more angles
thanks man and again good job

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 Post subject: Re: Shim's Constellation Six
PostPosted: Tue Jan 18, 2011 11:26 pm 
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I'm going to jump off into theory in a bit so if this need to be in a new thread let me know...

Timur wrote:
It's a vertex-turning trigonal dipyramid that has 3 complanar axes of rotation.

To me its this plane that defines the orientation of the puzzle. As such this IS a vertex-turning puzzle. If you try to look at this as an edge turn puzzle you combine a global rotation of the whole puzzle and a vertex turn.

Timur wrote:
If you consider Constellation Six as a vertex-turning puzzle, then it's a "normal" puzzle - it always has the same shape.

Its a doctrinaire puzzle so it doesn't jumble.

Timur wrote:
Can we say it shapeshifts or gets bandaged? You decide.

It doesn't shapeshift... but what is the definition of bandaged again? Since it does have cuts in locations where you can't perform rotations in the doctrinaire state I'm tempted to say its bandaged.... but I'm not sure. If you say its bandages that implies it can be un-bandaged and if you try to unbandage it and end up cutting it to dust that means it jumbles. Correct? What happens here? Just due to the odd geometry I'd be very surprised if this could be unbandaged. Yet, I've already come to the conclusion that it doesn't jumble!? Oh the head hurts.

We need Bram in this thread.

I'm thinking this puzzle could/should be considered as a fudged puzzle as the geometry doesn't work out exactly. As seen by the need for a circular face center, etc. Maybe its the combination of fudged and bandaged that is keeping this puzzle from being able to be unbandaged neatly.

Timur wrote:
3. The circle piece in the center could actually be 13-gon, because each 90° vertex turn turns it by 138.59°, and this repeated 13 times makes it turn by 1801.67°, which is only slightly different from 5 full turn-arounds (5x360° = 1800°).


Ummm... if we draw a line in a face center. Lets put it in the 12:00 position and then rotate a vertex by 360°. The circle piece is back in its starting position... correct? But are you saying its been rotated by 4*138.59° or 1 full turn plus 194.3°. Is the line we just drew now in the ~6:30 position? I'm a bit lost I think.

Some other questions...

Is this puzzle deep cut? One definition says if all the cut planes meet at single point in the center of the puzzle it's deep cut. Here the 3 planes meet in a line.... not a point. Still based on that definition I'd be inclined to say yes, its deep cut. Think of a 1x2x2. I'd say that has 2 deep cut planes and they meet along a line.

However the other definition of deep cut states that each cut plane divides the puzzle into two isomorphic groups of pieces. Is that the case here? If so it certainly isn't obvious to me. The part that rotates has 25 pieces:

16 small trianglular faces
4 circular face centers
4 edges
1 corner

The other half has 23 pieces that I see:
14 small triangular faces
2 circular face centers
5 edges
2 corners

How can this be considered two isomorphic groups? Yet at the same time how can this NOT be deep cut?

Help...
Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 12:00 am 
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Is there any chance we will see this on Shapeways Timur?

Thanks!

Chris Hemerich


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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 1:48 pm 
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Constellation Six - $ 82.51 | (€ 61.48)
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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 1:55 pm 
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wwwmwww wrote:
Its a doctrinaire puzzle so it doesn't jumble.
I believe that this fantastic novel design DOES jumble. After some turns, the puzzle can get blocked. Unbandaging would then be required. If I read the numbers and angles correctly, infinitely much unbandaging could be done.

Oskar

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 2:00 pm 
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I'm sorry, but I can't see how this puzzle is bandaged or jumbles. Could someone give me a picture of it in the bandaged state?

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 3:16 pm 
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I think the circles are fudged making it not jumble. The moving edges can be considered the same thing as mixup cubes moving center pieces being active and unactive. Both puzzles aren't bandaged and don't jumble yet have unused cuts.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 4:35 pm 
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wwwmwww: this isn't a doctrinaire puzzle - a doctrinaire puzzle always looks the same after every turn when it has no stickers IIRC.


