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 Post subject: An incomplete picture... a theory thread
PostPosted: Tue Nov 27, 2012 2:37 pm 
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Andreas asked me at the last IPP about my picture for the Complex NxNxN puzzles... particulary about the case with N=even. Andreas please feel free to re-state your question here as I'm not sure I remember all of it. But basically I feel the picture I detail here:

http://www.twistypuzzles.com/forum/viewtopic.php?f=1&t=18651

Is the most logically consistant way to view the series of Complex NxNxN puzzles. This means they include all the real AND all the imaginary pieces. Note none of the NxNxN puzzles contain ANY virtual pieces and this is due to the 3 axes all being orthogonal.

I think where Andreas was going with his question as I recall was that I was still leaving out possible piece types. And yes he is correct. The current picture we are operating under is one where I think everyone assumes all pieces are either real, virtual, or imaginary. However that is incomplete. I don't think we yet have an all inclusive theory of twistability. I don't yet have a good proposal for such a theory but I can easily point out a few 'gaps' in the "current picture".

Let's look at the Order=1 Face-Turn Hexahedron. For everone else that is the 2x2x2. Under the current picture this puzzle has 8 real corners, zero virtual pieces, and zero imaginary pieces so its the simplest example to look at. So does this mean ALL Order=1 Face-Turn Hexahedrons can only have this one piece type... a corner? The answer is NO. One just need to look at Timur's Clockwork 4x4x4 to see an example of an Order=1 Face-Turn Hexahedron which has 3 types of pieces. These other 2 'new' piece types are not virtual or imaginary. The simplest way I can think to explain them is to think of them as a subset of a very oddly bandaged 4x4x4... which is pretty much what they are. I don't feel one should consider these as part of the Complex 2x2x2. The Fused Cube is an even simpler example of an Order=1 Face-Turn Hexahedron that contains pieces not found in the Complex 2x2x2. In this case, they are a subset of a bandaged 3x3x3. Oskar's Enabler Cube is another very interesting example and I'm sure the list could go on.

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. But again these are NOT all of the pieces one might find in an Order=2 Face-Turn Hexahedron. An example would be my Mercury Uniaxial 3x3x3. It's an Order=2 Face-Turn Hexahedron yet it contains pieces not found in the Complex 3x3x3. Again the simplest way to think of them is as a subset of a bandaged 5x5x5.

Comments... thoughts...
Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Tue Nov 27, 2012 4:31 pm 
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Thoughts on the ultimate order=1 hexahedron:

Would "overlapping" every possible order=2 hexahedron do the trick? "Overlapping" meaning that making an R turn on the ultimate hexahedron means making an R turn on every "sub-puzzle". The puzzle would consist of, among others, a complex 2x2x2, eight Fused Cubes (bandaged corner block in all different positions), eight Enabler Cubes (ditto), a Clockwork 4x4x4... (would Oskar's Gear Cube be allowed in? Or Geared Mixup? Or Geary Cube?)

Which leads to this thought: there is a Clockwork 4x4x4. Similarly a Clockwork 6x6x6 could be made. And a clockwork 8x8x8. And so on, ad nauseum. Does every added pair of layers introduce new piece types?

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Tue Nov 27, 2012 6:01 pm 
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Coaster1235 wrote:
Thoughts on the ultimate order=1 hexahedron:

Would "overlapping" every possible order=2 hexahedron do the trick? "Overlapping" meaning that making an R turn on the ultimate hexahedron means making an R turn on every "sub-puzzle".
In principle... I think the answer is yes. However the issue I think you'd run into is there may be an infinite number of sub puzzles. I suspect... but have no idea how to prove it... is that if you did have such an ultimate order=1 hexahedron, that the only way to solve it would be to reverse the moves used to scramble it.
Coaster1235 wrote:
The puzzle would consist of, among others, a complex 2x2x2,
The Complex 2x2x2 is just the normal 2x2x2 at least the way I have things defined..
Coaster1235 wrote:
eight Fused Cubes (bandaged corner block in all different positions), eight Enabler Cubes (ditto), a Clockwork 4x4x4... (would Oskar's Gear Cube be allowed in?
I'm not sure I'd include the gears themselves as they simply provide the bandaging which reduces the 3x3x3 to an Order=1 puzzle and I'm sure in principle there are many different teeth configurations that could be used to provide the same effect... maybe even an infinite number. But yes I'd certainly include the Geary Cube. I guess you could also now consider that the Clockwork 3x3x3.
Coaster1235 wrote:
I'm not sure I'd consider the Geared Mixup a face turn only puzzle. It is certainly an Order=1 puzzle but I'm not sure it should be included in the Face Turn Hexahedron family. Note its possible for an Edge to be in the position that pieces are turning around it. I guess I could agrue that one both ways.
Coaster1235 wrote:
Yes, I consider this to be an Order=1 Face-Turn Hexahedron.
Coaster1235 wrote:
Which leads to this thought: there is a Clockwork 4x4x4. Similarly a Clockwork 6x6x6 could be made. And a clockwork 8x8x8. And so on, ad nauseum. Does every added pair of layers introduce new piece types?
You skipped the Clockwork 5x5x5... but this is a very interesting question. I suspect this process alone would only produce a finite number of new pieces until such a point was reach that all the pieces on a higher order clockwork puzzle already existed. Note the corners on a Clockwork 5x5x5 couldn't be scrampled so this pieces could be considered equivalent to a core. It does change orientation however so maybe its a new piece. However the corners on a Clockwork 9x9x9 are ALWAYS in the exact same state as the core so that isn't a new piece. Similarily the face centers on a Clockwork 3x3x3 (Oskar's Geary Cube) would be repeated as face centers on a Clockwork 11x11x11. Figuring things out for the other pieces gets more complicated but I strongly suspect they are all repeated at some point.

You also left off MANY MANY Order=1 Face-Turn Hexahedrons. There are 20 different Order=1 Face-Turn Hexahedrons which can be built using my Multi Gear Cube Kit. And you'd need many copies of each in your setup. For example the one which uses the 1:2 on one axis, the 1:3 on one axis, and the 1:4 on one axis; I think would require 48 copies. There are the 24 possible orientations of the cube, plus each of those would have a mirror image version which could be made. There is also Oskar's Cube Bouchon and Oskar's Variomatic Cube where Oskar has applied the same treatment to each axis. In principle these could be added to the Gear Cube Kit and you'd get many more options... think Bouchon gears on one axis, Variomatic gears on another axis, and a 1:2 gear ratio on the other. Things get out of hand very very fast. Even with the Bouchon gears on all axes I believe there are options as to which state you start in for the solved state so you'd need multiple copies of just that one even though the gearing on all 3 axes was the same. And I'm pretty sure I haven't yet named all the Order=1 Face-Turn Hexahedrons which have been made... and I'm sure there are others yet to be conceived as well.

Carl

P.S. Just thought of many more Order=1 Face-Turn Hexahedrons. Think of my Uniaxial Puzzles like the Mercury Uniaxial 3x3x3. These are Order=2 puzzles but they could be reduced to Order=1 via something akin to my Gear Cube kit which tied opposite faces together via a 1:1, 1:2, 1:3 or 1:4 gear ratio and you could add the Bouchon and Variomatic like gears to this picture as well. And the Uniaxial picture just uses 2 different cut depths for the 3x3x3. Here is an animation I made that uses 3 different cut depths. In principle you could have a different cut depth on each face and have a bandaged 13x13x13 which was an Order=2 puzzle and then use gears to reduce this to Order=1. I really think there could easily be an infinite number of Order=1 Face-Turn Hexahedrons.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Tue Nov 27, 2012 8:40 pm 
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I'm glad to see this thread. I've been wanting to re-hash some of this stuff for a while now. I have quite a few questions about your approach and methodology (Andreas's too). What I really need to do is go back and read all of the background material. I wasn't around for some of the original threads on this topic. Until I've gone back and done a bunch of reading I'll have to wade into this thread slowly.


For the Complex 3x3x3 I prefer to think of it as an abstract object and not weighed down by physical / mechanical constraints like cut depth or whether pieces have any volume or could serve as a holding point. I also like to think of puzzles as cuts per face rather than cuts per axis. So in this model the Complex 3x3x3 is better described as the "Complex 1-cut-per-face turning Cube". That cut is less about a physical property and more about a choice. The choice being whether or a given piece moves when you turn the face.

