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 Post subject: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Sep 13, 2008 2:49 am 
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I was watching an extremely detailed analysis on how to assemble the V-cube 6x6x6 on youtube. I observed that the 6x6x6 has a fully functional 7x7x7 mechanism built into it, albeit totally invisible on an assembled puzzle. Then I looked at my solved Professor Cube sitting next to the computer, and imagined it as a stack of 125 identical cubes. I came up with the notion that the theoretical model of a cube puzzle would operate in a different manner than it's real-world counterparts.

:idea: A perfect N-cube would actually consist of N^3 cubes, each with its own unique position/orientation. As an example, the hypothetical Professor cube model (5x5x5) would be cast from 125 perfect cubes, which rotate about 3 axis. Imagine that this cube consists of 5 layers of 25 cubes, with 5 rows of 5 cubes on each layer. Visible to the outside are the same 98 cubes that we see on the surface of a standard Professor cube. What is left inside this cube model that does not get represented, is a smaller cube of 3x3x3, and at the center of which we find the core. While it is true that the Professor does contain an internal 3x3x3 mechanism, that 3x3x3 moves in conjunction with the corner and center edge cubes, and not with the intermediate layers. This would mean that a total of 26 hypothetical (non-existant) cubes would move about in conjunction with the inner slices of the Professor. In other words, the faces on the hypothetical internal 3x3x3 rotate only when the inner slices of the 5x5x5 are rotated.

My question is, assuming that the Professor is also a supercube (all pieces have an exact position/orientation), once solved, what would the state of the hypothetical inner 3x3x3 be like? I imagine that it would require some fairly involved and sophisticated algorithms simply to rotate a single face of the inner 3x3x3 without disrupting the positions/orientations of the "outer 98". This also means that the centers first method would be a poor approach towards solving it, seeing that once all 12 of the edge trios are aligned, the inner cube is not disturbed throughout the rest of the solving process, meaning that its chaotic state would most likely remain.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Sep 13, 2008 4:08 am 
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How I would go about solving something like this would be to solve the internal 3x3 first. From there you should be able to gather the 5x5 centers fairly easily while retaining the internal solved section. From there simple algorithms can be done to solve the 5x5 edges and continuously return the internal 3x3 to solved. From there it would only be single slice turns to solved.

The only thing that would pose an issue for me would be the '5x5 edge parity'. I'm sure I could come up with something if the case were to show up, but without a good applet or a physical puzzle to tinker with, I'm drawing a blank.

Unless it is normally a 'odd' number of turns is afflicting the hidden 3x3 that causes the 5x5 edge parity which would make sense as people such as Pembo often say, it's a 'permutation' parity, not an orientation parity, while a single slice move would solve the case, that would account for the single hidden 90 degree turn.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Sep 13, 2008 5:08 am 
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If you look at a 3x3 in a mathematical way, all states are either odd or even in terms of quarter turns, because any quarter turn will toggle both corner and edge parity. You see this in blindfold cubing where edges (or corners) can only be solved with 3-cycles half of the time. Since doing a quarter turn on the inner 3x3 means doing a slice quarter turn on the outer 5x5 (and vice versa), and since the 5x5 edge parity [and +center parity, but nobody ever notices that except with supercube solving] is switched when you do a slice quarter turn on the 5x5, an interesting conclusion is reached: solving the inner 3x3 means that there will be no parity on the outer 5x5, and solving the outer 5x5 means that there will be no parity on the inner 3x3 (that is, it can be solved using only corner and edge 3-cycles).

As for solving it, Noah's way works fine (inner 3x3 and then outer 5x5), although the centers of the 5x5 will be inefficient because you always have to realign the inner 3x3 after you do any slice move. Solving the outer 5x5 first also works, though, because you can solve the inner 3x3 using only 3-cycles, that is, with a blindfold method. Many of the familiar 3x3 3-cycles still work: r u' l' u r' u' l' u, e' r u r' e r u' r', r2 d2 r u r' d2 r u' r, and so on actually have no effect on the outer 5x5, even if it is a supercube, but can be used to move around the pieces on the inner 3x3.

Per made a program called CubixPlayer2 that simulated what he called "Super-SuperCubes" (a supercube with a smaller cube inside it and so on) of up to 7x7x7. You can probably still download it from the speedsolvingrubikscube Yahoo! group.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Sep 13, 2008 6:43 am 
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Wow! That was really informative and insightful. It makes perfect sence that the parity of the face turns on the 3x3x3 would affect the parity of the middle slice pieces on the 5x5x5. It just seemed odd that two single pieces could be swapped without messing something up in the puzzle; now I know why. I just did a google search for that app you mentioned, and it pulled exactly one result (known as a "google-whack"), strait from the twistypuzzle forums:
viewtopic.php?t=5719

Tony Fisher wrote:
Mark Longridges recent topic reminded me of the small article I wrote for CFF (the newsletter for the Dutch Cube Club) several years ago. I haven't seen the subject mentioned again so I will repeat my idea here. Forgive me if this is old hat.