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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 4:38 pm 
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PuzzleMaster6262 wrote:
I think the circles are fudged making it not jumble.

If the puzzle is doctrinaire... which I believe it is... then by definition it does NOT jumble.

PuzzleMaster6262 wrote:
The moving edges can be considered the same thing as mixup cubes moving center pieces being active and unactive. Both puzzles aren't bandaged and don't jumble yet have unused cuts.

The unused cuts on the mixup cube are implied/internal and not seen on the surface. If you examine the mech of a mixup cube these "cuts" don't really exist as there aren't actually pieces that are cut in two by these "cuts". Here its different. Look at this picture...

Attachment:
Bandaged.png
Bandaged.png [ 215.44 KiB | Viewed 7020 times ]


The red lines are the 3 cut planes along which this puzzle can turn. However what about the blue cut plane? It can't turn... so I think its fair to say its blocked/bandaged. However what happens when one tries to unblock/unbandaged all similiar cut planes? I actually haven't tried but unless something very odd happens I don't see how its possible.

In short this puzzle seems to have found a loop-hole in our definitions of doctrinaire, bandaged, and jumbling. And I think it has something to do with the fudging aspects of this puzzle.

However I'm actually more at odds with the definition of deep cut at the moment. Half of me feels it should be and the other half tells me it isn't. I can actually argue it both way. If it's not deep cut is it possible to have a deep cut puzzle with this geometry? If not.... why not?

Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 4:58 pm 
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How difficult do you guys think this would be? To me it looks like it would be fairly difficult although with only 3 turning axises it seems very simple. :roll:

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 5:46 pm 
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wwwmwww wrote:
The red lines are the 3 cut planes along which this puzzle can turn. However what about the blue cut plane? It can't turn... so I think its fair to say its blocked/bandaged....

Carl


I don't think that this is a fair statement at all. Is a starminx bandaged because it does not turn like the starminx II and III and the dino dodecahedron? I dont think so. There are quite a few puzzles out there that have the same visual appearance, but turn differently.

I am going to design this puzzle later on tonight and I will post any findings here.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 5:57 pm 
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Jared wrote:
wwwmwww: this isn't a doctrinaire puzzle - a doctrinaire puzzle always looks the same after every turn when it has no stickers IIRC.


You recall correctly. However if this puzzle looks different after a turn I'm not seeing it. What am I missing?

Also speaking of stickers, you reminded me of something. Lets assume we add a sticker to the circular face center that gives it orientation. If I understand things correctly (not sure I do) then this piece has a continuum of orientation states. All other twisty puzzles that I'm aware of only allow discrete orietation states. This means this puzzle could have an infinite number of states, or if solved from a random starting position could require an infinite number of moves to solve. Since this is a real world puzzle without infinite precision I'm sure you could get to a state you couldn't distinguish from the solved state in a reasonable number of turns... but still this is a very odd property for a twisty puzzle to have.

Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 6:06 pm 
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gingervergo wrote:
I don't think that this is a fair statement at all. Is a starminx bandaged because it does not turn like the starminx II and III and the dino dodecahedron? I dont think so. There are quite a few puzzles out there that have the same visual appearance, but turn differently.

Take the starminix (it can be either versions I, II, or II) and show me two seperate pieces on the surface that don't allow the plane between them to turn. Granted the pieces are different and thus the planes between them are different between the 3 versions but there are no 2 pieces that have a cut between them that is blocked in the doctrinaire state.

This puzzle IS different. That blue cut plane I showed you does actually cut pieces in two and yet it doesn't allow rotation. I'm tempted to say that particular cut is blocked/bandaged. What I'm not sure about is if that makes this by definition a bandaged puzzle? If all bandaged puzzles that don't jumble can be unbandaged with a finite number of cuts I think we may have a few definitions that are now at odds with each other.

Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 6:25 pm 
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wwwmwww wrote:
Is this puzzle deep cut? One definition says if all the cut planes meet at single point in the center of the puzzle it's deep cut. Here the 3 planes meet in a line.... not a point. Still based on that definition I'd be inclined to say yes, its deep cut. Think of a 1x2x2. I'd say that has 2 deep cut planes and they meet along a line.