Under this definition you can think of pieces as a binary string.
Define the places in the string: UFRLBD. I chose that order so that opposite faces are pared up:

Code:
U  F  R  L  B  D
^  ^  ^  ^  ^  ^
|  |  |__|  |  |
|  |________|  |
|______________|


Now if you want to define the face centers of a Rubik's cube, there are 6 of them:
100000 -> U center
010000 -> F center
001000 -> R center
000100 -> L center
000010 -> B center
000001 -> D center

And if you want to describe edges here are a few:
110000 -> UF edge
101000 -> UR edge
[...]
000011 -> DB edge

Since there are 6 choices and each choice is either "turn or don't turn" (2 options) you have 2^6 = 64 total pieces. What I've described here is closest to what Carl would call a order-2 puzzle (3x3x3) because there is a middle slice implied by some of the piece definitions. Any piece that has a 0 in both faces that are opposite each other lies in a slice between those faces. Take the UF edge for example:

Code:
U  F  R  L  B  D
1  1  0  0  0  0
^  ^  ^  ^  ^  ^
|  |  |__|  |  |
|  |________|  |
|______________|


The U, D pair is 1, 0. The F, B pair is 1, 0. The R, L pair is 0, 0. Because the UF edge does not move when either R or L is turned, it is in the slice between R and L. You can "turn" that slice by doing R, L' and then reorienting the puzzle. There are 16 pieces in that slice (because there are 2^4 ways to set the other 4 bits while leaning the R and L bits 0).

Up to this point I think everything I've said is non-controversial.

Using my "cuts per face" perspective, the next puzzle up in order would be 2 choices per face which would basically be the "Complex 5x5x5". If we want to try to extend this to even-layered puzzles I think the thing we need to avoid is the the middle slice.

To get rid of the middle slice in the bitmask notation we need to look at the faces that are opposite each other in pairs (there are three pairs, one for each axis). We can have both faces, or one face move the piece. That is, 11, 01, 10. This is because the 00 case where you can turn both opposite faces led to their being a slice layer. For this, there are 3 axes and each axis has three choices which leads to 3^3 = 27 pieces.

Put another way, the 2x2x2 is a subset of the 3x3x3 where all pieces that lie in a slice have been removed. The Complex 2x2x2 is a subset of the Complex 3x3x3 where all pieces that lie in a slice have been removed. This leaves 27 pieces.

There's actually another way you could simulate a slice on the Complex 3x3x3 besides just opposite faces both being 0 in the piece definition. If they're both 1 then you can create a different type of slice. Here is U, D' performed on a puzzle with UFRD pieces (definition 111001):
Attachment:
3.1.31_effective_slice.png
3.1.31_effective_slice.png [ 14.07 KiB | Viewed 6234 times ]

Notice that you can simulate a slice relative to the corners but it's a different type of slice than the one the 3x3x3 edges are in. If your definition of the Complex 2x2x2 also can't have this slice then you must eliminate all pieces where opposite faces are both turn the piece or opposite faces where both do turn the piece. In that case, you have two options per axis: 01 and 10 and 3 axes for a total of 2^3 = 8 pieces. In this case the Complex 2x2x2 would be the standard 2x2x2.


Now let me suggest a way to try to extend this concept to the Complex 4x4x4 (the subset of the Complex 5x5x5 where you've eliminated the middle slice layer).

The first way would be to eliminate just the pieces where both opposite faces and opposite slices all don't move the piece. In this case there are 15 ways options per axis and you'd get a puzzle with 15^3 = 3375 pieces. If you also want to eliminate the other slice (illustrated above) then the options for the opposite faces and slices are:
0001
0010
0011
0100
1000
1100

Or 6 choices per axis for a total of 6^3 = 216 pieces.

Unfortunately I don't know how to manually count the number of unique pieces for these subsets. I'd have to write code to do it.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Tue Nov 27, 2012 9:23 pm 
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bmenrigh wrote:
Put another way, the 2x2x2 is a subset of the 3x3x3 where all pieces that lie in a slice have been removed. The Complex 2x2x2 is a subset of the Complex 3x3x3 where all pieces that lie in a slice have been removed. This leaves 27 pieces.
The problem I have with this picture is this view of the Complex 2x2x2 does NOT produce what I consider an Order=1 puzzle. This would still be order 2. Your U and D turns are still independant. My view of a Complex 3x3x3 is a puzzle which would only have U D R L F and B as valid turns. Turns of the slice layer are combinations of these plus global rotation. So with this picture in mind the Complex 2x2x2 should only have U R and F turns which are independant. The same extends to the Complex 4x4x4. One layer on each face is dependant... you only have 3 independant players per axis. You could call them U 1 D R 2 L F 3 and B if you like. Based on these independant turns I note the following:

The Complex 2x2x2 has 2^3 pieces or 8 pieces. This would look just like a normal 2x2x2.
The Complex 3x3x3 has 2^6 pieces or 64 pieces. You are familiar with this one.
The Complex 4x4x4 has 2^9 pieces or 512 pieces. This would look like an 8x8x8 and I show a possible construction in this thread.

I can make a table for the Complex 4x4x4 if you like.

The Complex 5x5x5 which should be less controversial would have 2^12 pieces or 4096 pieces due to the addition of one additional independant layer on each of 3 axes.

Why break the 2^(the number of total independant layers) formula for the Complex NxNxN where N is even puzzles?

I think your 216 piece puzzle may be a subset of my 512 piece puzzle... not really sure at the moment. But it leaves out pieces I'd expect to find in an Order=3 puzzle. Your 3375 piece puzzle I believe is still an Order=4 puzzle and therefore can't be the Complex4x4x4. But yes, it is a subset of the Complex 5x5x5. Your method of counting cuts per face doesn't address rather the layers produced are independant or dependant... and I believe you really need to look at the whole axis to cover this point.

Trying not to be too controversial,
Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Tue Nov 27, 2012 10:11 pm 
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wwwmwww wrote:
bmenrigh wrote:
Put another way, the 2x2x2 is a subset of the 3x3x3 where all pieces that lie in a slice have been removed. The Complex 2x2x2 is a subset of the Complex 3x3x3 where all pieces that lie in a slice have been removed. This leaves 27 pieces.
The problem I have with this picture is this view of the Complex 2x2x2 does NOT produce what I consider an Order=1 puzzle.
I didn't realize you were trying to preserve order. I figured you were trying to preserve a lack of a middle layer. I need to think about preserving order and I need to read more history of the things you've already written.
wwwmwww wrote:
Trying not to be too controversial
I'm up for a healthy, respectful debate. Under those conditions my feelings can't be hurt so I say unleash the controversy :)

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 1:08 pm 
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1. I have a plea.
Even we theorists couldn't agree onto a definition of order.
You might have seen that I quitted using that term.
I ask you to do the same.
Say "X cuts per axis" or "X cuts per face" or whatever makes it clear what you are writing about.
I apologize for any case in the future where I act against my own plea. I am only human too.

2. Since the IPP was in August I do not remember correctly what I asked Carl.
But I remember that I stated a condition. Any definition of a Complex NxNxN with even N must result in a set of pieces which is a subset of a Complex (N+1)X(N+1)X(N+1).
The 2x2x2 can be viewed as subset of the 3x3x3. The Skewb is a subset of the Master Skewb. You can generate the table for HC[N] (the table of all HoldingPoint pieces for the cornerturning hexahedron with N cuts per axis) by deleting some lines from HC[N+1].
If there is no way of creating a smaller ComplexNxNxN by making a subset I (only me!) won't accept it.

3. I have a problem with treating the Fused Cube as a puzzle out of HF1.
I consider it as a restricted (which is defined more narrowly compared to 'bandaged') HF2.
The Fused Cube does not deserve a treatment different from the other variants in this thread.
All of them are just restricted 3x3x3s with one possible (!) exception: B12C111
Compared with the other variants the restriction in B12C111 is perfectly symmetrical like the Gear Cube and the Clockwork 4x4x4.


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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 1:47 pm 
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I looked at nxnxn complex cubes before for even n, and came up with several different ways to define one, with several people preferring different definitions. The conclusion that I took from the discussion was that there are multiple valid ways to define these puzzles, and that we had to agree to disagree, though someone may even disagree with this conclusion :lol: . For what it's worth, I prefer the approach of eliminating the middle layer.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 1:55 pm 
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As I think about the Complex NxNxN where N > 3 I have a questions about what's in scope.

Can a piece ever move with two different face/slices on a single face? For example, could we ever define a special new face center on the Complex 5x5x5 that turns with both the U face and u slice? This seems to create something of a physical contradiction if you want to imagine holding this puzzle and turning both the U and u face/slice at the same time. The piece would have to "move twice as fast". Also, it seems to deviate from what a slice is on all other puzzles with slices. On every puzzle that I'm aware of, the set of pieces moved with a slice under a face is completely disjoint with the set of pieces moved with the face.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 2:52 pm 
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bmenrigh wrote:
Can a piece ever move with two different face/slices on a single face? For example, could we ever define a special new face center on the Complex 5x5x5 that turns with both the U face and u slice?


I've personally always viewed these puzzles as being abstract enough that this doesn't bother me. Also, the usual way of emulation (?) which for the complex 5x5x5 uses a 9x9x9 provides all of these pieces and doesn't seem to have any bother with this, I'm sure a suitable diagram/table has been posted before by someone.

There always seems to be a problem with turning two layers at once, I suggest only allowing to apply one twist of one layer at a time, since in theory each is a separate twist, and twists should be applied in a certain order in my opinion. It is just a convenient bonus that we can apply more than one twist simultaneously on the 5x5x5.


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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 4:58 pm 
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bmenrigh wrote:
As I think about the Complex NxNxN where N > 3 I have a questions about what's in scope.