Take a puzzle like a 4x4x4. Why do we call it a 4x4x4? Well it appears to be a stack of 64 little cubes in a 4x4x4 grid. We all know it isn't really but what if it was and still moved the same? That would mean there are 8 little cubes hidden in the center. If they were coloured the same as the 4x4x4 exterior then we would have a 2x2x2 puzzle waiting to be solved. Only slice moves on the 4x4x4 would affect it of cause but simply solving the 4x4x4 exterior would not be enough. Taking this idea further to a 7x7x7 for example, not only would you have an internal 5x5x5 to solve but also a central 3x3x3 as well. It is fairly obvious that the only way to play with such a puzzle is on a computer. You could have the movable 7x7x7 next to the 5x5x5 and 3x3x3 which would automatically move when appropriate. I must admit I haven't given the solution to such a puzzle much thought but I would love to see this computer simulation made by someone. Anyone up for the challenge? Below I have posted some photos of how a 5x5x5 would be with it's internal 3x3x3 pictured along side.


So apparently, that same question (maybe worded a little differently) came strait out of the mouth of Tony Fisher, one of the greatest puzzle modders of all time!

I feel proud :wink:

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Tue Sep 16, 2008 3:02 am 
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Hi :-)

Seeing a nxnxn cube as simply a stack of 6-sided cubes is not a very original idea. I might have been one of the first to actually do something about the idea, and trying to see how that would look. The solution is actually very obvious. Solve the innermost cube (2x2x2 or 3x3x3) depending on whether the cube at hand is even or odd size, What then remains is a series of outer shells to complete the cube. These shells must be constructed such that no inner disturbance is allowed. What i cam up with is that the outer shells could be solved with pure commutators. Sometimes one migth want to "shoot a piece" to a better position first. The actual solving is not very interesting after having a few goes. It is however very very easy to get lost and mess up some inner cubies. I know Chris Hardwick played with my Cubixplayer2 simulation and solved the 7x7x7x super-supercube couple of times. I always messed up and then got bored with it. It is a nice challenge however. But more of a concentration "do-no-mess-up" character than a real challenge. Because the solution is actually very simple.

- Per

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Tue Sep 16, 2008 10:41 pm 
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Hi Per - I want to ask you a question about that CubixPlayer program. I've been able to solve the outer and inner cubes, but I can't supersolve them because I can't tell which centers are which. What's the secret to making that visible?

Also, what does the "Explode" feature do?

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Wed Sep 17, 2008 12:49 pm 
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qqwref wrote:
Hi Per - I want to ask you a question about that CubixPlayer program. I've been able to solve the outer and inner cubes, but I can't supersolve them because I can't tell which centers are which. What's the secret to making that visible?

Also, what does the "Explode" feature do?


Hi :-)

There's 2 programs with that name downloadable from the Yahoo group file section. The first was added already in 2004, with my old user. The second added this year is with my current user. The old version had the explode feature disabled. I cannot recall why now :shock: Just download the 2008 version. The explode feature is fully working :-)

- Per

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Wed Sep 17, 2008 2:51 pm 
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Are there any super-cube solvers out there? I'm working on an animation and I want to solve the super cube from a given state and I'd like a fairly optimal solution to keep the animation wieldy. The animation will be of precisely the 3x3x3 inside the 5x5x5.

Thanks,
Carl

P.S. I'll be using the model I made here:
http://twistypuzzles.com/forum/viewtopic.php?p=47425

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Thu Sep 18, 2008 7:46 pm 
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I'm pretty decent at supercubes. Is this a specific state you're talking about or do you want a general method?

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Fri Sep 19, 2008 9:56 am 
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qqwref wrote:
I'm pretty decent at supercubes. Is this a specific state you're talking about or do you want a general method?


Its a specific state... let me explain. I'm wanting to put two sets of initials on a supercube like this:

Image

The specific colors don't mater but I want monochromatic leters on monochromatic backgrounds. The "H" and the "B" should be the same color as they need to be on the same face and both contain the face center. The "T" and the "C" should be on the same face and the background color of the "C" should be the same color as the "T" as that is where the face centers are. And the backgrounds of the "L" and the "N" need to be the same as again that is where the face centers are. The "L" and the "N" don't need to be the same color and I don't think they can be. The 3 faces that aren't seen I don't care about. The animation I want to start from a solved cube and end in this state. So if you can solve the cube from this state I can just run that backwards to get my animation. Pretend the other 3 sides of each is un-stickered and a face is considered solved if its stickered faces are all the same color. Then I can sticker the faces at that point and just run it backwards. And my rendering time is very slow so I'd like to have this in as few moves as possible. It doesn't need to be optimal as proof of that I think would be rather hard but the shorter the better.

If others are interested I might be willing to offer a prize for the shortest solution. I'm far from rich so I can't offer too much money but I could offer something like an animation of your initials getting formed on a cube (if you provided the solution).