However the other definition of deep cut states that each cut plane divides the puzzle into two isomorphic groups of pieces. Is that the case here? If so it certainly isn't obvious to me. The part that rotates has 25 pieces:
I would definitely prefer the first definition. The second one seems to have a clear bias against puzzles with an odd number of axis. Under that system, you could never have a deepcut triangular/pentagonal floppy but I think we can all imagine what they would look like.

wwwmwww wrote:
gingervergo wrote:
I don't think that this is a fair statement at all. Is a starminx bandaged because it does not turn like the starminx II and III and the dino dodecahedron? I dont think so. There are quite a few puzzles out there that have the same visual appearance, but turn differently.
...This puzzle IS different.
Here's a better example which I hope we can agree is not bandaged. Like your picture, red cuts are turnable, but the blue one is not.
Attachment:
turnstile.png
turnstile.png [ 119.98 KiB | Viewed 6932 times ]

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 6:34 pm 
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wwwmwww wrote:
PuzzleMaster6262 wrote:
I think the circles are fudged making it not jumble.

If the puzzle is doctrinaire... which I believe it is... then by definition it does NOT jumble.

PuzzleMaster6262 wrote:
The moving edges can be considered the same thing as mixup cubes moving center pieces being active and unactive. Both puzzles aren't bandaged and don't jumble yet have unused cuts.

The unused cuts on the mixup cube are implied/internal and not seen on the surface. If you examine the mech of a mixup cube these "cuts" don't really exist as there aren't actually pieces that are cut in two by these "cuts". Here its different. Look at this picture...

Quote:
3. The circle piece in the center could actually be 13-gon, because each 90° vertex turn turns it by 138.59°, and this repeated 13 times makes it turn by 1801.67°, which is only slightly different from 5 full turn-arounds (5x360° = 1800°).

A non fudged version would not be doctrinaire. The 1801.67 degree angle is not perfectly 1800. Also if the piece was left as a pentagon it would cause the puzzle to jumble by blocking rotations. The fudging fixes this.

As for the mixup cube, the cuts do exist.
Attachment:
Mixup Cube.jpg
Mixup Cube.jpg [ 25.7 KiB | Viewed 6926 times ]

This picture is what each side of a mixup cube would look like if it was sanded down. The hidden pieces between the edges and centers become visible showing the blocked rotations.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 10:42 pm 
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GuiltyBystander wrote:
I would definitely prefer the first definition. The second one seems to have a clear bias against puzzles with an odd number of axis. Under that system, you could never have a deepcut triangular/pentagonal floppy but I think we can all imagine what they would look like.

I'm not sure I follow. A deepcut triangular floppy would looks something like this.... wouldn't it?

Attachment:
TriCut.png
TriCut.png [ 10.08 KiB | Viewed 6888 times ]


Each red cut divided the puzzle up into 2 groups of 3 isomorphic pieces. Please elaborate as I must be missing something.
GuiltyBystander wrote:
Here's a better example which I hope we can agree is not bandaged. Like your picture, red cuts are turnable, but the blue one is not.

NICE!!! That is a much better example. Thank You. So I guess we can agree that a puzzle with a blocked cut (one that doesn't allow rotation) isn't necessarily a bandaged puzzle. So what is the best definition we have for "bandaged puzzle" that we have at the moment?

Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 10:51 pm 
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wwwmwww wrote:
Attachment:
Bandaged.png

In short this puzzle seems to have found a loop-hole in our definitions of doctrinaire, bandaged, and jumbling. And I think it has something to do with the fudging aspects of this puzzle.
Carl


What I believe you're talking about are puzzles with asymmetry... puzzles that, after each rotation, are asymmetrical based on INTERNALS (3x3 mods don't count)

Asymmetry, I feel, belongs in its own class. There is a puzzle I was thinking about but recently threw away on the advice of some pro designers that specifically utilizes asymmetrical turns, and therefore would jumble.