Can a piece ever move with two different face/slices on a single face? For example, could we ever define a special new face center on the Complex 5x5x5 that turns with both the U face and u slice?
Yes. And this issue exists for Complex NxNxN where N=3 too. Think about a slice layer move on the Complex 3x3x3. You have pieces turning at different rates there as well. As bobthegiraffemonkey states the easiest way to deal with this is just not to make those turns. You can always just turn one layer at a time. On the Complex 3x3x3 just limit yourself to face turns.
bobthegiraffemonkey wrote:
I've personally always viewed these puzzles as being abstract enough that this doesn't bother me. Also, the usual way of emulation (?) which for the complex 5x5x5 uses a 9x9x9 provides all of these pieces and doesn't seem to have any bother with this, I'm sure a suitable diagram/table has been posted before by someone.
I agree with most of this. However the standard model of the Complex 5x5x5 has 4096 pieces and a Multi-9x9x9 has only 729 pieces so I think something is wrong with your picture. My model of the Complex 4x4x4 uses a Multi-8x8x8. I'd have to think a bit about the Complex 5x5x5. A Multi-16x16x16 would have the correct number of pieces but you'd need at least a Multi-17x17x17 to represent the Complex 5x5x5 and I'm not 100% sure that is adequate.

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 5:52 pm 
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Andreas Nortmann wrote:
1. I have a plea.
Even we theorists couldn't agree onto a definition of order.
You might have seen that I quitted using that term.
I ask you to do the same.
Say "X cuts per axis" or "X cuts per face" or whatever makes it clear what you are writing about.
I apologize for any case in the future where I act against my own plea. I am only human too.
Fair enough... I have been using Order=X to mean X independant layers per axis. The working model of the Complex 3x3x3 is an oddly bandaged Multi-5x5x5 yet I consider it Order=2. Yet it could be viewed as having 4 cuts per axis. How do you define cuts? How many cuts per axis does the Fused Cube have?
Andreas Nortmann wrote:
2. Since the IPP was in August I do not remember correctly what I asked Carl.
That is mostly my fault for waiting so long to try and answer it. IPP was the week before my move so life got in the way there for a while.
Andreas Nortmann wrote:
But I remember that I stated a condition. Any definition of a Complex NxNxN with even N must result in a set of pieces which is a subset of a Complex (N+1)X(N+1)X(N+1).
And I think my model fits that.
Andreas Nortmann wrote:
The 2x2x2 can be viewed as subset of the 3x3x3. The Skewb is a subset of the Master Skewb. You can generate the table for HC[N] (the table of all HoldingPoint pieces for the cornerturning hexahedron with N cuts per axis) by deleting some lines from HC[N+1].
If there is no way of creating a smaller ComplexNxNxN by making a subset I (only me!) won't accept it.
Hmmm... not sure this is always true for HF for all N. For N=odd you have a problem. Is a 3x3x3 a subset of a 4x4x4... not even talking about the complex version? The 4x4x4 doesn't have face centers for example. Are you saying the Complex 4x4x4 MUST have the 3x3x3 face centers?
Andreas Nortmann wrote:
3. I have a problem with treating the Fused Cube as a puzzle out of HF1.
I agree that the pieces of the Fused Cube shouldn't be considered as part of the Complex 2x2x2. But how do you define HF1. The Fused Cube is a Face-turn Hexahedron with just 1 independant layer per axis. And using Bram's definitions of doctrinaire it is considered a doctrinaire puzzle so its not bandaged. But yes it agree its best viewed as a 'restricted' subset of HF2... but I'm not certain I've seen an agreed upon definition of 'restricted' in this context.
Andreas Nortmann wrote:
All of them are just restricted 3x3x3s with one possible (!) exception: B12C111
Compared with the other variants the restriction in B12C111 is perfectly symmetrical like the Gear Cube and the Clockwork 4x4x4.
I don't have your program on my work PC at the moment. Is B12C111 the slice-turn-only 3x3x3? I would also view this as a 'restricted' subset of HF2 that was also a Face-turn Hexahedron with just 1 independant layer per axis. So its restricted in a symmetrical fashion but I don't see an need to make any exceptions for it. If B12C111 is something else let me know... I could easily be out in left field by this point.

As for the question you asked me at IPP... my take on it was how do you determine which pieces should and shouldn't belong in the model. And I think I'll safe that thought for my next post.

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 5:57 pm 
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ok... make that the post after this one...

bobthegiraffemonkey wrote:
I looked at nxnxn complex cubes before for even n, and came up with several different ways to define one, with several people preferring different definitions. The conclusion that I took from the discussion was that there are multiple valid ways to define these puzzles, and that we had to agree to disagree, though someone may even disagree with this conclusion :lol: . For what it's worth, I prefer the approach of eliminating the middle layer.
Arg!!! That is NOT the conclusion I want to agree with. I see this as more clear cut then most seem to be making it. I understand many here have their own definition of order and I've accepted that. And I think that has clouded the way people see this problem. Andreas may be correct... I might make more headway if it took the word order out of the equation.

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 6:19 pm 
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wwwmwww wrote:
I agree with most of this. However the standard model of the Complex 5x5x5 has 4096 pieces and a Multi-9x9x9 has only 729 pieces so I think something is wrong with your picture. My model of the Complex 4x4x4 uses a Multi-8x8x8. I'd have to think a bit about the Complex 5x5x5. A Multi-16x16x16 would have the correct number of pieces but you'd need at least a Multi-17x17x17 to represent the Complex 5x5x5 and I'm not 100% sure that is adequate.

Carl


My bad, I seem to have remembered incorrectly, a 9x9x9 is clearly not large enough. By my analysis a 19x19x19 would be needed. Here's a table showing one way of doing this, each twist affects these layers on a 19x19x19:

U 2 4 5 6 7 9 10 11 15
u 3 4 5 7 8 10 12 14 16
D 4 9 10 11 13 14 15 16 18
d 5 6 8 10 12 13 15 16 17

Can someone check this is correct now? I realise it's probably not worth writing down, but it was bugging me; I guess I like problem solving or I wouldn't be on this forum at all.


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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 6:55 pm 
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Ok... let's start at the very beginning and work our way back up again.

We all agree the 3x3x3 (also known as the Rubik's Cube) is a puzzle which has 2 independant layers per axis of rotation. (I hope... correct me if I'm wrong) These two independant layers are typically taken to be the face layers and they are typically named U D R L F and B. The slice layers can be thought of as a rotation of the two adjacent faces and a global rotation of the entire puzzle. This typically produces a picture where the core cubie is viewed as stationary, and it serves as what is considered a holding point. All the other pieces move about this one in this picture and no degrees of freedom are lost.

Let's look at two ways of counting the pieces in this puzzle.

(1) It's a 3x3x3 so that means 3x3x3=27 and we see this gives us 1 core, 6 faces, 12 edges, and 8 corners. These are what I call the "real" pieces of the 'unrestricted' Face-turn Hexahedron puzzle with two indepant layers per axis.

Can there be other... what I call "imaginary" pieces? Yes...

(2) This puzzle has 2 independant layers per axis and it has 3 axes so that is a total of 6 independant layers for this puzzle. Can a piece exist in more then one layer? Yes, look at the intersection of the U face and the F face... you find 3 pieces which are in both layers. So a more general way of counting the pieces which could exist in a 3x3x3 is to think of each layer as a binary flag for the piece in question. The piece is either in that layer or it is not in that layer. This produces 2^6 potential pieces. The Complex 3x3x3 by definition must contain ALL of these pieces, so it has 64 pieces. The normal 3x3x3 is a subset of these pieces and its not hard to identify which ones they are regardless on the method used to visualize the Complex 3x3x3. Several different visualizations have been presented by myself and others so I don't want to get too sidetracked there at the moment but it is important to know that at a base level they are all equivalent.

So let's now move on to the (apparently much more complicated) 2x2x2. What makes this puzzle different from the 3x3x3 is that it only has 1 independant layer per axis of rotation. As many here will note that it still just has one layer per face just like the 3x3x3 has... yet this puzzle ISN'T a 3x3x3. Can we all agree on that? If so then that difference is due to the fact that the layer on one face is no longer independant of the layer on the opposite face. We can NO longer list the independant layers as U D R L F and B. I'll call the independant ones U R and F but that choice is arbitrary... all that really maters is that there are only 3.

So now lets apply the same two methods of counting the pieces.

(1) 2x2x2 = 8 so we can all agree this puzzle has 8 real pieces

Now let's look for imaginary pieces.

(2) This puzzle has 3 independant layers... a piece must either be in or out of each layer. This produces 2^3 potential pieces and 2^3 also happens to be 8. So I conclude the Complex 2x2x2 doesn't have any imaginary pieces and in this case the Normal 2x2x2 and the Complex 2x2x2 are the same puzzle.

Everyone following? If so let's move on to the 4x4x4. I don't really feel its necessary to move to the 5x5x5 and come back down as I believe its rather clear what's going on here.