Thanks,
Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Sep 20, 2008 12:01 am 
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That's an interesting color scheme, seems like it isn't the BOY scheme (since in that case two of those three colors would be switched). I don't think you can make that particular 5x5 pattern because a 5x5 has fixed centers, so if the red and yellow centers are in their original places the blue one must be too.

You can make something like that 3x3 pattern with R2 U2 R2 B R2 D2 R2 B, though.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Sep 20, 2008 8:17 am 
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qqwref wrote:
That's an interesting color scheme, seems like it isn't the BOY scheme (since in that case two of those three colors would be switched). I don't think you can make that particular 5x5 pattern because a 5x5 has fixed centers, so if the red and yellow centers are in their original places the blue one must be too.

You can make something like that 3x3 pattern with R2 U2 R2 B R2 D2 R2 B, though.


I just made that picture rather quickly to show you the "pattern" I was after. In the text I explain the specific colors don't mater. What I'm after is the letter "H" behind a "B", the letter "C" behind a "T", and a letter, "N" behind a "L" as shown. On any given face the letter needs to be a solid color and the background needs to be a different solid color. That is the only requirement. I'm pretty sure... though not 100% sure that this pattern is possible. I'll try to pull out my 5x5x5 and my 3x3x3 this weekend and see if I can get some specific colors for you if that will help.

Thanks,
Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Sep 20, 2008 5:03 pm 
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I'm pretty sure it's possible as well but the colors on the 3x3 and 5x5 won't be the same, because that is impossible.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Tue Sep 23, 2008 11:59 am 
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Thanks qqwref. That R2 U2 R2 B R2 D2 R2 B sequence works great for the center 3x3x3.

Here is a better set of puctures now with some colors that I think are doable.

Image

Image

Do you think how can come up with a similiar sequence for this super cube?

Thanks,
Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Mon Oct 06, 2008 5:34 pm 
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Ok... I've been thinking a bit more and I have a few questions...

(1) I think I've been calling the puzzle of a 3x3x3 inside a 5x5x5 a super cube. I now realize this is what is typically refered to as a 5x5x5 super cube.

Image

And I see the version of a 3x3x3 inside a 5x5x5 where all the face orientations are important is called the 5x5x5 super super cube. Is there a name for just your standard 3x3x3 inside a standard 5x5x5? The one where the face orientations aren't important?

(2) I'm still after a sequence of moves that will produce the above pattern. I'm going to start playing with Per's program but I was wondering if it records the sequence of moves you make? If not could the program be modified to output an ascii file containing the move sequence made?

(3) I tend to think the most efficient solution would be found by starting from the puzzle in the desired state and assuming the 3 unseen sides of each puzzle were unstickered. Then solve the inner cube and finally solving the outer cube and then finally stickering the unstickered faces. Then to get the state I want I can just play the moves backwards. However I don't think I can use Per's program to do this. Can I?

(4) The other option would be to start from the fully solved cube. And then solve to the state I want. Working on the inner cube first say "R2 U2 R2 B R2 D2 R2 B" gets it to where I want and then finally working on the outer 5x5x5. This obviously won't give me the same move sequence as above as this would produce an animation where the 3x3x3 arrived at its desired state very early in the animation. The above method when played backwards would appear to show the 3x3x3 and the 5x5x5 arriving at their desired states at the same time. If Per's programs can output the move sequence I think I can solve it this way. I just don't think it would be as optimal as the other method.

Anyways... I'll plug away at this until I get something. If anyone beats me to the punch I'll offer a free Void Cube for the first to offer me a move sequence that produces the pattern above. And even if I already have a solution I'll offer a free Void Cube for a more efficient solution. Max 2 Void Cubes to anyone that takes this on and I'd have to limit this to no more then say 5 total Void Cubes to give away (provided I'm able to get them here in the Sates) as I suspect someone could find a solution with 1000 moves and others could peck the solution down 1 move at a time for a very long time. I don't expect to have that problem though as I suspect it will be harder to find people interested in this specific challenge.

Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Mon Oct 06, 2008 5:48 pm 
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Haara wrote:
Johannes Laire wrote:
First solve 3x3 and then 5x5 using only conjugates XYX' where Y is an outer layer turn, like r U r'. The 3x3 remains solved.

Yeah, that will work, but probably there might be a more "speedy" method that solves every part simultaneously. But I guess that would be very hard to actually implement since I think that the number of aglorithms involved would be a bit to high...


I saw this in the thread started by Tony... anyone know of anyone ever started on a set of algorithms for effecient solving of a super-super cube as Haara mentions above? Or any sites specifically for solving super-super cubes? Just trying to learn as much as I can.

Thanks,
Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Thu Oct 09, 2008 3:01 am 
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Hi :-)

Actually my cage method is quite suitable for super-super cubing. But the intuitive first layer solving must be replaced by a more tedious commutator style solving, especially for the edges. Solving the corners with only outer layer (for that shell) turns does not disturb the inner shells already solved.

