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 Post subject: Simulating Complex Puzzles
PostPosted: Fri Aug 16, 2013 2:01 pm 
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Hey guys,
I recently figured out a way to simulate complex twisty puzzles using Wouter Meesen's Ultimate Magic Cube software. Unfortunately, it does have its limitations:
1) The puzzles are all super (meaning every pieces' orientations are visible)
2) This can only be done for vertex-turning puzzles.
3) You cannot scramble the puzzles using the scramble button.
4) Reproducing any solves is impossible.
However, it will finally allow others to create commutators for and "solve" puzzles such as the complex vertex-turning cube.
Here, I'll start by simulating the complex face-turning cube. I mentioned earlier that we can only simulate vertex-turning puzzles, so instead we must create a complex vertex-turning octahedron (which is really the same puzzle).
We'll start with our octahedron:
Attachment:
complex 1.jpg
complex 1.jpg [ 168.11 KiB | Viewed 1584 times ]

We must now create a puzzle which simulates all of the piece types of the vertex-turning multioctahedron. To do this, we'll start by adding cuts which creates a trajber's octahedron:
Attachment:
File comment: Cut depth=150
complex 2.jpg
complex 2.jpg [ 204.62 KiB | Viewed 1584 times ]

Now, we need to make the core visible. To do this, we'll add "circles" to the puzzle:
Attachment:
File comment: Cut depth=90
complex 3.jpg
complex 3.jpg [ 221.21 KiB | Viewed 1584 times ]

But our original centers have disappeared, so we will need to add one final set of cuts to make them visible:
Attachment:
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complex 4.jpg
complex 4.jpg [ 225.83 KiB | Viewed 1584 times ]

There! Now to play with the puzzle properly, we are only allowed to turn the first and third layers at the same time, like this:
Attachment:
complex 5.jpg
complex 5.jpg [ 232.97 KiB | Viewed 1584 times ]

If we only scramble and solve the puzzle using these turns, the puzzle behaves like a vertex-turning multioctahedron which contains the core, centers, edges, and corners from the complex 3x3x3. I will always turn the right corner in these examples. By turning on color fade, we now have a super version.
But what about the imaginary pieces? To make this work, we simply change which layers we turn from 1 and 3 to 1, 3, and 4:
Attachment:
complex 6.jpg
complex 6.jpg [ 235.46 KiB | Viewed 1584 times ]

Now we have another puzzle. By making these turns only, this puzzle contains the edges, corners, anti-edges, anti-centers, UDF pieces, and UD^-1 pieces from the complex 3x3x3.
However, we still don't have all of the piece types, so we must change which layers we turn for a third time, from 1, 3, and 4 to 1, 2, 3, and 6:
Attachment:
complex 7.jpg
complex 7.jpg [ 234.52 KiB | Viewed 1584 times ]

Now, we have yet another puzzle. By making these turns only, this puzzle contains the centers, edges, corners, anti-edges, UDF pieces, and UD pieces from the complex 3x3x3.
If you must be complete, we can change the layers we turn to 1, 2, 3, 4, and 6:
Attachment:
complex 8.jpg
complex 8.jpg [ 236.77 KiB | Viewed 1584 times ]

We get the corners, anti-edges, anti-centers, and anti-core from the complex 3x3x3.
So what do we do with all of these puzzles? Well, solving puzzle 1, puzzle 2, puzzle 3, and puzzle 4 at the SAME time (that is, all puzzles turn at the same time) would be equivalent to solving the super complex 3x3x3. In essence, we've successfully simulated a complex puzzle!
But what about other examples? In this next one, I will demonstrate the complex FTO. However, because only vertex-turning puzzles can be used, we instead will simulate the complex vertex-turning cube (which is again the same puzzle).
We'll start with our cube:
Attachment:
complex 9.jpg
complex 9.jpg [ 166.17 KiB | Viewed 1584 times ]

Now, we need to create a puzzle with all of the pieces from the vertex-turning multicube. To do this we'll add the cuts from a Master Skewb:
Attachment:
File comment: Cut Depth=150
complex 10.jpg
complex 10.jpg [ 218.24 KiB | Viewed 1584 times ]