As I realized, this type of puzzle is impossible to build (EDIT: I do have an idea, but it would by nature be very unstable) because , after each rotation, the piece would have to somehow cut into the core. Any asymmetrical puzzle, which does not retain its mechanical shape, I believe would either be very unstable or simply not exist.

The blue line you have highlighted, if I'm not mistaken, can be classified as asymmetry.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 11:10 pm 
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PuzzleMaster6262 wrote:
A non fudged version would not be doctrinaire. The 1801.67 degree angle is not perfectly 1800. Also if the piece was left as a pentagon it would cause the puzzle to jumble by blocking rotations. The fudging fixes this.

Agreed... the non-fudged version of this does deserve looking into. However so far I'm been focusing on the fudged version presented here as its turned my understanding of a few commonly used terms on their ear.

PuzzleMaster6262 wrote:
As for the mixup cube, the cuts do exist.
This picture is what each side of a mixup cube would look like if it was sanded down. The hidden pieces between the edges and centers become visible showing the blocked rotations.

Have you examined the pictures of the mech of the mixup cube Oskar made? You can see a pic here:

http://www.shapeways.com/model/31732/mixup_cube.html?gid=mg

This pieces simply aren't there. At least not in Oskar's design. And speaking of Oskar's design... Oskar have you changed the design of your Mixup Cube? Unless my memory is failing me the picture you have up now shows a different core then the one you had up while I was designing my Thorny Cube. Which is itself a Mixup Cube with the addition of some Circle Cube ideas. Again the pieces you describe are NOT present in my mech either. If you glue the green (circle) pieces to the gray (face centers and edges) pieces and cut the pegs off the core that keep the face centers from rotating you would have a Mixup Cube.

Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 11:49 pm 
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I recently designed my own mixup cube. This is the video. In my version these pieces do exist making the puzzle solid/no fudging needed. It becomes a doctrine puzzle, atleast to the best of my understanding.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Wed Jan 19, 2011 11:58 pm 
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wwwmwww wrote:
I'm not sure I follow. A deepcut triangular floppy would looks something like this.... wouldn't it? Each red cut divided the puzzle up into 2 groups of 3 isomorphic pieces. Please elaborate as I must be missing something.
Yeah, I should have clarified that and I'm kicking myself because saw that question coming. If you think about them in the same way as the corners of the skewb, you'll see that there is kind of 2 types of pieces. These two types of pieces cannot switch places with each other ever. If you also think about the Multi-Triangular Floppy, you'll see that the 6 pieces come from 2 distinct other pieces too. Dunno if you think interpreting like this is "cheating."
Attachment:
triangular-floppy.png
triangular-floppy.png [ 2.59 KiB | Viewed 6846 times ]
I'm not exactly sure what the proper definition of "isomorphic" is and that could easily change you interpretation of a puzzle.
Scratch what I said about the pentagonal floppy though, this logic doesn't apply there.

wwwmwww wrote:
GuiltyBystander wrote:
Here's a better example which I hope we can agree is not bandaged. Like your picture, red cuts are turnable, but the blue one is not.
NICE!!! That is a much better example. Thank You. So I guess we can agree that a puzzle with a blocked cut (one that doesn't allow rotation) isn't necessarily a bandaged puzzle. So what is the best definition we have for "bandaged puzzle" that we have at the moment?
I want to say "a bandage is anything that blocks turning of an axis." But then you need to worry about what an "axis" is and oddballs like the Knob cube where the axes move. It also doesn't cover doing a 45 degree turn on a 3x3x3 which "bandages" the 4 adjacent faces. I don't think my vocabulary is good enough to concisely disallow this. What definition of bandaging were you using?