To "reverse" the challenge try to come up with algorithms that disturb some inner shell while having no visible effect on a normal supercube :)

I know a few and they are all commutator based :P

Per

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Thu Oct 09, 2008 9:28 am 
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perfredlund wrote:
Hi :-)

Hello...
perfredlund wrote:
Actually my cage method is quite suitable for super-super cubing. But the intuitive first layer solving must be replaced by a more tedious commutator style solving, especially for the edges. Solving the corners with only outer layer (for that shell) turns does not disturb the inner shells already solved.

Do you have a site on this "cage method". I'm far from what most here would probably consider an experienced solver. I can solve a 3x3x3 using what I think you are refering to as the first layer solving method but even that I got by memorizing a set of moves from a book. So in the big scheme of things I'd probably still be considered a novice.
perfredlund wrote:
To "reverse" the challenge try to come up with algorithms that disturb some inner shell while having no visible effect on a normal supercube :)
I know a few and they are all commutator based :P

Can you PLEASE share them here?

Thanks,
Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Thu Oct 09, 2008 3:11 pm 
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Hmm, found a pattern that does an R2 on the inner 3x3 while leaving the 5x5 okay: using the old 5x5 notation (lowercase letter = slice move):
r U2 r2 U2 r2 U2 r U2 r2 U2 r2 l' B2 l F2 l' B2 l2 F2 l' B2 l F2 l' B2 F2 U2 (27? moves)
Or one that does an M slice:
(r U2)4 r U2 D2 l' (U2 l)4 U2 D2 (22 moves)
Or an M2 slice:
r2 F2 U2 r2 U2 F2 r2 l2 F2 U2 l2 U2 F2 l2 (14 moves)
There are also the fun little slice-turn-only commutators:
r u r' d r u' r' d' (8 moves)

Anyway wwwmwww your 5x5 pattern is still impossible because of the fixed centers. The URF corner that you show (blue-orange-yellow) means the color scheme of the centers must be the same, but the centers in your diagram are on a white-orange-yellow color scheme, which is impossible. I was able to make a *similar* pattern though. In the Randelshofer notation (applet):
R2 F2 U' R2 B2 L2 D' MR2
U R' U' D' L' D M1R2 M1L2 D' L D U R U'
F' M1U2 WF2 WR2 M1U2 WR2 WF2 F (30 btm)
Of course it messes up the back but I don't think you will mind that. This pattern should leave the inner 3x3 unchanged (I tested it). Doing them both at the same time is a little tricker. I guess before you do the 5x5 pattern you could set up the 3x3 pattern without disturbing the 5x5. So one pattern that does this would be the following. The first part should set up the 3x3 without disturbing the 5x5, and the second should set up the 5x5 without disturbing the 3x3. Here's an applet.
(M1U' M1D' M1R2 M1U M1D MR M1U' M1D' M1R2 M1U M1D MR'
M1R2 F2 U2 M1R2 U2 F2 M1R2 M1L2 F2 U2 M1L2 U2 F2 M1L2
MF2 M1B M1R M1B' M1R' MF2 M1R M1B M1R' M1B'
M1R2 M1B M1R M1B' M1R' MF2 M1R M1B M1R' M1B' MF2 M1R2
MF2 M1B M1R' M1B' M1R MF2 M1R' M1B M1R M1B')
(R2 F2 U' R2 B2 L2 D' MR2
U R' U' D' L' D M1R2 M1L2 D' L D U R U'
F' M1U2 WF2 WR2 M1U2 WR2 WF2 F) (88 btm)

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Thu Oct 09, 2008 4:24 pm 
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qqwref wrote:
Hmm, found a pattern that does an R2 on the inner 3x3 while leaving the 5x5 okay: using the old 5x5 notation (lowercase letter = slice move):
r U2 r2 U2 r2 U2 r U2 r2 U2 r2 l' B2 l F2 l' B2 l2 F2 l' B2 l F2 l' B2 F2 U2 (27? moves)
Or one that does an M slice:
(r U2)4 r U2 D2 l' (U2 l)4 U2 D2 (22 moves)
Or an M2 slice:
r2 F2 U2 r2 U2 F2 r2 l2 F2 U2 l2 U2 F2 l2 (14 moves)
There are also the fun little slice-turn-only commutators:
r u r' d r u' r' d' (8 moves)

Interesting stuff... thanks for sharing. I can't wait to play with these.
qqwref wrote:
Anyway wwwmwww your 5x5 pattern is still impossible because of the fixed centers. The URF corner that you show (blue-orange-yellow) means the color scheme of the centers must be the same, but the centers in your diagram are on a white-orange-yellow color scheme, which is impossible.

Good catch. I never noticed that.
qqwref wrote:
I was able to make a *similar* pattern though. In the Randelshofer notation (applet):
R2 F2 U' R2 B2 L2 D' MR2
U R' U' D' L' D M1R2 M1L2 D' L D U R U'
F' M1U2 WF2 WR2 M1U2 WR2 WF2 F (30 btm)
Of course it messes up the back but I don't think you will mind that.