Next, we need to make the fixed centers visible, so we add "circles" once again:
Attachment:
File comment: Cut depth=60
complex 11.jpg
complex 11.jpg [ 240.09 KiB | Viewed 1584 times ]

Finally, we add one more tiny cut to get our original Master Skewb corners back:
Attachment:
File comment: Cut Depth=15
complex 12.jpg
complex 12.jpg [ 244.04 KiB | Viewed 1584 times ]

So here we are: a vertex-turning multicube. To play with this properly, we must, once again, only turn the first and third layers in unison:
Attachment:
complex 13.jpg
complex 13.jpg [ 258.05 KiB | Viewed 1584 times ]

This puzzle contains the fixed centers, dino edges, wing pieces, the 2 orbits of Offset Skewb centers, the square centers, the MS corners, and the "special" MS corners from the complex VTC. I will always turn the top right corner in these examples. Turning on color fade gives us our super version.
Note that the core is not visible, but the orientation of the core is determined by the permutation of the fixed centers, so we really don't need it.
Now, we can change the layers that turn from 1 and 3 to 1, 3, and 4:
Attachment:
complex 14.jpg
complex 14.jpg [ 266.26 KiB | Viewed 1584 times ]

Now, we have another puzzle. By making these types of turns only, this puzzle contains the square centers, the 2 types of master skewb corners, the anti-dino edges, the anti-wings, and 3 other new piece types. I'll let you figure out names for those :lol:
Next we can change the layers that move once again: from 1, 3, and 4 we change them to 1, 2, 3, and 6:
Attachment:
complex 15.jpg
complex 15.jpg [ 261.44 KiB | Viewed 1584 times ]

At this point we have another new puzzle. By making these turns only, this puzzle contains the (I'm getting tired of this) square centers, 1 type of master skewb corners, wing pieces, dino cube edges, and (I think) 4 new types of pieces, which I will let you name :)
One more to go. We just need to change the layers we turn to 1, 2, 3, 4, and 6:
Attachment:
complex 16.jpg
complex 16.jpg [ 270.23 KiB | Viewed 1584 times ]

We now have our final puzzle which contains the square centers, anti-dino cube edges, anti-fixed centers, anti-wings, and 3 new piece types which you must name for me :mrgreen:
Now, PLEASE correct me if I am wrong, but I think solving puzzle 1, puzzle 2, puzzle 3, and puzzle 4 at the SAME time (again, all puzzles turn at the same time) would be equivalent to solving the super complex vertex-turning cube. I might have missed some piece types, but I think this is all of them.
I hope all of this makes sense, and I hope I didn't make any mistakes. Let me know what you think!

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 Post subject: Re: Simulating Complex Puzzles
PostPosted: Fri Aug 16, 2013 2:10 pm 
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Hi Ben,

If I understand your construction correctly, I think it's basically the same idea as what Carl did in this post:
http://twistypuzzles.com/forum/viewtopic.php?p=226302#p226302

And how Landon showed all of the pieces on a set of Rubik's cubes in his program:
http://twistypuzzles.com/forum/viewtopic.php?p=268645#p268645 (turn on all of the various layer combination cube views).

Can you comment on these techniques and how your suggestion may differ?

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 Post subject: Re: Simulating Complex Puzzles
PostPosted: Fri Aug 16, 2013 2:25 pm 
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bmenrigh wrote:
Can you comment on these techniques and how your suggestion may differ?

The nice thing about doing this in UMC is that you aren't just limited to the complex face-turning cube (as Landon has done in his program). You can also simulate other complex puzzles like the complex vertex-turning cube (which I demonstrated in the second part of the post) and even the complex face-turning dodecahedron (although this requires a different method).