The 4x4x4 is a Face-turn Hexahedron with 3 indepenant layers per axis of rotation. Agreed? So let's count the pieces.

(1) 4x4x4 = 64 so the Face-turn Hexahedron with 3 indepenant layers per axis of rotation contains 64 real pieces. For those of you looking at your 4x4x4's now and saying something is wrong as you only count 56... I agree. Something is wrong... you have defective 4x4x4's. Go out and buy a Circle 4x4x4 which allows you to play with the other 8 pieces which are hidden away inside. Oh and don't take your 4x4x4's apart looking for them... as I said they are defective. (My poor attempt at humor)

Now lets look for imaginary pieces in the 4x4x4

(2) This puzzle has a total of 9 independant layers regardless as to how we name them. Each part, real or imaginary, must either be in a given layer or outside of that layer. So again this binary approach can easily be applied. And it gives us 2^9 or 512 pieces. By definition the Complex 4x4x4 should contain ALL of them. Try to give me a reason why it should have less? If you find more, tell me which independant layers each piece is in... and you should find duplicate pieces that can't be seperated.

I have methods of classifying the pieces similiar to those used by Andreas for all Complex NxNxN but there is no point going there if we can't even agree on the number of pieces to clasify.

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 7:14 pm 
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bmenrigh wrote:
If you also want to eliminate the other slice (illustrated above) then the options for the opposite faces and slices are:
0001
0010
0011
0100
1000
1100

Or 6 choices per axis for a total of 6^3 = 216 pieces.
wwwmwww wrote:
I think your 216 piece puzzle may be a subset of my 512 piece puzzle... not really sure at the moment.
Having thought about this more... I don't think even your 216 piece puzzle is a subset of my Complex 4x4x4. You are allowing states which I believe require all 4 layers to be independant while ignoring some of the interaction between axes. I think this puzzle must also be considered as a subset of the Complex 5x5x5.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 7:57 pm 
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bobthegiraffemonkey wrote:
My bad, I seem to have remembered incorrectly, a 9x9x9 is clearly not large enough. By my analysis a 19x19x19 would be needed. Here's a table showing one way of doing this, each twist affects these layers on a 19x19x19:

U 2 4 5 6 7 9 10 11 15
u 3 4 5 7 8 10 12 14 16
D 4 9 10 11 13 14 15 16 18
d 5 6 8 10 12 13 15 16 17

Can someone check this is correct now?
I don't think that is a trivial question. I think a 17x17x17 should work though. This is the sequence I would try.
Attachment:
Complex5x5x5B.png
Complex5x5x5B.png [ 6.29 KiB | Viewed 5960 times ]

But I think a full check would require naming all 4096 pieces and then verifying they were present this this puzzle. With this set up the piece which never moved with any of the 12 independant layers would be the outer corners (all acting like a single piece) and the piece which moved with all of the 12 independant layers would be the central core. But I may be easily over looking something... probably am.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 7:59 pm 
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Thanks for backing up and explaining it this way. This is the first time I've understood where your 2^9 number for the Complex 4x4x4 comes from.

My approach is based on choices per face and your approach is based on choices per layer.

Complex 3x3x3:
Brandon:
There is a binary choice per face. There are 6 faces so 2^6 = 64.

Carl:
There is a binary choice per independent layer. 2 independent layers per axis, 3 axes so 2^(2*3) = 64.

Complex 2x2x2:
Brandon:
There is a binary choice per face. There are 6 faces so 2^6 = 64.

Carl:
There is a binary choice per independent layer. 1 independent layers per axis, 3 axes so 2^(1*3) = 8.

Complex 4x4x4:
Brandon:
There are two binary choices per face (one for each layer). There are 6 faces so 2^(6*2) = 4096.

Carl:
There is a binary choice per independent layer. 3 independent layers per axis, 3 axes so 2^(3*3) = 512.


For NxNxN where N is odd our different way of looking at things yields the same results. For N = even my method isn't expressive enough and yours is. I think I'm close to conceding but there are still a few things that trouble me about your model.

First, my gut tells me the Complex 4x4x4 should be a superset of the Complex 3x3x3. In general I'd like the Complex MxMxM to be a superset of the Complex NxNxN when M > N. Perhaps this is controversial since the 4x4x4 isn't a superset of the 3x3x3?

Second, ignore whether the superset property should hold, take a look at Gelatinbrain's 3.1.21:
Attachment:
3.1.21.png
3.1.21.png [ 7.22 KiB | Viewed 5966 times ]

In this puzzle the pieces inside of the circle do not turn with any layer. I know you will say that because of this, the puzzle has 4 independent layers instead of three but stay with me for a minute.

Suppose we want to define a piece that only turns with these two slices:
Attachment:
3.1.21_slice1.png
3.1.21_slice1.png [ 7.88 KiB | Viewed 5966 times ]
AND
Attachment:
3.1.21_slice2.png
3.1.21_slice2.png [ 7.76 KiB | Viewed 5966 times ]


I have highlighted the piece that has that definition:
Attachment:
3.1.21_slice1_2_piece.png
3.1.21_slice1_2_piece.png [ 14.27 KiB | Viewed 5966 times ]

This piece is equivalent to a 3x3x3 edge. There are 12 of them. The same happens with the Circle 2x2x2 where the circle pieces don't turn with any face. I think this piece (the 3x3x3 edges) must be included in the Complex 4x4x4 but I'm pretty sure following your definition, none of the 2^9 pieces are these 3x3x3 edges.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 8:37 pm 
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Note: after spending a long time typing and thinking about this post I'm a bit worried my logic is faulty. I'll post anyways because even if I'm wrong I should be able to learn from the responses.

wwwmwww wrote:
[...](apparently much more complicated) 2x2x2. What makes this puzzle different from the 3x3x3 is that it only has 1 independant layer per axis of rotation. As many here will note that it still just has one layer per face just like the 3x3x3 has... yet this puzzle ISN'T a 3x3x3. Can we all agree on that? If so then that difference is due to the fact that the layer on one face is no longer independant of the layer on the opposite face. We can NO longer list the independant layers as U D R L F and B. I'll call the independant ones U R and F but that choice is arbitrary... all that really maters is that there are only 3.

I'd like to focus on this concept for linear independence for a moment because I think it has a subtle problem that doesn't show up on the 2x2x2 but shows up on some other puzzles.

Take a 2x2x2 which is by almost anybodies definition, "deep-cut". I say it is because all moves are ambiguous -- for each move there is another move + re-orientation that achieves the same effect. As you've pointed out, this is essentially the same as your order-1 definition.

So lets look at the UD axis. If I turn U then that's the same as D' + a re-orientation around D'. If we'd chosen D then we could simulate a U turn instead.

In general, for the 2x2x2 if you pick one move from the sets {U,D}, {R,L}, {F,B} there are 8 possibilities and all 8 of these make the exact same puzzle. The choice for each axis is arbitrary.


Let's take a look at the 4x4x4 though. Looking at just one axis, we have the U face, u slice, d slice, D face. If you want to simulate a D turn just do [U', u', d] and then re-orient about U'. For each axis you have 4 choices for which layer is dependent on the other three. The problem is that different choices for the "dependent layer" yield different puzzles.

For example, if you decided that the {u, f, r} slices are the dependent layer then what'd you get would be fundamentally different than if you'd chosen the {U, F, R} face as the dependent layer.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 10:55 pm 
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bmenrigh wrote:
ignore whether the superset property should hold, take a look at Gelatinbrain's 3.1.21:
In this puzzle the pieces inside of the circle do not turn with any layer. I know you will say that because of this, the puzzle has 4 independent layers instead of three but stay with me for a minute.
Oh you are correct... that puzzle has 4 independent layers per axis. As such it should be a subset of the Complex 5x5x5 and it is. I have no problem with that. I just don't think the Complex 4x4x4 should contain the normal 5x5x5 which it would given your definitions. In fact the Complex 4x4x4 and the Complex 5x5x5 become the same puzzle which strikes me as wrong because you've added in the extra degrees of freedom that make the normal 5x5x5 a different puzzle then the normal 4x4x4.

Also a property that I like is the Normal 2x2x2 and the Complex 2x2x2 are both deep cut puzzles. Easy as they are the same puzzle. With your definition the Complex 2x2x2 is no longer deep cut.

Also note that the Normal 4x4x4 has 3 deep cuts and 6 shallow cuts. Using my definition of the Complex 4x4x4 these ALL become deep cuts. Each independent layer (all 9 of them) becomes an isomorphic half of the puzzle. Its simplest puzzle I'm aware of with more then one deep cut on the same axis. It appears to be a very interesting puzzle and I seem to struggle to convince anyone it exists. Even if it weren't for my objections about not keeping track of the independent layers I'd wonder why you'd want to define the Complex 4x4x4 as the same puzzle as the Complex 5x5x5 where there was an equally consistent way to define a new puzzle in its place.

Also in my picture all Complex NxNxN puzzles can be mapped to Multi-MxMxM puzzles. If N is odd so is M. And if N is even so is M.

A few other observations about Gelatinbrain's 3.1.21.