No I don't care what the back looks like and this pattern looks great! And I wasn't even aware of this website you are using to show me the moves. What a useful tool!!! Thank you!
qqwref wrote:
This pattern should leave the inner 3x3 unchanged (I tested it).

I'll have to take your word for that... I can't see the inner 3x3x3 at the moment but I'll test it out this weekend.
qqwref wrote:
Doing them both at the same time is a little tricker. I guess before you do the 5x5 pattern you could set up the 3x3 pattern without disturbing the 5x5. So one pattern that does this would be the following. The first part should set up the 3x3 without disturbing the 5x5, and the second should set up the 5x5 without disturbing the 3x3. Here's an applet.
(M1U' M1D' M1R2 M1U M1D MR M1U' M1D' M1R2 M1U M1D MR'
M1R2 F2 U2 M1R2 U2 F2 M1R2 M1L2 F2 U2 M1L2 U2 F2 M1L2
MF2 M1B M1R M1B' M1R' MF2 M1R M1B M1R' M1B'
M1R2 M1B M1R M1B' M1R' MF2 M1R M1B M1R' M1B' MF2 M1R2
MF2 M1B M1R' M1B' M1R MF2 M1R' M1B M1R M1B')
(R2 F2 U' R2 B2 L2 D' MR2
U R' U' D' L' D M1R2 M1L2 D' L D U R U'
F' M1U2 WF2 WR2 M1U2 WR2 WF2 F) (88 btm)

NICE!!!! Again I can't see the 3x3x3 but if it looks like what I think it does after this it looks like I owe you a Void Cube and you've more then earned it my friend. I bet I typically take over 100 moves just to solve a 3x3x3 and here you've solved a 3x3x3 inside of a 5x5x5 in under that. I'm very impressed. And a quick question... on that link I see this:
Quote:
Twists: 88 btm, 92 ltm, 147 ftm, 210 qtm

Can you clue me in on all the tla's... Three Letter Acronyms? From context I can make out they are different ways of counting moves but that's about it.

Oh and another question that I've been meaning to ask and this is as good a place as any... is there a standard coloring for these cubes? The red opposite orange, blue opposite white, and yellow opposite green pattern I tried to copy above I pulled off the oldest 3x3x3 in my collection which I think is an original 80's Rubik's Cube. The website you are using is different and I've seen other patterns as well. Not really a big issue but something I've always been curious about.

PM me when the Void Cubes are out and I'll be sure to get you one. Thanks again,
Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Thu Oct 09, 2008 5:00 pm 
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the -tm in the acronyms stands for turn metric. I'm not completely sure what all of the different ones are or what the difference is.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Thu Oct 09, 2008 5:53 pm 
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ftm = Face Turn Metric.
qtm = Quarter Turn Metric.
ltm = Layer Turn Metric.
btm = Block Turn Metric.

The Block and Layer metrics become important on higher order cubes.

For example, giving a half-turn to half a 6x6x6 would yield 1 ftm, 2 qtm, 1 btm, and 3 ltm.


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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Thu Oct 09, 2008 11:10 pm 
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Yeah, what VWP said. Pattern people usually use the block turn metric because a block turn signifies a block of consecutive slices all turned in the same direction, so it basically allows any move you could do all at once to have a count of one.

I think the Randelshofer applet uses the standard 'American'/western color scheme, which is white opposite to yellow, green opposite to blue, and red opposite to orange, and blue-orange-yellow around one of the corners (going clockwise around the corner). Most of the cubes sold recently have that color scheme, although sometimes people replace white with black if they're using a white-plastic cube (since white might be hard to see).

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Fri Oct 10, 2008 8:20 am 
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qqwref wrote:
Yeah, what VWP said. Pattern people usually use the block turn metric because a block turn signifies a block of consecutive slices all turned in the same direction, so it basically allows any move you could do all at once to have a count of one.

Ok... lets see if I got this correct.

ltm = btm*the number of layers that are actually turned.
ftm = btm except in the case where its the middle layer(s) turned in which case ftm = 2*btm.
qtm = btm except when its a 180 degree turn then qtm = 2*btm.

And another releated question about the ShowCubePlayer... I assume the Corner Permutation, Edge Permutation, and Side Permutation info somehow is a complete representation of the state of the cube. I don't quite understand the notation but what is the Order info that is given?

Thanks,
Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Fri Oct 10, 2008 11:04 pm 
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I'm not sure about the exact definitions of the turn metrics there, since I only use btm. If you want you can feed it one or two moves to see how much each counts as in the various metrics.