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 Post subject: Re: Simulating Complex Puzzles
PostPosted: Sat Aug 17, 2013 5:50 pm 
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benpuzzles wrote:
I recently figured out a way to simulate complex twisty puzzles using Wouter Meesen's Ultimate Magic Cube software. Unfortunately, it does have its limitations
Interesting... I've not used this software. I may want too look into it.
benpuzzles wrote:
1) The puzzles are all super (meaning every pieces' orientations are visible)
Personally I consider that a good thing. I like super puzzles.
benpuzzles wrote:
2) This can only be done for vertex-turning puzzles.
Looking at the pictures in this thread is it safe to assume this software can only deal with planer cuts? I see where you mention adding "circles"... and the circles on a Circle Cube can be modeled with planer cuts on a octahedron. Is that the reason you say this can only be done for vertex-turning puzzles?
benpuzzles wrote:
Now to play with the puzzle properly, we are only allowed to turn the first and third layers at the same time, like this
<SNIP>
Well, solving puzzle 1, puzzle 2, puzzle 3, and puzzle 4 at the SAME time (that is, all puzzles turn at the same time) would be equivalent to solving the super complex 3x3x3. In essence, we've successfully simulated a complex puzzle!
Ok... I haven't verified this accounts for all the pieces but it certainly may. But as this picture seems more complicated then necessary allow me to ask a few questions:
(1) You now have 4 puzzles each with layers which turn together. At this point is Wouter Meesen's Ultimate Magic Cube software needed because that is the only way you can play with these 4 puzzles?
(2) Does the software let you tie these puzzles together such that when you make a turn on one all the others turn appropriately?
(3) Does the software only allow valid turns or can you turn a layer without turning the layers its paired with?

The reason I ask is because its possible to simulate a Complex 3x3x3 with just 2 puzzles which are easily available... a 5x5x5 and a 3x3x3. If you define R as a right face plus middle slice layer turn on the 5x5x5 and a slice layer turn on the 3x3x3 you get to the Complex 3x3x3 as well and one could do this with a physical 5x5x5 and 3x3x3 and not need any software so I'm curious what the software does for you that makes this easier to play with?

Once my Real 5x5x5 is finished you could simulate the Complex 3x3x3 with a single physical puzzle. Enforcing the rule that the middle slice layer must turn with all face turns is something I'm still working on. I'm pretty sure in principle it can be done... I'm just not sure how practical such a design would be. If this software allows such rules to be enforced then I agree you are on to something. The problem with asking the solver to stick to a set of rules is humans make errors and if the software or the mechanism of the puzzle doesn't enforce the rules then it becomes too tedious for most to want to play with.
benpuzzles wrote:
However, because only vertex-turning puzzles can be used, we instead will simulate the complex vertex-turning cube (which is again the same puzzle).
<SNIP>
Now, PLEASE correct me if I am wrong, but I think solving puzzle 1, puzzle 2, puzzle 3, and puzzle 4 at the SAME time (again, all puzzles turn at the same time) would be equivalent to solving the super complex vertex-turning cube. I might have missed some piece types, but I think this is all of them.
Again I haven't check all this but your approach certainly seems valid. A way to check is to count the pieces. The complex 3x3x3 has (2*2)^3 or 64 pieces. That is (2 * the number of independent layers per axis)^the number of axes. The 2 comes from the binary choice as each piece is either in a given layer or it is not.

So your complex vertex-turning cube has (2*2)^4 or 256 pieces. If it took you 4 puzzles to get the 64 pieces of the Complex 3x3x3, I'm surprised you'd only need 4 puzzles for the complex vertex-turning cube.
benpuzzles wrote:
and even the complex face-turning dodecahedron (although this requires a different method).
The complex face-turning dodecahedron has (2*2)^6 or 4096 pieces. I don't think you are going to get there with 4 puzzles. And without knowing what rules this software allows you to put in place I think playing with 4 puzzles and asking the player to keep track of the rules is already a bit much. So I'm hoping the software can do this for you.

Any topic about complex puzzles is a good topic so I hope I didn't come across as overly critical.

Thanks,
Carl

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 Post subject: Re: Simulating Complex Puzzles
PostPosted: Sat Aug 17, 2013 7:08 pm 
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wwwmwww wrote:
Looking at the pictures in this thread is it safe to assume this software can only deal with planer cuts? I see where you mention adding "circles"... and the circles on a Circle Cube can be modeled with planer cuts on a octahedron. Is that the reason you say this can only be done for vertex-turning puzzles?