Boy, Timur is starting to make a habit of starting these weird discussions.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Thu Jan 20, 2011 12:25 am 
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This puzzle certainly is exciting! It's a puzzle just to analyze it! :lol:

My apologies for writing long blocks of text. I'm in new classes this semester that require lots of writing, so maybe I am just in the habit of writing too much... :roll:

I had to look at this for a while before I figured out what was going on. Along the way I also remembered why the new fudged Tuttminxwas confusing me so much. All other "fudged" puzzles are called fudged because pieces that don't "occupy" the same mathematical space are interchangeable. In nealry every example so far (including this one), the "near-miss" pieces are designed to be IDENTICAL, and the mech given wiggle room so that two holes of slightly different shapes can be filled by the same piece. From the Tuttminx picture, these "near-miss" pieces (hex-hex edges and penta-hex edges) are appear to be distinct shapes. In the case of the Tuttminx corners, that are geometrically biased, typical fudging techniques are to design a corner that really is 3-fold symmetry when the mathematical "spot" for it is actually not exactly 3-fold. I am SERIOUSLY concerned that a fully scrambled Tuttminx will lose functionality... :?

But back to this puzzle. In some ways, the peculiarities of this puzzle are very closely related to the spheres on Gelatinbrain and some puzzles mentioned by Carl here, like this one:
Image
They are related because a subset of the rotations on these puzzles form a pure twisty puzzle everyone is comfortable with. However on all these puzzles, the Constellation Six included, rotations can stop at additional points that were previously unattainable. I cannot help but think these types of moves are artificial in a sense. While the gelatinbrain spheres with euclidean rotations CAN be thought of as crosses between a 2x2x2 cube and a little chop, that really does not capture the essence of the types of interactions that are going on. I also am not sure all puzzles designed in this way necessarily can correspond to another in this fashion, for example the ConstellationSix. We might need to reanalyze a few of these and define a new, different type of rotation.....

Let me tell you what I do know about this puzzle. I do know that it absolutely is fudged. In fact, you can draw an "unfudged" version of the puzzle:
Attachment:
ConstellationSixUnfudged.png
ConstellationSixUnfudged.png [ 39.62 KiB | Viewed 6837 times ]

Hopefully you can see where the obvious "near-miss" piece is (there are actually several, but one stands out). The thing about this design is that it DOESN'T work using strict math. The angles, sizes, and shapes of certain pieces are very close, but not perfect. This "perfect" puzzle would be considered quite bandaged and in fact would not scramble very far. However, Shim's design allows for imperfections so that a single sort of "average shape" works in any of the little triangle spots. While this puzzle should technically have 3 distinct small triangle pieces, Shim's design has only one; a total of 30 identical small triangles. This is the fudged aspect of the puzzle. Although the math does not work out exactly, this design does not require it to work out exactly. So from this point on, mathematical calculations are worthless to prove or disprove anything about the puzzle.

This puzzle has only 3 rotating faces, each of which can stop at one of 4 positions, all 90 degree rotations. However, in a rest state, only 2 of the resulting cuts actually line up with any other rotation plane on the puzzle. The other two have been artificially added, exactly like the gelatinbrain spheres. If you take away these extra stopping points you have the following puzzle:
Attachment:
ConstellationSixRebandagedPure.png
ConstellationSixRebandagedPure.png [ 31.01 KiB | Viewed 6837 times ]

This can be described as a single layered Rubik's Ufo or cheese and is a very simple puzzle. The relation between the above picture and the ConstellationSix is exactly the relation between a 2x2x2 and this GelatinBrain Sphere, which is exactly the relation between the 3x3x3 and this GelatinBrain Sphere (which is much easier to understand). These extra moves have been inserted in without disrupting the base puzzles. Well actually in the case of Gelatinbrain's Spheres, they have been inserted perfectly without disrupting the base puzzles while here, that was only possible with fudging.

GuiltyBystander's picture was EXCELLENT. That simple 2D puzzle beautifully shows how in a rest state, some divisions between pieces simply lie dormant in a portion of the puzzle where they can not immediately separate. Imagine a 2D puzzle of this type that looks exactly like GuiltyBystander's picture except with an single, infinite string of circles in a straight line. Then fuse together any pieces that can not immediately be separated in a single move. The result can be thought of as the pure twisty base puzzle. When we go back and add in the extra 2 points the rotations can also stop, we have the original puzzle (repeated infinitely). The same thing is happening with the ConstellationSix. The more I think about it, the more I believe this puzzle can not be precisely defined with our current terms. Likewise Gelatinbrain's spheres and the puzzles in Carl's post referenced above, while techinically forced into our current definitions, are not adequately described. I think we need a new term here describing extra stopping points added into a base puzzle.