The order is the number of times the position has to be done before you return to a solved cube again. I know there's a lot of stuff displayed in that applet, but most of it isn't important unless you're trying to construct a pattern.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sun Oct 12, 2008 2:40 am 
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I suppose the same applies for the 2x2x3 inside the 4x4x5. :)

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sun Oct 12, 2008 8:02 am 
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wwwmwww wrote:
And another releated question about the ShowCubePlayer... I assume the Corner Permutation, Edge Permutation, and Side Permutation info somehow is a complete representation of the state of the cube. I don't quite understand the notation but what is the Order info that is given?

The permutation notation is used to describe location and orientation changes of cube parts.


Describing the location of a part:
The location of a part can be described by listing the faces on which the stickers of the part can be seen.
The faces are denoted by the letters: r, u, f, l, d, b (right, up, front, left, down, back).

e.g. r describes the center piece of the right face, fu the edge of the front-up faces and frd the corner of the front-right-down faces.


Describing location changes of parts:
Location changes always happen in cycles. If a part moves into the location of another part, then that part must move as well. This may trigger more location changes until one part closes the cycle by moving into the start location of the cycle.
To describe a cycle, the location of the parts are listed in the sequence of the change. Each cycle is enclosed in brackets. Each location description is separated by a comma.

e.g. (fu,fr,fd,fl) describes, that the edge part at fu moves to fr, fr moves to fd, fd moves to fl, and fl moves to fu.


Describing orientation changes of parts:
When the orientation of a part changes, then it stickers can be seen in a different sequence on the faces of the cube.
The face names in a cycle can be used to describe orientation changes as well. The first face name in a location description denotes where the first sticker goes. The second face name in a location description denotes where the second sticker goes, and so on.

e.g. (ufl,rfu, ...) describes, that the first sticker on face uยป at corner location ufl goes to face r at corner location rfu, the second sticker on face f at corner location ufl goes to face f at rfu, and the third sticker on l at ufl goes to u at rfu.


Describing the orientation change of the last part in a cycle:
To denote an orientation change of the last edge part in a cycle, a + is prepended to the location description of the first part of the cycle.
To denote an orientation change in clockwise direction of the last corner part in a cycle, a + is prepended to the location description of the first part in the cycle. To denote an orientation change in anticlockwise direction, a - is prepended.

e.g. (+fu,fr) describes that the sticker on face f of the last part fr goes to face u of the first part fu, and the sticker r of fr goes to f at fu.


Describing orientation changes of center pieces:
A center piece can have four different orientations. Since the orientation of a center piece from its location name can not be told, a prefix is used. No prefix denotes that the side part is in its default orientation. + denotes that the side part is oriented at 90 degrees in clockwise direction. ++ denotes that the side part is oriented at 180 degrees. - denotes that the side part is oriented at 90 degrees in anticlockwise direction.
An orientation change is denoted by the difference between the orientation of the part at the current location and the orientation of the part at the next location.
To describe the orientation change of the last part in a cycle, an orientation token to the first part in the cycle is prepended.

e.g. (f,+r) describes that the center piece at location f moves to r and changes its orientation by 90 degrees in clockwise direction. The side part at r moves to f and changes its orientation by 90 degrees in anticlockwise direction.

e.g. (++u) means that the upper center piece has been rotated at 180 degrees.
If you do have a cube with marked centers the sequence (U R L U2 R' L')2 will rotate the upper center exactly that way.


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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sun Oct 12, 2008 8:16 am 
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The Visual Order Metric (v) and Real Order Metric (r):
The order line of the applet shows the visual and real order of the current cube state.
The order tells, how many times the current permutation must be applied to the cube, to get the the state before the permutation was applied.

The Visual Order Metric is denoted with the letter v. This metric ignores orientation changes of center pieces.
The Real Order Metric is denoted by the letter r. This metric takes orientation changes of center pieces into account.

e.g. The algorithm F' L' F B' U B U R L U R L' B' R2 U' has the order 6 v, 12 r.
So it has to be applied 6 times to visually get its initial state.
Note, that now the permutation (++l) (++b) is displayed. The cube seems to look perfectly solved, but actually the left and back center pieces are both rotated at 180 degrees.
To get its real initial state, the algorithm has to be applied 12 times instead.


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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Mon Oct 13, 2008 9:40 am 
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Very interesting.... thanks for that Waran. Since I don't recall ever seeing any really large numbers in the Visual Order Metric (v) and Real Order Metric (r) fields it begs the question... is there an upper limit for want any given permutation can have? Or does the upper bounds on the order metric continue to increase as the number of turns in the permutation increases. Let's limit ourselves to the 3x3x3 as I suspect this could be a rather complicated answer... if the answer is even known.

Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Mon Oct 13, 2008 12:11 pm 
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Hi :-)

The highest order for a permutation (disregarding centers orientations) is 1260. The analysis for this takes into account how to break down the 12 edges and 8 corners down into cycles of different orders and then combining them to determine the lcm of the various lenghts. Detailed analysis can be found elsewhere. The centers will surely be restored with this order, so the centers add nothing with this respect.