Yes. It has to be vertex-turning in order for these shallow cuts to be allowed.
wwwmwww wrote:
Ok... I haven't verified this accounts for all the pieces but it certainly may. But as this picture seems more complicated then necessary allow me to ask a few questions:
(1) You now have 4 puzzles each with layers which turn together. At this point is Wouter Meesen's Ultimate Magic Cube software needed because that is the only way you can play with these 4 puzzles?

I'm pretty sure these puzzles aren't on any other programs, so...yes.
wwwmwww wrote:
(2) Does the software let you tie these puzzles together such that when you make a turn on one all the others turn appropriately?

Unfortunately you can't do this. You would need multiple windows open in order to play with each puzzle.
wwwmwww wrote:
(3) Does the software only allow valid turns or can you turn a layer without turning the layers its paired with?

A nice feature is that you can pair layers together so that when you do a turn, you can indicate which layers will turn at once.
wwwmwww wrote:
The reason I ask is because its possible to simulate a Complex 3x3x3 with just 2 puzzles which are easily available... a 5x5x5 and a 3x3x3. If you define R as a right face plus middle slice layer turn on the 5x5x5 and a slice layer turn on the 3x3x3 you get to the Complex 3x3x3 as well and one could do this with a physical 5x5x5 and 3x3x3 and not need any software so I'm curious what the software does for you that makes this easier to play with?

The method is intended to work with any axis system and geometry, so following these guidelines you could simulate any complex puzzle. There are certaintly easier ways to do it for some of them (like the complex 3x3x3) but this approach works for most, if not all of them.
wwwmwww wrote:
Once my Real 5x5x5 is finished you could simulate the Complex 3x3x3 with a single physical puzzle. Enforcing the rule that the middle slice layer must turn with all face turns is something I'm still working on. I'm pretty sure in principle it can be done... I'm just not sure how practical such a design would be.

That is very interesting...I've actually figured out a mechanism for the augmented 3x3x3 (which also requires these slice layers to turn with the faces) which I need to start designing soon.
wwwmwww wrote:
If this software allows such rules to be enforced then I agree you are on to something. The problem with asking the solver to stick to a set of rules is humans make errors and if the software or the mechanism of the puzzle doesn't enforce the rules then it becomes too tedious for most to want to play with.

I agree. The reason I really wanted to demonstrate this is because solvers can now figure out commutators for the complex puzzles. To be honest, it isn't practical for completely solving them.
wwwmwww wrote:
Again I haven't check all this but your approach certainly seems valid. A way to check is to count the pieces. The complex 3x3x3 has (2*2)^3 or 64 pieces. That is (2 * the number of independent layers per axis)^the number of axes. The 2 comes from the binary choice as each piece is either in a given layer or it is not. So your complex vertex-turning cube has (2*2)^4 or 256 pieces. If it took you 4 puzzles to get the 64 pieces of the Complex 3x3x3, I'm surprised you'd only need 4 puzzles for the complex vertex-turning cube.

I was a little surprised as well. I do know for a fact that the core and anti-core aren't visible, but the permutations of the fixed and anti-centers determine their orientations, so there isn't really any need to show them.
wwwmwww wrote:
The complex face-turning dodecahedron has (2*2)^6 or 4096 pieces. I don't think you are going to get there with 4 puzzles. And without knowing what rules this software allows you to put in place I think playing with 4 puzzles and asking the player to keep track of the rules is already a bit much. So I'm hoping the software can do this for you.

Of course this puzzle can't be made with 4 puzzles, which is why I mentioned a different method must be used. The method builds up the pieces in multiple layers and changes in cut depths (unlike the other puzzles) so I wouldn't expect less than 8 puzzles being required.

wwwmwww wrote:
Any topic about complex puzzles is a good topic so I hope I didn't come across as overly critical.

You're not being critical at all. I this will help me understand the puzzles a little better as well.

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Last edited by benpuzzles on Sun Aug 18, 2013 7:49 pm, edited 1 time in total.