(1) Its NOT a deep cut puzzle. No cut divides the puzzle into two isomorphic halves.

(2) The 3x3x3 edge that you highlight is a REAL piece. As such if it were to appear in the Complex 4x4x4 puzzle it SHOULD first appear in the NORMAL 4x4x4 as that is the collection of ALL the REAL pieces in the Complex 4x4x4. I see NO reason why a more general picture should force a REAL piece to become IMAGINARY.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 11:13 pm 
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wwwmwww wrote:
A few other observations about Gelatinbrain's 3.1.21.

(1) Its NOT a deep cut puzzle. No cut divides the puzzle into two isomorphic halves.

(2) The 3x3x3 edge that you highlight is a REAL piece. As such if it were to appear in the Complex 4x4x4 puzzle it SHOULD first appear in the NORMAL 4x4x4 as that is the collection of ALL the REAL pieces in the Complex 4x4x4. I see NO reason why a more general picture should force a REAL piece to become IMAGINARY.

I probably shouldn't have jumped straight to 3.1.21 because I know that middle slice layer is obvious and somewhat orthogonal to the point I'm trying to make.

Let me start and the beginning too. To me a Complex puzzle is a puzzle that has a piece for every subset of the turnable parts. There are 12 turnable parts on a 4x4x4 (4 per axis) and for any given subset of those parts, there is a piece that turns with those and only those parts.

The number of subsets for a set of 12 items is 2^12.

When I'm holding a 4x4x4 and trying to imagine various piece types for a Complex 4x4x4 (the puzzle that that has a piece for every subset), one of those pieces is the piece that only turns with the u slice and f slice. 3.1.21 just has the nice property that it shows this piece which is why I showed 3.1.21.

Your definition of the Complex 4x4x4 can not contain the piece that only moves with the u and f slice. Since this isn't a real piece on the 4x4x4 it would have to be an imaginary piece in order for it to be part of your Complex 4x4x4 definition.

Part of the issue is that I don't quite understand your definition of an imaginary piece so I don't understand why the uf piece isn't an imaginary piece.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 11:17 pm 
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bmenrigh wrote:
For example, if you decided that the {u, f, r} slices are the dependent layer then what'd you get would be fundamentally different than if you'd chosen the {U, F, R} face as the dependent layer.
You sure of that? The only difference between the two puzzles is which piece you define as the holding point. Other then that all the same pieces should appear. I haven't done all the math for the 4x4x4 but I have tested this exact same question on the 3x3x3. Lets assume that instead of holding the 3x3x3 by the core (the easy choice as there is only one) that I decide to hold the 3x3x3 by a corner. This way on each axis my independent layers become one face layer and one slice layer. The other face layer is the dependent one. Now the analysis is a bit more complicated as we must analyze the puzzle with respect to each of the 8 corners as they are in a sense all equivalent. You just run through the analysis once for each of the corners serving as a holding point. Here is the results you get in that case.

http://www.twistypuzzles.com/forum/download/file.php?id=19188&mode=view

Note you get the same Complex 3x3x3 with the same 64 pieces. I believe if I did the same thing for the 4x4x4 I'd get very similar results. I just haven't done that yet as there are 512 pieces to keep track of and not just 64 but I see no reason to expect something different to happen.

What is it that makes you think you get a fundamentally different puzzle?

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Wed Nov 28, 2012 11:31 pm 
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bmenrigh wrote:
Let me start and the beginning too. To me a Complex puzzle is a puzzle that has a piece for every subset of the turnable parts. There are 12 turnable parts on a 4x4x4 (4 per axis) and for any given subset of those parts, there is a piece that turns with those and only those parts.

The number of subsets for a set of 12 items is 2^12.
Ok... using this logic then the 3x3x3 has 9 turnable parts, 3 on each axis if you count the slice layer. So why does your picture of the complex 3x3x3 NOT contain 2^9 pieces?
bmenrigh wrote:
Part of the issue is that I don't quite understand your definition of an imaginary piece so I don't understand why the uf piece isn't an imaginary piece.
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: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 12:16 am 
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wwwmwww wrote:
bmenrigh wrote:
Let me start and the beginning too. To me a Complex puzzle is a puzzle that has a piece for every subset of the turnable parts. There are 12 turnable parts on a 4x4x4 (4 per axis) and for any given subset of those parts, there is a piece that turns with those and only those parts.

The number of subsets for a set of 12 items is 2^12.
Ok... using this logic then the 3x3x3 has 9 turnable parts, 3 on each axis if you count the slice layer. So why does your picture of the complex 3x3x3 NOT contain 2^9 pieces?
Good question. I think this really gets at my view of things.

I'd say the 3x3x3 doesn't have a directly turnable middle slice. Actually I'd say no order-2 puzzle (for example the Master Pentultimate or Master Skewb) has a directly turnable middle slice. The middle slice is this special thing that is the collection of pieces that don't turn with opposite grips on the same axis.

Take the core of a 3x3x3 for example. It is the piece that doesn't turn with any face. The piece with the definition 000000. The middle slice turns this piece though. Why? Because to turn the middle slice you actually turn two opposite faces and then re-orient the puzzle. All pieces move in a re-orientation regardless of their turning definition.

This is why on the Complex 2x2x2 I say there is an implied middle slice. It's the set of pieces that don't turn with opposite faces. The same for the Complex 4x4x4.

This is also why I suggested that if the objection to the Complex NxNxN where N is even puzzles is the middle slice, then you could remove all pieces that belong in a middle slice on any of the axes. For the Complex 2x2x2 this means eliminating the 37 pieces that either don't turn with both R and L, or both F and B, or both U and D. On the Complex 4x4x4 it's why I suggested the pieces that don't turn both opposite slice and face pairs. It is these pieces that imply the middle layer.

The other, non-middle slices on a puzzle are just additional, directly turnable options for each face.
wwwmwww wrote:
bmenrigh wrote:
Part of the issue is that I don't quite understand your definition of an imaginary piece so I don't understand why the uf piece isn't an imaginary piece.
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.

Carl

I was hoping you were constructing imaginary pieces directly but it sounds to me like in order to get a list of imaginary pieces you:

Come up with the definition of a Complex Puzzle and enumerate all of the pieces.
Determine which pieces are real and subtract them out.
Determine which pieces are virtual and subtract them out.
Everything left is an imaginary piece.

On the Complex 4x4x4 I think it should contain the pieces mentioned above that turns with the u slice and f slice. Since that piece isn't contained in your overall Complex 4x4x4 set, it can't be left over in the imaginary set when you subtract out the real and virtual pieces.


Edit: minor grammatical cleanups.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 12:31 am 
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wwwmwww wrote:
bmenrigh wrote:
For example, if you decided that the {u, f, r} slices are the dependent layer then what'd you get would be fundamentally different than if you'd chosen the {U, F, R} face as the dependent layer.
You sure of that?[...]
What is it that makes you think you get a fundamentally different puzzle?

I'm not entirely sure I'm correct here but this is the logic I'm using.

Your definition of the Complex 4x4x4 with its 512 pieces does not contain pieces equivalent to all 64 of the Complex 3x3x3 pieces.

So if I can can show a direct mapping from a careful selection of which layer of the Complex 4x4x4 I leave out for each axis that is equivalent to the Complex 3x3x3 then it would imply different selections for the dependent layer produce different sets of pieces.

Here is how I think you can do that. Choose the three dependent layers, one for each axis, as the u, f, r slices. Then each axis would be: {U, d, D}, {F, b, B}, {R, l, L}. These 9 options produce 2^9 = 512 pieces. Now remove from that list all 448 pieces that turn with either the d, b, or l slices. All that is left are the pieces that turn with the U, D, F, B, R, L faces, all 2^64 of them. This set of 64 pieces behaves identically to the set of pieces in the Complex 3x3x3.