Per

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Mon Oct 13, 2008 2:48 pm 
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I don't know any general solution to the largest possible order on an NxNxN cube, but I can confirm that 1260 is the highest order for a 3x3x3, and I think the highest possible order on a 5x5x5 (non-supercube) is 978120.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Mon Oct 13, 2008 6:42 pm 
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1260 is not the highest order. 2520 is.

Try this out and tell me what you get :wink:

x
U L D' L' U' L D L'
M' U M' U M' U M' U
U B' U' F2 M2 F2 M2 U B U'

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Mon Oct 13, 2008 11:42 pm 
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Yeah, but if you keep the centers constant then the maximum order is 1260. Orders of 2520 are only possible on 3x3 if you allow the centers to move around too.

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Tue Oct 14, 2008 3:24 am 
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Hi :-)

We already have furmuli for the number of possible permutations of any regular size cube. I believe Chris Hardwick may have been the first to come up with such formuli. I'd be interested in the highest possible order for a permutation on a nxnxn size regular cube. In the first instance i do not care about permutation of centers with same color (no supercubing). Those who like a bit of number crunching might have a go at this :)

Tips: the approach used for 3x3x3 still applies, but there is a much larger number of edges (and centers) to play around with for larger cubes. The corners analysis will stay the same.

Per

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Tue Oct 14, 2008 8:36 am 
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Wow, I've really started a fascinating discussion here. Something I've noticed:

I use the following (good version) algorithm to fix front edge parity on the 4x4x4:
r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2
Image
Source: http://www.speedcubing.com/chris/4speedsolve3.html

This seems to be the cleanest version of the maneuver that I have found. It is totally kosher on the 5x5x5, though it does swap opposite top-inner-face corners on the 4x4x4. I don't know if it does the edges as well on a 5x5x5 supercube; if so then a simple 3x3x3 permutation for half-rotation of Up-face center should suffice.

One might notice that this manuever perfectly swaps an odd number of pairs of intermediate cubies, whether performed on the Revenge or the Professor cubes. Notice what happens when I eliminate all of the outer-face rotations: (r2 l r' r r l' r2) these moves are highly redundant and you get left with this: (r+) So consequently, an odd face rotation occurs on the internal 2x2x2 or 3x3x3, thus satisfying the condition that the "odd parity" issue is not violated :mrgreen: it all makes perfect mathematical sense - you can't make the true model of the supercube complete without considering the theoretical internal cubes as well. Parity situations are possible because the internal cube is not accounted for. And yes, I answered my own question. Cheers...

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Wed Oct 15, 2008 2:27 am 
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wwwmwww wrote:
Are there any super-cube solvers out there? I'm working on an animation and I want to solve the super cube from a given state and I'd like a fairly optimal solution to keep the animation wieldy. The animation will be of precisely the 3x3x3 inside the 5x5x5.

Thanks,
Carl

P.S. I'll be using the model I made here:
http://twistypuzzles.com/forum/viewtopic.php?p=47425


Late reply i know .... Why not simply do a "backwards" solve?? I dont mean to reverse a scrambling video. Just practice doing say a 50-turn scramble backwards. This may not look too realistic for the trained eye. But maybe you don't care? 8-)

Per

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Fri Oct 17, 2008 2:30 pm 
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perfredlund wrote:
Late reply i know .... Why not simply do a "backwards" solve?? I dont mean to reverse a scrambling video. Just practice doing say a 50-turn scramble backwards. This may not look too realistic for the trained eye. But maybe you don't care? 8-)

Per


Hello Per,

I'm not sure I follow. Qqwref has already given me exactly what I needed above. I needed a path from A to B and I can see memorizing a given scramble from A and undoing it to get back to A but that still doesn't get me to point B. Am I missing something?

Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Fri Oct 17, 2008 4:07 pm 
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stardust4ever wrote:
Wow, I've really started a fascinating discussion here.

I couldn't agree more.
stardust4ever wrote:
I use the following (good version) algorithm to fix front edge parity on the 4x4x4:
r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2

Ok I see that here:
http://www.randelshofer.ch/test/ShowCubePlayer/RevengeCube.html?MR2B2U2MLU2MR-U2MRU2F2MRF2ML-B2MR2
stardust4ever wrote:
This seems to be the cleanest version of the maneuver that I have found. It is totally kosher on the 5x5x5,

This?
http://www.randelshofer.ch/test/ShowCubePlayer/ProfessorCube.html?M1R2B2U2M1LU2M1R-U2M1RU2F2M1RF2M1L-B2M1R2
Not 100% sure what you mean by totally kosher in this context.
stardust4ever wrote:
though it does swap opposite top-inner-face corners on the 4x4x4.

You mean the top face centers? There are face centers moved around on the 5x5x5 as well. So I'm not sure I follow.
stardust4ever wrote:
I don't know if it does the edges as well on a 5x5x5 supercube; if so then a simple 3x3x3 permutation for half-rotation of Up-face center should suffice.