As for whether this puzzle is bandaged: there are cuts that are not always accessible. This is clearly true. However there are never any blocked ROTATIONS or MOVES. This puzzle can only be rotated in 3 places (using the vertex turning viewpoint and keeping the orientation fixed in space). The other cuts can be thought of as being stored or set aside. They are not in active play right now. We as a community (or at least I) have never really considered or recognized this fact before even though it is clearly present on tthese 2D puzzles. As GuiltyBystander again pointed out in his picture, although the blue circle is not rotatable, it was never really considered a move in the first place, so intuition does not tell us the puzzle is bandaged. Based on this, I believe these added "stopping points" with "inactive cuts" should never alone count as a reason to call a puzzle bandaged. I really think it is important to make a distinction between moves and cuts here. Bandaged puzzles have blocked MOVES. Jumbling puzzles have moves that allow for succesive, interacting moves, but simultaneously block others. Remembering that the fudging causes all the little triangle pieces to be identical and thus interchangeable, there is no difference in the structure after a 90 degree turn of a face, so if all the stickers were removed, there is essentially only one move and only one position (is this what Bram calls doctrinaire? - I never really understood that, but if so, he has it exactly right). Therefore we can see a MOVE will NEVER be blocked and thus, I believe this should not be considered bandaged. Likewise, I do not believe it should be consiedered jumbling either.

On to the question of whether it is deepcut. I personally decided a long time ago that the quality of deepcut did not only mean a quantitative observation on how deep the cuts of a puzzle are, but rather a limiting condition of what happens when cuts from opposite sides of the puzzle meet. Deepcut puzzles do not have an explicit orientation in space, that is, moves can always be defined from 2 points of view. Construction-wise, this means a standard design would allow the core to move separate from the exterior so gluing is required. Deep cut puzzles have no middle layer, and shallowing the cuts ever so slightly brings the middle layer back in without eliminating any pieces from the deepcut state. Deepcut puzzles also have no limit on shape modding (one reason why Skewbs are such popular mod cores). These reasons are some of the many why I consider order differently than many on the forum: to me a 3x3x3 is a first order, a 5x5x5 is a second order, a 7x7x7 is a third order,etc. A 2x2x2 is a deepcut 3x3x3 and a 4x4x4 is a 5x5x5 where the inner most cuts have reached the limit of becoming deepcut.

One advantage of this viewpoint is found when considering tetrahedral cored puzzles. I think everyone agrees that the pyraminx (let's just ignore the tips) is not deepcut (the planes do not meet at a single point, there is a real-volumed core, etc.). Deepen the cuts a little and the cuts still do not go through the center of the puzzle, but you get new pieces - equivalent to the centers of a Halpern-Meier Pyramid (aka Jing Pyraminx). Now push the cuts all the way to the center of the puzzle (what I guess people would call deepcut?). The resulting puzzle is.... EXACTLY the same. It is STILL a Halpern-Meier Pyramid. Nothing significant happened by making the puzzle "deepcut". So why does this particular observation matter? The answer is it doesn't on tetrahedral puzzles because tetrahedrons don't have opposite faces. There is no cut coming in from the other side to meet in the center and drastically altering the puzzle. Therefore, I don't consider "deepcut" to even apply to puzzles that do not have opposite rotations. So for puzzles like ConstellationSix, Halpern-Meier Pyraminx, MoreMadness, etc. I don't think "deepcut" should even be defined. Because of this, I really think the whole discussion of "splitting the puzzle into isomorphic groups" is unneccessary.... but I certainly don't want to stop you guys :)

Just my 2 cents. Or, actually... considering the obscene length of this post... more like my $3.87 :lol:

