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 Post subject: 4d crazy cubesPosted: Fri Dec 16, 2011 11:27 am

Joined: Mon Jan 03, 2011 7:19 am
Hello everyone.

I was playing with 3x3x3x3 cube and noticed that it isn't actually 3^4. All moves we are making happen within 3-sphere. There's 1 tesseract left inside, and the bigger order puzzle we get the bigger number of tesseracts are left that way. It is similar to the situation in 3d when we have 3x3x3 cube but 26 movable pieces, 4x4x4 but 56 pieces. It can be easily solved by adding circles on faces, such as those here. I think same solution can be applied to 4d puzzle. Instead of circles on faces we would add spheres inside cells. That way 3d projection doesn't limit us and we can see those internal pieces. Such idea could be used also in other ways. Here is visualization of simplest crazy tesseract which shows how I imagine it.

Because I haven't found any software that simulates such a puzzle and don't have enough coding skills, I'm leaving the idea here so maybe someone will be interested in making an applet.

BTW, I was wondering if blue and green spheres should be fixed when making move like in posted gif, however in my opinion all surrounding cells should behave same. It could be also that my intuition tricks me here.

What do you think about it?
Krystian

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 Post subject: Re: 4d crazy cubesPosted: Fri Dec 16, 2011 11:54 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Very Very Cool!!!! Looks to me like you've started down the path of the multi-4D puzzles.
It could be also that my intuition tricks me here.
I don't trust my intuition when it comes to higher dimentions either. I'm sort of surprised there appears to be an inside to the 2x2x2x2. You add circles to a 2x2x2 and you don't really add any new pieces... unlike the case with the 3x3x3 which makes the center cubie visable.

Very interesting... and very curious to see where this goes,
Carl

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 Post subject: Re: 4d crazy cubesPosted: Fri Dec 16, 2011 12:47 pm

Joined: Mon Jan 03, 2011 7:19 am
wwwmwww wrote:
I'm sort of surprised there appears to be an inside to the 2x2x2x2.

Actually, there's no inside. Maybe I wasn't clear enough. It's same situation as in Pocket Cube. You have only 8 corners. In case of hypercube 16 'corners'.
As multicube I understand every cubic puzzle which has another puzzle(s) with smaller order within it and pieces of inside puzzle are made visible by adding circles on the faces. For example mentioned above crazy 4x4x4 or your Real5x5x5 Cube. As such multicubes are subset of crazy cubes which have circles on faces. Let's take crazy 4x4x4 II. It has additional pieces that don't have representational cubes inside. It is crazy cube, but not multi cube. The puzzle I've posted in animation it's also crazy but not multi, the simplest case.
Internal hypercubies starts from order=3. Just as in 3d.

Krystian

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 Post subject: Re: 4d crazy cubesPosted: Fri Dec 16, 2011 3:14 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Actually, there's no inside. Maybe I wasn't clear enough. It's same situation as in Pocket Cube. You have only 8 corners. In case of hypercube 16 'corners'.
In that case the puzzle you have shown should be equivalent to a 2x2x2x2. The pieces inside the sphere should behave just as some of the pieces outside the sphere and you haven't introduced anything new. At least with this particular puzzle. I do believe you've shown a way to bring out the insides of the higher order NxNxNxN puzzles. So the method used to construct this is certainly new and I believe has great value.
As multicube I understand every puzzle which has another puzzle(s) with smaller order within it and pieces of inside puzzle are made visible by adding circles on the faces. For example mentioned above crazy 4x4x4 or your Real5x5x5 Cube. As such multicubes are subset of crazy cubes which have circles on faces.
Actually its the other way around... atleast for the NxNxN puzzles. The crazy (or circle) cubes are a subset of the multicubes. The multi-NxNxN puzzle contains all the possible real pieces of that order puzzle. The circle puzzles can contain all the real pieces too but many just contain some of the real pieces particulary when you get to the 5x5x5 and greater puzzles. This isn't always true for puzzles with the non-cubic geometry as the circles can introduce virtual pieces which aren't present in the 'multi' version.

This also isn't true for the "crazy plus" puzzles. They are subsets of bandaged higher order multi-NxNxN puzzles. I discuss that here:

http://twistypuzzles.com/forum/viewtopic.php?f=1&t=21467

Let's take crazy 4x4x4 II. It has additional pieces that don't have representational cubes inside. It is crazy cube, but not multi cube.
Not true. Look at this picture:

The piece which you think is new is actually a 4x4x4 Face X-center. True it's not inside but its there on the outside of a normal 4x4x4. You've just stickered different faces of the cubie.
The puzzle I've posted in animation it's also crazy but not multi, the simplest case.
Internal hypercubies starts from odder=3. Just as in 3d.
Well the normal 2x2x2x2 is a multi-2x2x2x2 if there is no inside and I believe your puzzle is equivalent. I believe if you solve the outside 2x2x2x2 then your new volumes inside the spheres will also be solved. However go to the 3x3x3x3 and the 4x4x4x4 and I do believe you'll be introducing new pieces. If the 5x5x5x5 is like the 5x5x5 you'll need 2 concentric spheres inside each (4D equivalent of 3D term "Face") to make the Multi-5x5x5x5.

Carl

P.S. I don't use "Multi-" to just mean show the pieces of the puzzle inside. I mean show all the possible volumes inside 3D space created by the cut planes apparent on the surface of the puzzle. A multi-megaminx if I were to ever use that term would contain the pieces of a master pentultimate. Those pieces however would be in the space "outside" the megaminx. To avoid that confusion I use the term face turn multidodecahedron. Another example... the Multi3x3x3 has 27 pieces and contains the 1x1x1 core cubie. Not sure everyone here actually considers the 1x1x1 a "puzzle".

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 Post subject: Re: 4d crazy cubesPosted: Fri Dec 16, 2011 7:59 pm

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
You're not the first one to think of these, I've had them in my head for a while, and I would be surprised if other people hadn't. There's a couple of interesting variations which have been mentioned before for circle cubes but would be hard to physically implement, but would be fairly easy to apply to 4D in a sim. I also have a couple of other puzzles I want to simulate, so I intend to learn Java when I have the free time and implement them. The main other idea is from 3D (mentioned in the thread about how to visualise pieces of a complex nxnxn, I gave a very undeveloped description there which I will expand on when I'm ready to), but it can be extended to 4D and I intend to do that too in a sim. You might have to wait a little while, but I've got some crazy and awesome 4D puzzles to play with on my (rather long) to-do list. In the meantime, there's already several insane and difficult 4D puzzles already implemented, those should keep you entertained for a decade or two .

@Carl: In case you were going to ask, I will be implementing a complex 3x3x3x3 as a bandaged multi-5x5x5x5 with spheres. I still want the complex 3x3x3 to physically exist, and also want to eventually own one.

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