This is the inner 3x3x3 of that 5x5x5 super-supercube:
http://www.randelshofer.ch/test/ShowCubePlayer/RubiksCube.html?R2LR-RRL-R2
stardust4ever wrote:
One might notice that this manuever perfectly swaps an odd number of pairs of intermediate cubies, whether performed on the Revenge or the Professor cubes.

The Professor cube shows this after the above move:

Edge Permutation:
(fu1,uf2)
Side Permutation:
(u1,++u3) (u2,++u4) (u5,++u7)

Isn't that an even number of pairs?
stardust4ever wrote:
Notice what happens when I eliminate all of the outer-face rotations: (r2 l r' r r l' r2) these moves are highly redundant and you get left with this: (r+) So consequently, an odd face rotation occurs on the internal 2x2x2 or 3x3x3, thus satisfying the condition that the "odd parity" issue is not violated :mrgreen: it all makes perfect mathematical sense - you can't make the true model of the supercube complete without considering the theoretical internal cubes as well. Parity situations are possible because the internal cube is not accounted for. And yes, I answered my own question. Cheers...

I'm not 100% sure I know what your question was and that may be partly to blame for why I'm having problems following your answer and being a big fan of math and these puzzles I think you are onto something I'd very much love to understand better. Maybe I need a better understanding of exactly how parity is defined in this case.

For example... looking at the inner 3x3x3 I see:

Corner Permutation:
(ubr,bdr,dfr,fur)
Edge Permutation:
(ur,br,dr,fr)
Side Permutation:
(+r)

The move isn't broken down into pair swaps so how is this counted... odd or even?

Thanks,
Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Fri Oct 17, 2008 4:19 pm 
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A few questions about the Randelshofer applet ShowCubePlayer.

(1) Does the person who made this hang out here?
(2) Could they be persuaded to make a super-super cube version?
(3) I did a little digging here http://www.randelshofer.ch/ and I think I see where these aplets can be downloaded but I don't see direct links to the one's we've been using on-line. Is there a page that contains these links and explains some of the notation?
(4) For example http://www.randelshofer.ch/test/ShowCubePlayer/RevengeCube.html?R2MR2 shows up as 2 block turns. What is the notation for doing this as 1 block turn?

Thanks,
Carl

P.S. Just found the answer to number 4 above.
http://www.randelshofer.ch/test/ShowCubePlayer/RevengeCube.html?TR2

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Oct 18, 2008 7:51 am 
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(1) Yes. The player is actually made by two people. The user names in this forum are Waran and rawcoder.
(2) Do you have cubes with textures like this in mind? Super 5x5
(3) The applets used in this forum are exactly the same you can download here (see for the download link 'Applets').
The cube state on the right can be activated with the applet parameter:
Code:
<param name="showInfo" value="true">

(4) I described the superset notation in the 6x6 and 7x7 patterns! thread when the new V-Cube 6 and V-Cube 7 applets have been added in august.


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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Oct 18, 2008 8:28 am 
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Waran wrote:
(1) Yes. The player is actually made by two people. The user names in this forum are Waran and rawcoder.

Wow! Not only is one of the creaters here but he's replying to my questions. Great!! Thank you for these great tools. I wasn't aware of them until this thread and I'm enjoying exploring them very much.
Waran wrote:
(2) Do you have cubes with textures like this in mind? Super 5x5

No, that isn't what I'm talking about. Your current 5x5x5 appet keeps track of the face centers so that really wouldn't convey any new info though it would make it easier to see on the cube what was going on without having to look at the notation.

What I would be interested in seeing is the Super-Super 5x5x5. Your link is a picture of a Super 5x5x5. I envision a Super-Super 5x5x5 applet would show the 5x5x5 puzzle and just to its right would be a 3x3x3. Inner slice turns of the 5x5x5 would show up on the 3x3x3 and outer slice turns would not. And there would be 2 sets of corner, edge, and side permutations to the right of that. One for the outer 5x5x5 and one for the inner 3x3x3. It might even be beneficial to add notation that tracks the orientation of the center 1x1x1.

There could be a similiar applet for the 6x6x6 that shows a 4x4x4 AND a 2x2x2 enclosed. And the 7x7x7 applet would show the enclosed 5x5x5, 3x3x3, and optional 1x1x1.

Do you follow?
Waran wrote:
(3) The applets used in this forum are exactly the same you can download here (see for the download link 'Applets').
The cube state on the right can be activated with the applet parameter:
Code:
<param name="showInfo" value="true">

(4) I described the superset notation in the 6x6 and 7x7 patterns! thread when the new V-Cube 6 and V-Cube 7 applets have been added in august.

Thanks! Looks like I have some more playing to do.

Carl

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 Post subject: Re: Cube Theory: The 3-cube inside the 5-cube
PostPosted: Sat Oct 18, 2008 9:56 am 
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Hi Carl,

You can add multiple applets on the same page. With some JavaScript, or by assigning a special notation for the inner cube, you can let both applets perform the same script.

Cheers,
Werner


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