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 Post subject: Re: Simulating Complex Puzzles
PostPosted: Sat Aug 17, 2013 11:40 pm 
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benpuzzles wrote:
That is very interesting...I've actually figured out a mechanism for the augmented 3x3x3 (which also requires these slice layers to turn with the faces) which I need to start designing soon.
Interesting... before I get to your augmented 3x3x3 I need to start with some definitions so I did some digging. And these should be added to Oskar's site if they aren't their already. Here I found these:

Multi - X = is the puzzle that contains all the pieces created by the volumes of space that the cut surfaces (usually planes but not always) of puzzle - X creates.
Super Multi - X = Has all the pieces of Multi - X plus each piece has a fixed orientation and position in the solved state.
Augmented - X = Has all the pieces of Multi - X plus all the virtual pieces of puzzle - X.
Super Augemented - X = Has all the piece of Augmented - X plus each piece has a fixed orientation and position in the solved state.
Complex - X = Has all the pieces of Multi - X plus all the imaginary pieces of puzzle - X. Virtual pieces are a subset of the imaginary pieces so they are included here.
Super Complex - X = Has all the piece of Complex - X plus each piece has a fixed orientation and position in the solved state.

Where X = the name of the puzzle being discussed. The 3x3x3 in this case.

I first coined the term Augmented in this context here.

And I found this related quote here.
wwwmwww wrote:
Let's look at the Order=2 Face-Turn Hexahedron. Here I believe there is general acceptance as to what the Complex 3x3x3 actually is. This is a puzzle with 64 pieces across 10 piece types. It's numer of permutations has been calculated as seen here. And its even been solved as seen here. This puzzle still contains no virtual pieces but it does have the 6 imaginary piece types.
Oh and I'll also need this definition from here:
Holding Point = A point that is unaffected by a complete set of linearly independent twists for a given puzzle. Example, the core of a 3x3x3 never moves with this set of 6 twists [RLUDFB].

However what I was looking for were clear and concise definitions of virtual and imaginary in this context which I felt sure I had done before but maybe I hadn't. If someone finds them PLEASE point me to them. Anyways let me take a shot at it now.

Virtual Piece = The set of pieces defined by Andreas Nortmann's analysis of twistability that exist outside of the set of real physical pieces. As discussed in this thread. Note this analysis looks at the stationary point (aka the holding point) under all possible sets of linearly independent twists for a given puzzle. In other words it looks at all the possible ways a piece can NOT move. Andreas himself has since moved away from using the term virtual and now prefers the term ZHP for Zero volume Holding Point.

Imaginary Piece = The set of pieces defined by Matt Galla's analysis of twistability that exist outside the set of real physical pieces. This is also discussed in the same thread but Matt uses a totally different analysis. He makes no assumption of a holding point and instead looks at all possible way a piece CAN move. For example he looks at pieces that move with BOTH the L and R layers on a 3x3x3. Such pieces doesn't exist in Andreas's method. But Matt goes on to prove such pieces "MUST exist mathematically". While not immediately obvious it turns out the Imaginary Pieces are a superset of the Virtial Pieces. Andreas now prefers the term NHP for Non-Holding Point over Imaginary.

Personally I still prefer the terms Virtual and Imaginary as they extrapolate to Augmented and Complex in such a nice way but I'm ok with the terms ZHP and NHP too.

And now having said all this I realize we also need a definition of Real Piece so let me think about that a bit. Again I think I've already defined it but maybe that's just in my head.

Anyways I don't need to define Real Piece to get to the point of this post... well at least the initial point...

What is your Augmented 3x3x3? Using the definitions above the 3x3x3 contains NO virtual pieces... only real and imaginary pieces. The Real3x3x3 contains 27 pieces counting the core and the Complex3x3x3 contains 64 pieces as talked about above. So the above definitions tell us that the Augmented3x3x3 and the Real3x3x3 are the same puzzle.

And yes... the secondary point of this post was to put these definitions in one place to facilitate putting them in Oskar's Twistypedia.

Carl

P.S. Found the post I was thinking of... I think. Its here:
wwwmwww wrote:
My definitions basically are as follows...

Real Piece = Take the cut planes that define the puzzle (in this case the 4x4x4). Those cut planes divide up all 3D space into given volumes. Any non-zero volume (some are infinite) relate to real pieces.

