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 Post subject: Number of permutations of the Complex3x3x3
PostPosted: Fri Aug 13, 2010 12:07 pm 
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This is some kind of late follow-up to
viewtopic.php?p=191376#p191376
and this (both from the same thread)
viewtopic.php?p=191464#p191464

Today I had the idea to calculate the number of permutations of what is called "complex 3x3x3" in that thread. I should mention that I calculated it for the Super Complex3x3x3 which means orientations for every piece is taken into consideration.
The well known 3x3x3 has 88580102706155225088000 permutations.
The Complex 3x3x3 has 261873301579133051001433349178932006108528640000000 permutations.
The main point for me is that these pieces indeed make a difference. They are not redundant with the exception of the piece UD.
If anybody is interested in the underlying GAP-file, I can post it.


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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Fri Aug 13, 2010 12:13 pm 
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I don't have too much knowledge of this sort of things, but I'm actually taking a class at uni involving permutations, symmetry and group theory in a few months. So I would like to see that file posted, as I might be able to understand and learn from it later.


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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Fri Aug 13, 2010 5:28 pm 
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Nice, I took some interest in that thread (I didn't post because I wasn't a member yet), and I may try to see if I can calculate the same answer. I also thought of trying to use a circle 5x5x5 with 2 circles to make a physical version (using the fact that it can be emulated with a 5x5x5 multicube), but I have little experience with CAD and I moved on to other cube things before I figured out the mechanism. I even made a paint file showing where the circles could be to bring each of the 10 pieces of a 5x5x5 multicube to the surface. I wonder if anyone else here is interested enough and has the skill to actually make it.

Matt


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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Sun Aug 15, 2010 2:09 am 
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@Matt: Good luck there.
For those who are interested I have attached the file. The extension should be renamed to .g


Attachments:
Complex3x3x3.txt [6.88 KiB]
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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Tue Oct 05, 2010 10:07 am 
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How did I miss this thread!!!! I know how... next to zero free time since starting my new job is the answer. Anyways, sorry for the bump but I HAD to comment. This thread deserved more attention then it got.
Andreas Nortmann wrote:
The well known 3x3x3 has 88580102706155225088000 permutations.
The Complex 3x3x3 has 261873301579133051001433349178932006108528640000000 permutations.
If anybody is interested in the underlying GAP-file, I can post it.
I just downloaded the GAP file... and I'm still not sure what I have. What is GAP? And how do I learn how to use it. The ascii file looks a bit like my POV-Ray code so I think I have a chance of making sense of it.
bobthegiraffemonkey wrote:
I even made a paint file showing where the circles could be to bring each of the 10 pieces of a 5x5x5 multicube to the surface. I wonder if anyone else here is interested enough and has the skill to actually make it
A physical Multi5x5x5 is doable. That is talked about a bit here:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=14868
Adding the 5x5x5 face centers while keeping the center 1x1x1 and 3x3x3 face centers will be hard but I think it could be done. Forcing this 5x5x5 Multicube to then act like a Complex3x3x3 would be almost impossible. I'd love to be proven wrong... but I suspect we'll see a simulated Complex3x3x3 that can be played with on Gelatinbrain or elsewhere before we see a physical Complex3x3x3 or even a Super-Multi5x5x5.

Carl

P.S. 5x5x5 Multicube = Multi5x5x5. I just realized I used both notations above.

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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Tue Oct 05, 2010 10:34 am 
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Next time I will point you towards such a thread if you don't answer after some time...
wwwmwww wrote:
I just downloaded the GAP file... and I'm still not sure what I have. What is GAP? And how do I learn how to use it.
You read this
http://www.gap-system.org/
http://www.gap-system.org/Doc/Examples/rubik.html
and afterwards you know almost as much as I do.


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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Tue Oct 05, 2010 11:00 am 
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Andreas Nortmann wrote:
]You read this
http://www.gap-system.org/
http://www.gap-system.org/Doc/Examples/rubik.html
and afterwards you know almost as much as I do.
Thanks!!! That helps alot and actually doesn't look too hard to use.

Carl

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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Tue Oct 05, 2010 1:44 pm 
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wwwmwww wrote:
bobthegiraffemonkey wrote:
I even made a paint file showing where the circles could be to bring each of the 10 pieces of a 5x5x5 multicube to the surface. I wonder if anyone else here is interested enough and has the skill to actually make it
A physical Multi5x5x5 is doable. That is talked about a bit here:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=14868
Adding the 5x5x5 face centers while keeping the center 1x1x1 and 3x3x3 face centers will be hard but I think it could be done. Forcing this 5x5x5 Multicube to then act like a Complex3x3x3 would be almost impossible. I'd love to be proven wrong... but I suspect we'll see a simulated Complex3x3x3 that can be played with on Gelatinbrain or elsewhere before we see a physical Complex3x3x3 or even a Super-Multi5x5x5.