Either the assumption that your Complex 4x4x4 does not contain a set of pieces equivalent to the Complex 3x3x3 is wrong or by choosing the dependent layers for each axis as the {u, f, r} slices has produced a different set of 512 pieces.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 2:27 pm 
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bmenrigh wrote:
Good question. I think this really gets at my view of things.
Oh and I'm not sure this helps. Back when we started I was inclinded to think of your view as equally valid... just much less interesting and less general. The more I read the more I think its just wrong.
bmenrigh wrote:
I'd say the 3x3x3 doesn't have a directly turnable middle slice. Actually I'd say no order-2 puzzle (for example the Master Pentultimate or Master Skewb) has a directly turnable middle slice. The middle slice is this special thing that is the collection of pieces that don't turn with opposite grips on the same axis.
You are happy to admite that the 3x3x3 is an Order=2 puzzle. Or to keep Andreas happy I need to say you recognize the 3x3x3 has 2 independant layer per axis of rotation. However your model seem to restrict the dependant layer to being the slice layer. In effect this only acknowledges the core as a valid holding point, or piece with definition 000000. My model is general enough to allow any of the REAL pieces to serves as this holding point... to hold the definition 000000 (in a way) and it still produces the same picture of the Complex 3x3x3.
bmenrigh wrote:
Take the core of a 3x3x3 for example. It is the piece that doesn't turn with any face. The piece with the definition 000000. The middle slice turns this piece though. Why? Because to turn the middle slice you actually turn two opposite faces and then re-orient the puzzle. All pieces move in a re-orientation regardless of their turning definition.
No disagreement here. But you can also consider a corner as the cubie that isn't turned with any of the independant layers and it still produces the same puzzle. My model is the same as yours if I happen to pick the FACE layers but that choice isn't required. You seem unwilling to let go of that choice and I feel you are missing some interesting stuff as a result.
bmenrigh wrote:
This is why on the Complex 2x2x2 I say there is an implied middle slice. It's the set of pieces that don't turn with opposite faces. The same for the Complex 4x4x4.
It hits me as you are just unwilling to acknowledge the the 2x2x2 has ONLY 1 independant layer per axis of rotation. As such your holding point is in the OTHER face layer. Your unwillingness to acknowledge that OTHER face layer as dependant is why you believe there must be an implied slice layer. In my model this is no other layer available to stick your holding point into. This does make the analysis of the 2x2x2 more tricky as there is now no single copy of any one piece type that can be picked to serve as a UNIQUE holding point. Here is my analysis of the 2x2x2.

http://www.twistypuzzles.com/forum/download/file.php?id=19189&sid=9328312b763f26aa81cf9dca81ff1723&mode=view

The notation [] indicates the piece with definition 000.
bmenrigh wrote:
This is also why I suggested that if the objection to the Complex NxNxN where N is even puzzles is the middle slice, then you could remove all pieces that belong in a middle slice on any of the axes. For the Complex 2x2x2 this means eliminating the 37 pieces that either don't turn with both R and L, or both F and B, or both U and D. On the Complex 4x4x4 it's why I suggested the pieces that don't turn both opposite slice and face pairs. It is these pieces that imply the middle layer.
But even with the pieces you remove you get different pieces counts then I do and I suspect your puzzle is still stuck with more independant layers then it should have. I don't mind talking about these other subsets of the Complex 5x5x5 for example but in my mind these are NOT the Complex 4x4x4. The pieces of my Complex 4x4x4 are there in your Complex 5x5x5 so there may be something to learn from looking at which pieces I'm excluding from the Complex 5x5x5 and comparing them with the pieces you are wanting to excluded but as there are SO many pieces in the Complex 5x5x5 I must confess I don't have a good grasp on exactly what that might be at this stage.
bmenrigh wrote:
The other, non-middle slices on a puzzle are just additional, directly turnable options for each face.
See this just strikes me again as you are wanting to call all four layers of the 4x4x4 as independant as they are "non-middle" slices. This to me is just flat... incorrect.
bmenrigh wrote:
I was hoping you were constructing imaginary pieces directly but it sounds to me like in order to get a list of imaginary pieces you:

Come up with the definition of a Complex Puzzle and enumerate all of the pieces.
Determine which pieces are real and subtract them out.
Determine which pieces are virtual and subtract them out.
Everything left is an imaginary piece.
Yes... this is mostly true. But I can say something about the properties this gives the imaginary pieces. An imaginary piece is NEVER a holding point. Another way of looking at this is to say that an imaginary piece MUST be in at least 2 (or more) distinct layers (can be either dependent or independent) on at least 1 axis. So your 3x3x3 edge piece can NEVER be imaginary. If its going to be part of the Complex 4x4x4 then it most also be a part of the Normal 4x4x4. And in effect you are now calling the Normal 4x4x4 and the Normal 5x5x5 the same puzzle when I hope we can agree they are different.
bmenrigh wrote:
On the Complex 4x4x4 I think it should contain the pieces mentioned above that turns with the u slice and f slice. Since that piece isn't contained in your overall Complex 4x4x4 set, it can't be left over in the imaginary set when you subtract out the real and virtual pieces.
True... in my model that piece exists as a part of the Complex 5x5x5 and not in the Complex 4x4x4. You think the Complex 4x4x4 should contain it because you are giving the 4x4x4 the same number of independant layers as the 5x5x5 and I think that is incorrect.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 3:06 pm 
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bmenrigh wrote:
Either the assumption that your Complex 4x4x4 does not contain a set of pieces equivalent to the Complex 3x3x3 is wrong or by choosing the dependent layers for each axis as the {u, f, r} slices has produced a different set of 512 pieces.
I'll have to dig into this. Don't know enough at the moment to comment.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 5:11 pm 
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bmenrigh wrote:
I'm not entirely sure I'm correct here but this is the logic I'm using.
Ok... I think I now see the hole in this logic.
bmenrigh wrote:
Your definition of the Complex 4x4x4 with its 512 pieces does not contain pieces equivalent to all 64 of the Complex 3x3x3 pieces.
Correct. Emphasis placed on all. You still have the 2x2x2 corners which have been around since the Complex 2x2x2 for example.
bmenrigh wrote:
So if I can can show a direct mapping from a careful selection of which layer of the Complex 4x4x4 I leave out for each axis that is equivalent to the Complex 3x3x3 then it would imply different selections for the dependent layer produce different sets of pieces.
Correct. You are allowed to select any 1 of the 4 layers on each axis as the dependant one. I still believe the choice as to which you pick doesn't matter.
bmenrigh wrote:
Here is how I think you can do that. Choose the three dependent layers, one for each axis, as the u, f, r slices. Then each axis would be: {U, d, D}, {F, b, B}, {R, l, L}. These 9 options produce 2^9 = 512 pieces.
I follow you and agree with you to this point.
bmenrigh wrote:
Now remove from that list all 448 pieces that turn with either the d, b, or l slices.
So you are now picking d, b, and l as your dependant slices? Correct. You are aware that at this point you then need to put back in the 448 pieces that will relate to the pieces that move with the u, f, and r slices if you are to maintain a list of pieces that are found in a Face-turn Hexahedron with 3 independant layers per axis. What you have now done is simply say that you have two dependant layers per axis. I can do this with a real 4x4x4 too. Just bandage all the face centers together on each face and what do you have. Note the wing pieces can no longer be seperated either. So I now have a 3x3x3 puzzle with REAL face centers and REAL edges. That still doesn't mean those pieces exist on the 4x4x4 (Normal or Complex) once the bandaging you've just applied is removed.
bmenrigh wrote:
All that is left are the pieces that turn with the U, D, F, B, R, L faces, all 2^64 of them. This set of 64 pieces behaves identically to the set of pieces in the Complex 3x3x3.
That is because this is a Complex 3x3x3. So yes, you've proven that my Complex 4x4x4 can be bandaged into a Complex 3x3x3. This property is also true for the Normal 4x4x4 and the Normal 3x3x3 so I agree that is a nice property for it to have.
bmenrigh wrote:
Either the assumption that your Complex 4x4x4 does not contain a set of pieces equivalent to the Complex 3x3x3 is wrong or by choosing the dependent layers for each axis as the {u, f, r} slices has produced a different set of 512 pieces.
I'd simply say there is a 3rd option you over looked. That being that it is possible to bandage (remove degrees of freedom... in this case make one other layer dependant per axis) to re-create the pieces found in the Complex 3x3x3.

To more easily see what you've just done let's look at the Complex 2x2x2. It has 8 pieces. If my turning layers are U, F, and R and I remove one dependant layer per axis, then I need to remove all the pieces that turn with these layers. That leaves me with the single corner cubie in the DBL position. And you are more then welcome to call this piece the Complex 1x1x1 if you like. (Note 2^0 = 1. So I'd expect the Complex 1x1x1 to contain just 1 piece.) Without the pieces that have just been removed that is basically what it is. That doesn't mean that in the presence of the other 7 corners that that piece doesn't just behave like the others and you have a Normal 2x2x2.

Follow?
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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 5:39 pm 
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wwwmwww wrote:
bmenrigh wrote:
Here is how I think you can do that. Choose the three dependent layers, one for each axis, as the u, f, r slices. Then each axis would be: {U, d, D}, {F, b, B}, {R, l, L}. These 9 options produce 2^9 = 512 pieces.
I follow you and agree with you to this point.
bmenrigh wrote:
Now remove from that list all 448 pieces that turn with either the d, b, or l slices.
So you are now picking d, b, and l as your dependant slices? Correct. You are aware that at this point you then need to put back in the 448 pieces that will relate to the pieces that move with the u, f, and r slices if you are to maintain a list of pieces that are found in a Face-turn Hexahedron with 3 independant layers per axis. What you have now done is simply say that you have two dependant layers per axis.[...]
No I was trying to define a subset of the 512 pieces.

Choose the u, f, r slices as the dependent slices and the remaining independent layers are {U, d, D}, {F, b, B}, {R, l, L}. We both agree this describes 512 unique pieces. Now I want to look at just a subset of these 512 pieces. The 64 I want to look at are all of the pieces that don't turn with {d, b, l} at all. That is, all of the pieces who's definition has a 0 in the d, b, and l places. There are 448 pieces that turn with at least one of these slices and 64 pieces that don't turn with any.