Peace,
Matt Galla

PS I know my posts are always long, and I always feel a little guilty about that, but I only write so much because I am so passionate about this stuff and honestly believe I have some pretty good ideas. I guess I will never have the genius to make history altering posts that are less than 5 lines long :roll:


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 Post subject: Re: Shim's Constellation Six
PostPosted: Thu Jan 20, 2011 10:33 am 
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PuzzleMaster6262 wrote:
I recently designed my own mixup cube. This is the video. In my version these pieces do exist making the puzzle solid/no fudging needed. It becomes a doctrine puzzle, atleast to the best of my understanding.

Interesting... does your puzzle have its own thread? I think I missed it. And personally I don't consider Oskar's Mixup Cube design fudged and I consider it a doctrine puzzle too. But clearly there is more then one mech that makes a working Mixup Cube. I'd love to see the details of yours. Do you have a 3D pdf you'd be willing to share?

Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Thu Jan 20, 2011 10:52 am 
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GuiltyBystander wrote:
I'm not exactly sure what the proper definition of "isomorphic" is and that could easily change you interpretation of a puzzle.

Isomorphic as it applies to deep cut twisty puzzles, simply means there exists is a 1-to-1 mapping between the pieces on one side of the deep cut to the other side. Your triangular floppy DOES have this propery and IS deep cut. The two blue pieces on one side map to the two red pieces on the other and the one red piece maps to the one blue piece on the other. Make a shape mode of your triangular floppy in the shape of the star of David and you'll see these pieces are the same... they just exist in two orbits. Your 1-to-1 mapping doesn't have to exist within each orbit but to the puzzle as a whole.

GuiltyBystander wrote:
I want to say "a bandage is anything that blocks turning of an axis." But then you need to worry about what an "axis" is and oddballs like the Knob cube where the axes move. It also doesn't cover doing a 45 degree turn on a 3x3x3 which "bandages" the 4 adjacent faces. I don't think my vocabulary is good enough to concisely disallow this. What definition of bandaging were you using?

In my head I was thinking any puzzle with a bandaged cut plane is a bandaged puzzle. I.e. all cut planes must go all the way through the puzzle in its doctrinaire state for it to be un-bandaged. Not sure what is the commonly accepted definition here but I'm sure some have been posted before.

GuiltyBystander wrote:
Boy, Timur is starting to make a habit of starting these weird discussions.

Agreed... he and Oskar are constantly forcing us to re-think our understanding of twisty puzzles. Just when you think you see some nice order to things they throw some chaos back into the mix.

Carl

P.S. Ok.... Time to read Matt's post...

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 Post subject: Re: Shim's Constellation Six
PostPosted: Thu Jan 20, 2011 11:53 am 
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Allagem wrote:
Remembering that the fudging causes all the little triangle pieces to be identical and thus interchangeable, there is no difference in the structure after a 90 degree turn of a face, so if all the stickers were removed, there is essentially only one move and only one position (is this what Bram calls doctrinaire? - I never really understood that, but if so, he has it exactly right).

Wow!!! Lots of nice ideas there Matt. Thanks. I'll try to get back more when I have time but I wanted to answer this question....

Yes, this is exactly what I've takens Bram's definition of doctrinaire to mean.

And one other quick thought...

As far as deep cuts splitting the puzzle into isomorphic groups... let's assume all puzzles can be shape modded into a sphere. Is that true for Shim's Constellation Six? If we cut the sphere in half through its center the two halfs are isomorphic... interchangeable. As we continue to add cuts through the center I feel this property should always be true. Sphere_f0 over at Gelatinbrain appears to be isomorphic as best I can tell. I haven't been able to play with any of the puzzles there since upgrading to Windos 7. Regardless of rather this concept is needed for twisty puzzles or not I still find it an interesting property of geometry. Consider the deep cut Halpern-Meier Pyraminx versus the shallow cut Halpern-Meier Pyraminx you describe. Sure the puzzle is the same but the geometry is different. In one the cuts define a volume you could call the core and in the other the volume of this piece goes to zero. It's only with this piece out of the picture that the pieces on each side of the cut can be divided into two isomorphic groups. So it appears to me that Shim's Constellation Six has broken a rule of geometry... not just twisty puzzle terminology. How do you cut something in half and NOT form two isomorphic groups? I'm sure it has something to do with fudging being more the just simple cutting... or does it? The un-fudged version you posted... are it's two halves isomorphic? If not then it must be due to these "inactive cuts" not going all the way through the puzzle.