I have already figured out how to have the face centers for 5x5x5, 3x3x3 and 1x1x1 in the same puzzle, as I have said I have all 10 pieces visible: 6 from the 5x5x5, 3 from the 3x3x3, and the 1x1x1. I just didn't get around to making the mechanism, and I have other projects I am working on just now so I will let someone else try and make it. I don't know how to add images to a post though, or I would show the result. I will email to images to Carl and he can add them for me? Bandaging the puzzle to turn like a complex 3x3x3 would be tricky, but I have a rough idea which might work. Imagine if there were 3 rods which could slide between each pair of centers (obviously they can't all go through the very center of the puzzle, they would need to be offset a little) which have some sort of blocks attached which can block pieces and thus bandage layers together (sort of like an eastsheen 4x4x4). One end of each rod would be sticking out of either center at any one time, and it can be slid backwards and forwards as necessary to bandage the layers on either side of the puzzle as necessary in order the emulate the complex 3x3x3. I am not sure if this will work, or which method of bandaging into a complex 3x3x3 will be best suited for the task, but I'm sure someone here can figure it out.

Matt

PS. Did anyone see my posts here? I didn't seem to get any feedback.


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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Tue Oct 05, 2010 10:42 pm 
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bobthegiraffemonkey wrote:
I will email to images to Carl and he can add them for me?
Matt

Here is the image Matt emailed me.
Image

So some questions. I assume this is the top face of a Multi5x5x5. If I turn the 3rd layer down the black piece should turn as its in the center layer. So should this piece be a circle?

If I just turn the 2nd layer down the gray and both shades of red pieces should turn. They are in the 3x3x3 face layer off in this direction. Your pic seems to show orange and yellow pieces moving but the brighter red pieces appear to be in a different circle.

All the other colors should move by themselves when just the outer layer is turned and again I don't see how that is accomplished.

Confused as I'm now not sure if I'm looking at this correctly,
Carl

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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Wed Oct 06, 2010 8:13 am 
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wwwmwww wrote:
bobthegiraffemonkey wrote:
Image

So some questions. I assume this is the top face of a Multi5x5x5. If I turn the 3rd layer down the black piece should turn as its in the center layer. So should this piece be a circle?

If I just turn the 2nd layer down the gray and both shades of red pieces should turn. They are in the 3x3x3 face layer off in this direction. Your pic seems to show orange and yellow pieces moving but the brighter red pieces appear to be in a different circle.

All the other colors should move by themselves when just the outer layer is turned and again I don't see how that is accomplished.

Confused as I'm now not sure if I'm looking at this correctly,
Carl


Thanks for posting that for me Carl.

Yes, you are looking at it wrong: just as the pieces which correspond to the centers on a circle 3x3x3 don't appear on the same face as the center they are equivalent to, pieces which correspond to one face might be shown on another face (for example, all three types of 5x5x5 center pieces appear on an adjacent face in this puzzle to their usual position on a normal 5x5x5). If the face shown is U, the green pieces shown correspond to some of the X-centers of F, R, B and L.

The right side of the picture shows the surface pieces of the 5x5x5, 3x3x3 and 1x1x1 sections of the puzzle, and equivalent pieces have the same colour throughout the pic. On the left we have a circle 5x5x5 with two circles. Pieces inside the inner circle move with the third layer, if the arms of the spider mech for the puzzle don't rotate you have the necessary bandaging. Pieces between the two circles move with the second layer. As far as I can tell, the only way to achieve this is to make the light red pieces (3x3x3 corners) one physical piece, since they are present on the second layer on two axes, and between the two circles (which corresponds to the second layer) of the other axis. Pieces outside both circles turn with the face. If you examine how each piece is affected by this definition, you will see that all pieces on the right side of the picture are represented by the circle 5x5x5. To mark the orientation of each piece will require super-stickering of some sort, though as far as I can tell it is only a problem for the dark red pieces (3x3x3 edges).