This set subset of 64 pieces is isomorphic to the 64 pieces in the Complex 3x3x3 which means the overall set of 512 pieces described above contains 64 pieces that are identical to the Complex 3x3x3 which would make a Complex 4x4x4 defined in this way a superset of the Complex 3x3x3.


I don't have a huge problem with the Complex 4x4x4 being a superset of the Complex 3x3x3 but I think if we choose a different set of dependent layers we'll end up with other things.

For example, instead of choosing the dependent layers as {u, f, r} instead choose them as {U, f, r} so that the independent layers are {u, d, D}, {F, b, B}, {R, l, L}. This also defines a set of 512 pieces. The question is, is this set of 512 pieces really isomorphic to the previously described set of 512 pieces? A necessary but not sufficient condition for that to be true is if some subset of 64 pieces of this new set of 512 is also isomorphic to the Complex 3x3x3. I can't find one. Perhaps there is a clever mapping that I'm not spotting. It wouldn't surprise me.


The other thing I'd like to point out about picking subsets relates to the Complex 2x2x2. Take the Complex 4x4x4 and choose the dependent layers to be {U, F, R} so that the independent layers are {u, d, D}, {f, b, B}, {r, l, L}. This defines a set of 512 pieces. Now take the subset of 64 pieces that only turn with the {u, d, f, b, r, l} slices and don't turn with the {D, B, L} faces. This means discard the 448 pieces that do turn with any of the {D, B, L} faces. This subset of 64 pieces is also isomorphic to the Complex 3x3x3. What's troubling though is that the {u, d, f, b, r, l} slices all share a deep cut just like the 2x2x2 is deep cut. The subset of the Complex 4x4x4 defined in this way is isomorphic to the Complex 3x3x3 but picking the pieces that turn with just the {u, d, f, b, r, l} slices seems to be equivalent to how the 2x2x2 behaves and this doesn't match up with your Complex 2x2x2 calculation.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 6:09 pm 
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wwwmwww wrote:
You are happy to admite that the 3x3x3 is an Order=2 puzzle. Or to keep Andreas happy I need to say you recognize the 3x3x3 has 2 independant layer per axis of rotation. However your model seem to restrict the dependant layer to being the slice layer. In effect this only acknowledges the core as a valid holding point, or piece with definition 000000. My model is general enough to allow any of the REAL pieces to serves as this holding point... to hold the definition 000000 (in a way) and it still produces the same picture of the Complex 3x3x3.
bmenrigh wrote:
Take the core of a 3x3x3 for example. It is the piece that doesn't turn with any face. The piece with the definition 000000. The middle slice turns this piece though. Why? Because to turn the middle slice you actually turn two opposite faces and then re-orient the puzzle. All pieces move in a re-orientation regardless of their turning definition.
No disagreement here. But you can also consider a corner as the cubie that isn't turned with any of the independant layers and it still produces the same puzzle. My model is the same as yours if I happen to pick the FACE layers but that choice isn't required. You seem unwilling to let go of that choice and I feel you are missing some interesting stuff as a result.

I understand your idea of the holding point and I can see how any of the real 3x3x3 pieces can be used as the holding point and the resulting puzzle to be equivalent to a 3x3x3.

I'm having trouble doing that in my head for the Complex 3x3x3 so let me describe where I'm stuck.

Now I'm trying to imagine the definition and behavior of the UD piece relative to the U and D pieces. On the "traditional" Complex 3x3x3, if you turn the U face, the UD piece and U center piece turn and their orientations stay linked together through the turn but their orientation changes relative to the D center. If you turn the D face the D center and UD piece turn and their orientations stay linked, but their orientations change relative to the U center. I'd expect this property / behavior to have an equivalent for any other definition of the Complex 3x3x3.

We both agree that a Complex 3x3x3 has two independent layers per axis. I want to choose those layers as {U, D, F, B, R, L} and not use the middle slice but let me try to use a middle slice instead. I will introduce the middle slice between the U and D faces as M. Instead of M being the dependent layer, choose U and M to be the independent layer and D to be the dependent layer. Leave {F, B, R, L} as the independent layers for the other axes.

I believe using this definition the holding-point piece is the face center on the D face (it has the 000000 definition).

With this definition of the Complex 3x3x3 I don't know how to define the UD piece. It still must have the U bit set. Is the other bit set the M bit? Is it now the UM piece? Perhaps the M bit is not set?

Let's look at simulating a D turn by instead doing a U and M turn and then a re-orientation.

Since I don't know if the UD piece should have the M bit set lets look at both cases.
First with M set:
Do U, M and the re-orient to simulate a D turn. The UD piece moved with the U turn (relative to the D center) and then again with the M turn (relative to the D center). The re-orientation doesn't change the relative orientations of things so the UD piece orientation is no longer linked to the D center which means the D move we tried to simulate did not properly simulate a D move on the Complex 3x3x3.
Now with M unset:
Do U, M and the re-orient to simulate a D turn. The UD piece moved with the U turn (relative to the D center). When M was turned it did not. The re-orientation did not change the orientations relative to each other so the UD piece orientation still does not match the D center orientation which means this procedure did not properly simulate a D turn.


I see two possibilities for resolving this. The first is that changing the definition of the Complex 3x3x3 so that D is a dependent layer changes the resulting puzzle. The second is that I haven't setup the UD piece properly or simulated the D turn properly. If this is the case I'd like to understand how the UD piece works using the {U, M, F, B, R, L} definition of the Complex 3x3x3.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 6:13 pm 
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bmenrigh wrote:
Choose the u, f, r slices as the dependent slices and the remaining independent layers are {U, d, D}, {F, b, B}, {R, l, L}. We both agree this describes 512 unique pieces. Now I want to look at just a subset of these 512 pieces. The 64 I want to look at are all of the pieces that don't turn with {d, b, l} at all. That is, all of the pieces who's definition has a 0 in the d, b, and l places. There are 448 pieces that turn with at least one of these slices and 64 pieces that don't turn with any.
Agreed... but in this picture of the Complex 4x4x4 there are ATLEAST 8 copies of each piece type. This method of taking a subset doesn't keep all the copies of each type. Let's apply this method of taking a subset to just the 64 REAL pieces in the NORMAL 4x4x4 and see what happens. Your subset now includes just one copy of the 8 central corner pieces. Does that make this corner piece isomorphic to a 3x3x3 core? Let's look at the 6 face center pieces that remain. Surely these can't be considered as isomorphic to a 3x3x3 face piece can they if you are NOT removing independent layers. Because I can perform independant turns and get 4 face centers onto the same side of the puzzle. Let's see a 3x3x3 do that.
bmenrigh wrote:
This set subset of 64 pieces is isomorphic to the 64 pieces in the Complex 3x3x3 which means the overall set of 512 pieces described above contains 64 pieces that are identical to the Complex 3x3x3 which would make a Complex 4x4x4 defined in this way a superset of the Complex 3x3x3.
This EXACT same argument would make the Normal 4x4x4 a superset of the NORMAL 3x3x3. A Normal 4x4x4 has 64 pieces. If I look at the subset of pieces that ONLY turn with U, D, F, B, R, and L that will leave you with just 27 pieces. Ask yourself are these 27 pieces IDENTICAL to the NORMAL 3x3x3. I think you will see there this fails.
bmenrigh wrote:
I don't have a huge problem with the Complex 4x4x4 being a superset of the Complex 3x3x3 but I think if we choose a different set of dependent layers we'll end up with other things.

For example, instead of choosing the dependent layers as {u, f, r} instead choose them as {U, f, r} so that the independent layers are {u, d, D}, {F, b, B}, {R, l, L}. This also defines a set of 512 pieces. The question is, is this set of 512 pieces really isomorphic to the previously described set of 512 pieces? A necessary but not sufficient condition for that to be true is if some subset of 64 pieces of this new set of 512 is also isomorphic to the Complex 3x3x3. I can't find one. Perhaps there is a clever mapping that I'm not spotting. It wouldn't surprise me.