That's all the time I have for now,
Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Thu Jan 20, 2011 11:56 am 
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On the subject of what "deep-cut" means, my definition is nearly the same as Matt's. That is:

On a deep-cut puzzle, the effect of any possible twist can be achieved through a different twist + a puzzle re-orientation.

Using this definition we don't have to worry about if the twists are opposite each other or if one is a vertex twist and another is a face twist, etc. On the canonical deep-cut puzzles such as the 2x2x2, Skewb, Pentultimate, etc, the definition seems non-controversial.

Put another way, everybody agrees a Skewb is deep-cut. The shape modes (Skewb Diamond, Skewb Ultimate) are also deep cut. So why is the tetrahedral shape mod not also deep-cut? Using this definition avoids the "core volume problem" on the H-M Pyramid as long as you consider a vertex twist is equally as valid as a face twist. That is, it doesn't matter if the core has volume or if all of the cuts meet at a point. As long as the external puzzle has this twist indistinguishability problem (both visually and functionally) it is deep cut.

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 Post subject: Re: Shim's Constellation Six
PostPosted: Thu Jan 20, 2011 12:15 pm 
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bmenrigh wrote:
On the subject of what "deep-cut" means, my definition is nearly the same as Matt's. That is:

On a deep-cut puzzle, the effect of any possible twist can be achieved through a different twist + a puzzle re-orientation.


NICE!!!! I actually like that one. Using it Oskar's Slice Kilominx is deep cut. I've been arguing that it should be considered deep cut since here but I don't think I convinced anyone else.

Still even in that puzzle the two halves are isomorphic. So let me ask the question differently... under what circumstances are the two halfs of a deep cut twisty puzzle NOT isomorphic? There is a method to my madness. See if we can understand how Shim's Constellation Six broke that mold maybe there are other deep-cut non-isomorphic puzzles waiting to be discovered.

Carl

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 Post subject: Re: Shim's Constellation Six
PostPosted: Thu Jan 20, 2011 1:47 pm 
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Argh!
What a discussion!
I am afraid you have lost me. Except one point:
wwwmwww wrote:
So what is the best definition we have for "bandaged puzzle" that we have at the moment?
Bram defined bandaging here:
viewtopic.php?f=1&t=11126&p=171260#p171260
Interesting about that definition is that this makes the FusedCube a non-bandaged puzzle since he acknowledged my assumption here
viewtopic.php?f=1&t=17707&p=218378#p218378
that the FusedCube fullfills his definition of doctrinaire.

I say:
doctrinaire -> See Brams definition
restricted (easy definition) -> a subgroup of a doctrinaire puzzle where restrictions do NOT depend of permutation.
restricted (more precise Definition) -> a subgroup of a doctrinaire puzzle which generators can be written as a subset of the doctrinaires generators. Example: FusedCube [U,L,F] is a subset of [U,D,L,R,F,B].
truly bandaged -> a subgroup of a doctrinaire puzzle where the allowed moves depend on the permutation.
bandaged -> truly bandaged plus restricted

Please note:
Bandaging(in terms of building) -> gluing pieces together.

To complete the confusion:
A restricted puzzle can be made by bandaging(in terms of building) a doctrinaire one. :shock:


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 Post subject: Re: Shim's Constellation Six
PostPosted: Thu Jan 20, 2011 1:49 pm 
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The fused cube fits the definition yet isn't isomorphic.

Oskar a while ago I thought said something about how his mixup cube wasn't a "pure" puzzle. Also I made a thread about mine in the puzzle building and modding section.

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