Matt


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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Wed Oct 06, 2010 9:34 pm 
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bobthegiraffemonkey wrote:
Thanks for posting that for me Carl.
You are MORE then welcome. Thank YOU!
bobthegiraffemonkey wrote:
Yes, you are looking at it wrong
WOW!!! When I read this earlier today it was like a eureka moment. Not only did I see the Multi5x5x5 before my eyes... I think I saw how to make it too. Arg!!! And I'm already trying to get 2 other puzzles up on shapeways with what little I have in the way of free time. I know now this will be the 4th complete puzzle I'll design in POV-Ray. Bonus points if you can name my other 3.
bobthegiraffemonkey wrote:
To mark the orientation of each piece will require super-stickering of some sort, though as far as I can tell it is only a problem for the dark red pieces (3x3x3 edges).
Yes, its only the 3x3x3 edges that have a problem but its NOT with orientation. You have sets of pieces that will equate to four edge 3x3x3 cubies which are stickered on opposite faces, so the 4 in each slice layer will look identical. You can tell if each is in the correct orientation but as you can't tell them apart you won't know if you have the correct placement.

NICE!!!!
Carl

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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Wed Oct 06, 2010 10:24 pm 
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Hmmm I still have a feeling it is impossible to design a working, physical Complex 3x3x3.

Think of it conceptually. No matter what, there will be a piece that is moved by both U and D, right? That means either a slice move is physically impossible or must cause this particular piece to move in the angular sum of the two turns that are actually happening (U and D' simultaneously). This would mean that some pieces are FORCED to move twice the speed as a slice during a slice move. The only think I can think of to make this happen is some sort of gear system. But then we would also need both U and D available separately (in addition to the other similar pieces regarding L and R, and F and B).

Any way you approach this you have this tremendous design challenge. While I can't say with 100% certaintity that this is downright impossible, it certainly seems like it. I would be VERY curious to hear any idea Matt (or anyone else) may have. :)

Nevertheless, it looks like you have a good start on the idea Matt, I wish you the best of luck! 8-)

Peace,
Matt Galla


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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Thu Oct 07, 2010 11:44 am 
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Allagem wrote:
Hmmm I still have a feeling it is impossible to design a working, physical Complex 3x3x3.
I'd stop short of saying impossible. But a physical Multi5x5x5 I think is doable and I think I have an idea that will work... need some time to play with POV-Ray.

As for the Complex 3x3x3 I think its possible but not with typical techniques and its probably not practical. Think of a normal 3x3x3 that acted as a controller for a few other 3x3x3's. The solver only manipulates the normal 3x3x3 controler and the imaginary pieces are present on the other 3x3x3's (or maybe a 5x5x5) and they're being moved by servos. The controller only needs to accept face turn inputs so a slice turn would be two sequential face turns. They don't have to be done at the same time. And the objective of solving the Complex 3x3x3 becomes one of solving all of these puzzles at the same time while just manipulating the normal 3x3x3.

Can something like this be built? I think the answer is yes and I really don't think it would be that hard to do but it would be expensive. The 3x3x3's would probably be rather large so they had room for the electronics. And I free much of the "fun" of solving it would be lost doing it this way.

If I'm able to design a working Multi5x5x5 using typical techniques maybe I'll have a better idea then.

Carl

P.S. Would a 3x3x3 with a USB cable coming out of it that once plugged into a PC allowed the solver to play with a Complex 3x3x3 on screen count as a "physical Complex 3x3x3"? Make that a 3D screen and have the view track the 3x3x3's orientation so to see the back of the Complex 3x3x3 you just rotate the 3x3x3 in your hand. And while we are at it let's go bluetooth and drop the USB cable. It may not be totally physical but I actually like this idea even more...

A physical Complex 3x3x3 if it could be built with typical techniques might look like a Multi5x5x5 and it might not be obvious one is really playing with a type of 3x3x3. The bluetooth 3x3x3 controller idea in that sense I might like more then a physical puzzle. In 20 years, maybe less, you may be able put a holographic projector in one of the face centers of the 3x3x3 controler and not even need the PC and screen.

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 Post subject: Re: Number of permutations of the Complex3x3x3
PostPosted: Thu Oct 07, 2010 2:44 pm 
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Ok, I've thought about the bandaging required a little more, and it seems like I have a rough idea which could work. Sticking with my idea of 3 sliding rods going through each axis of the core, what we need is 3 blocks: one in the middle which can somehow bandage the middle layer to either adjacent layer (sliding the rod will change where the block is, and thus what layers are bandaged), and one at either end which bandages the two outer layers together, and one of them to their adjacent layer. This means that we can only turn the second and third layers at any one time. A slight issue is that the two types of block have to work separately, ie. if the rod is one solid piece then nothing can move. The solution (I think) is to make the middle block able to rotate around the rest of the rod. I'm thinking that the layers might have blocks built into them which collide with the blocks from the rods.

I may be getting mixed up and this won't work. It might be a theoretical solution but be completely useless in practice. I will leave it to people who know more about designing mechs than I do to figure that one out. For now, I will simply throw out half-baked ideas and hope someone turns them into something which actually works!

Matt


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