Virtual Piece = Any piece that can serve as a holding point (a property that all the REAL pieces have) yet it has no volume using the picture described above. None of the NxNxN puzzles have any.

Imaginary Piece = Any piece in a Complex puzzle which isn't Real or Virtual.

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 Post subject: Re: Simulating Complex Puzzles
PostPosted: Sun Aug 18, 2013 12:25 am 
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wwwmwww wrote:
benpuzzles wrote:
That is very interesting...I've actually figured out a mechanism for the augmented 3x3x3 (which also requires these slice layers to turn with the faces) which I need to start designing soon.

What is your Augmented 3x3x3? Using the definitions above the 3x3x3 contains NO virtual pieces... only real and imaginary pieces. The Real3x3x3 contains 27 pieces counting the core and the Complex3x3x3 contains 64 pieces as talked about above. So the above definitions tell us that the Augmented3x3x3 and the Real3x3x3 are the same puzzle.

Oops :oops: I thought augmented meant something completely different. Basically, the puzzle I am referring to contains the centers, edges, corners, anti-edges, and anti-centers of the complex 3x3x3. I called it augmented cube because I though augmented pieces were the deeper-than-origin equivalents of real pieces. Sorry for my ignorance...but what could I call the puzzle? I think the deeper-than-origin piece types, which are often called "anti-x" pieces (where x is the name of the original piece) should have a name. Perhaps anti-cube would be good?

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 Post subject: Re: Simulating Complex Puzzles
PostPosted: Sun Aug 18, 2013 1:38 am 
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benpuzzles wrote:
Oops :oops: I thought augmented meant something completely different. Basically, the puzzle I am referring to contains the centers, edges, corners, anti-edges, and anti-centers of the complex 3x3x3. I called it augmented cube because I though augmented pieces were the deeper-than-origin equivalents of real pieces. Sorry for my ignorance...but what could I call the puzzle? I think the deeper-than-origin piece types, which are often called "anti-x" pieces (where x is the name of the original piece) should have a name. Perhaps anti-cube would be good?
I just made a big update to the Twistypedia adding all the above definitions and many pictures and animations. As far as the term "deeper-than-origin piece types" I'm not sure that is well defined. Anti-piece types comes closer however even there you have to be careful. There is the core and anti-core. There are face centers and anti-face centers. There are edges and anti-edges. But an anti-corner is simply another corner, this piece type is self-anti if you will. So would you consider a corner a deeper-than-origin piece type? What about the other imaginary pieces? So yes I think you could make a case for calling your puzzle the AntiCube or Anti3x3x3. Another way to look at it is that mater and anti-mater combine to form nothing (ok energy). A number plus its opposite equals zero. As your puzzle would have each piece and its opposite you could also call it the Null Cube. Just a suggestion...

Carl

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 Post subject: Re: Simulating Complex Puzzles
PostPosted: Sun Aug 18, 2013 10:57 am 
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wwwmwww wrote:
So would you consider a corner a deeper-than-origin piece type?

I would consider corners to be deep-turning pieces, since half of them move per turn and they are isomorphic to each other.
wwwmwww wrote:
What about the other imaginary pieces?

I've never really thought about this one. I think that the UD pieces and UD^-1 pieces can't really be determined as shallow or deeper-than-origin. They behave like the centers in a slice-turn-only cube and an opposite-faces-turn-only cube. So perhaps we could call them slice turning...
For the UDF pieces, in my opinion these are deep-turning. I know this sounds strange, but when turning the complex 3x3x3 exactly half of these pieces move, and the 2 halves are isomorphic to each other (I think). The centers, edges, and core are shallow-turning, and the anti-edges, anti-centers, and anti-core are deeper-than-origin turning. Again, this is my opinion.
wwwmwww wrote:
So yes I think you could make a case for calling your puzzle the AntiCube or Anti3x3x3. Another way to look at it is that mater and anti-mater combine to form nothing (ok energy). A number plus its opposite equals zero. As your puzzle would have each piece and its opposite you could also call it the Null Cube. Just a suggestion...

I LOVE it! The looks of the puzzle would actually make this a more fitting name...hint hint :wink:

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