The other thing I'd like to point out about picking subsets relates to the Complex 2x2x2. Take the Complex 4x4x4 and choose the dependent layers to be {U, F, R} so that the independent layers are {u, d, D}, {f, b, B}, {r, l, L}. This defines a set of 512 pieces. Now take the subset of 64 pieces that only turn with the {u, d, f, b, r, l} slices and don't turn with the {D, B, L} faces. This means discard the 448 pieces that do turn with any of the {D, B, L} faces. This subset of 64 pieces is also isomorphic to the Complex 3x3x3. What's troubling though is that the {u, d, f, b, r, l} slices all share a deep cut just like the 2x2x2 is deep cut. The subset of the Complex 4x4x4 defined in this way is isomorphic to the Complex 3x3x3 but picking the pieces that turn with just the {u, d, f, b, r, l} slices seems to be equivalent to how the 2x2x2 behaves and this doesn't match up with your Complex 2x2x2 calculation.
I think I've already shown the assumption you made going into this argument are wrong. However at this point the correct way to answer ALL these questions would be to create a table like the following:

http://www.twistypuzzles.com/forum/download/file.php?id=19188&mode=view

for each of the possible holding points. That would be:

(1) The 8 outer corners
(2) The 8 inner corners
(3) The 24 face centers
(4) The 24 wings

So that would be a total of 4 tables with 512 rows. Note the last 2 would require 24 columns. I made the above table for the Complex 3x3x3 above by hand and it took me a few days to double check everything. I really don't want to do this for the Complex 4x4x4 by hand so I'd have to write a program to generate these tables and I'm not sure when I might have the time to do that. But I feel very confident that ALL four tables will produce the EXACT same 512 pieces. If they don't... this is the approach you'd have to take to prove it to me. [Not so coy an attempt to get you do to the leg work... ;)]

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 6:45 pm 
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bmenrigh wrote:
I see two possibilities for resolving this. The first is that changing the definition of the Complex 3x3x3 so that D is a dependent layer changes the resulting puzzle. The second is that I haven't setup the UD piece properly or simulated the D turn properly. If this is the case I'd like to understand how the UD piece works using the {U, M, F, B, R, L} definition of the Complex 3x3x3.
This is easy for me to tell you how I'd approach the problem. You'd make a table very similar to this one:

http://www.twistypuzzles.com/forum/download/file.php?id=19188&mode=view

On the left you would have your 64 pieces. The first column would be labeled UMFBRL. However you would also need 5 more columns, one for each of the other 5 face centers serving as the holding point. Once all 6 columns were filled it would be easy to see which pieces were of the same type and you could add that info on the right (as I did) and you would see that this produced the same 10 pieces types and that you have the same Complex 3x3x3. Would you like me to make that table for you? That shouldn't take me more then a few hours... I might be able to do that this weekend.

As I've said above the picture gets more complicated when you pick a piece type to serve as your holding point which isn't unique. Look at my analysis of the Complex 2x2x2 which I've linked to above but here it is again as well.

http://www.twistypuzzles.com/forum/download/file.php?id=19189&sid=9328312b763f26aa81cf9dca81ff1723&mode=view

If you pick RUF as your independant layers it appears you may have 4 different piece types. Left most column. There is a piece that doesn't turn with any layers, there are 3 pieces that turn with 1 layer, there are 3 pieces that turn with 2 layers, and another piece which turns with all 3 layers. However you must realize that the choice of corner you pick to hold is arbitrary. So to get the full picture you must repeat this analysis for all 8 corners. Once you've done that and looked at things as a whole you'll see the signature of all 8 pieces is the same. I've done this with the corners and core of the 3x3x3 and shown they produce the same Complex 3x3x3. I have been lazy and I have not yet done this for the face centers and the edges but I would really be amazed if they both didn't produce the exact same picture for the Complex 3x3x3.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 6:56 pm 
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wwwmwww wrote:
Would you like me to make that table for you? That shouldn't take me more then a few hours... I might be able to do that this weekend.
This would be much better to do programmatically. I've already looked at your table some so let me try to figure out a easy way to generate it with a program. Can you describe which slice is which? From your edges it looks like:
3 is the slice between F and B (turns same direction as F)
2 is the slice between U and D and turns the same way as U
1 is the slice between R and L and turns the same way as R.

Does this sound right?

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 7:06 pm 
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bmenrigh wrote:
This would be much better to do programmatically.
I agree. But then I'd need to get QuickBasic4.5 installed and running under Windows8. That may be doable but that would take me a few hours by itself. Writing the program and debugging would take me the better part of a day too. My programming skills are far below yours...
bmenrigh wrote:
I've already looked at your table some so let me try to figure out a easy way to generate it with a program. Can you describe which slice is which? From your edges it looks like:
3 is the slice between F and B (turns same direction as F)
2 is the slice between U and D and turns the same way as U
1 is the slice between R and L and turns the same way as R.

Does this sound right?
Yes, I think so... as I picked all 3 slice layers in my example I don't really think it makes a difference. You should be able to use any naming convention that you prefer.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 7:55 pm 
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wwwmwww wrote:
bmenrigh wrote:
This would be much better to do programmatically.
I agree. But then I'd need to get QuickBasic4.5 installed and running under Windows8. That may be doable but that would take me a few hours by itself. Writing the program and debugging would take me the better part of a day too. My programming skills are far below yours...
If we need a bigger table I'll be happy to do the coding work needed to generate it.

Looking at your table closely I think I understand it now and I think it illustrates the same problem I was trying to show with the UD piece using the D center as the holding point.

Following your table, the first column has the independent layers RUF123 which is the same thing as saying the L, D, B layers are dependent (which is why the LDB corner is the [] piece).

I want to turn the UD piece and D center the same direction relative to the U center. On the "traditional" Complex 3x3x3 this would take just turning the D face which would turn the UD and D center relative to the U center but keep their orientation the same relative to each other.

For your RUF123 column, the relevant pieces are:
U center -> [U13]
D center -> [13]
UD piece -> [U123]

We can't use the 1 or 3 slice because that would move / permute all three pieces relative to the holding point and not change their orientations relative to each other.

The only remaining turning layers that could affect any of the piece are the U and 2 layers. The U will turn the U center and UD piece relative to the Dcenter but keep their orientations linked relative to each other.

The 2 slice will turn the UD piece relative to both the U and D centers which also doesn't achieve the goal and isn't even a possible position on the traditional Complex 3x3x3.

If I've read and understood the table correctly then I'm not sure what the puzzles are that your table describes. They have 64 pieces but they don't match up with the Complex 3x3x3 with only face turns.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 8:40 pm 
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bmenrigh wrote:
If I've read and understood the table correctly then I'm not sure what the puzzles are that your table describes. They have 64 pieces but they don't match up with the Complex 3x3x3 with only face turns.
Ok... my brain is fried for the day so I'm too lazy to see what is wrong with this picture at the moment. However something must be wrong with it. The pieces you are talking about exist on a normal 3x3x3. So let's take the Complex part out of the equation... are you really trying to convince me that a 3x3x3 held by the core produces a completely different puzzle then a 3x3x3 held by a face center. I believe we both know better then that.

As for the 4x4x4 (Complex or Normal) defined by {U, d, D}, {F, b, B}, {R, l, L}.... if you still believe the subset of pieces left after you remove the pieces turned by {d, b, l} is a 3x3x3 (Complex or Normal) let me ask you to do the following. Perform the operation F followed by the operation d on this subset. Do you still think they are a 3x3x3?

Carl

P.S. Ok... reading this again you want to hold the D face center as your holding point. Correct? Now you are asking me how you can turn the D face and the UD piece (which is imaginary so ignore what I just said above) . Well the answer should be to turn U and 2 in sequence... both in the opposite direction that you wanted to turn D and then to follow this with a global rotation of the entire puzzle. The U turn moves the layer containing the piece UD by 90 degrees. The 2 turn which is a slice turn moves the UD piece by -90 degrees. Note there are other pieces in this 2 turn that are moving by +90 degrees. You have to remember how the 2 turn is defined. On the normal face turn Complex 3x3x3 it is a linear combination of a U turn and a D turn followed by a global rotation of the entire puzzle. So after these two turns the D piece and the UD piece are aligned and the global rotation gives you what you want.

Take a 5x5x5 and defined the Complex 3x3x3 turns as a face turn and the central slice turning with it in the same direction. Now perform a 90 degree U turn and a 90 degree D turn rotating them the same direction relative to the axis. Now grab the axis and rotate that back -90 degrees. What you see is the effect of one 2 turn.

I hope my fried brain kept that all strait

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 8:57 pm 
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wwwmwww wrote:
bmenrigh wrote:
If I've read and understood the table correctly then I'm not sure what the puzzles are that your table describes. They have 64 pieces but they don't match up with the Complex 3x3x3 with only face turns.
Ok... my brain is fried for the day so I'm too lazy to see what is wrong with this picture at the moment.
I know what you mean :? Brain has been pretty much incapable of higher level thought on anything other than this problem for about two days now. Even if I turn out to be wrong in the end I'm really enjoying trying to think deeply about these abstract puzzles. I think debate and disagreement push creativity and logical boundaries a lot faster consensus.

wwwmwww wrote:
However something must be wrong with it. The pieces you are talking about exist on a normal 3x3x3. So let's take the Complex part out of the equation... are you really trying to convince me that a 3x3x3 held by the core produces a completely different puzzle then a 3x3x3 held by a face center. I believe we both know better then that.
No I'm not talking about purely 3x3x3 pieces here. The UD piece is an imaginary piece on the complex 3x3x3. There are three of them, they have 8 orientations (although relative to the traditional core piece, each one can only reach 4 orientations).

I'm pretty sure all of the "real" pieces that can serve as a holding point are well defined and behave properly with each other no mater how you define the holding points. This includes all of the pieces on a 3x3x3 including the core. I believe it's this UD piece that has trouble when the holding point is defined as one of the corners of the corners.

wwwmwww wrote:
As for the 4x4x4 (Complex or Normal) defined by {U, d, D}, {F, b, B}, {R, l, L}.... if you still believe the subset of pieces left after you remove the pieces turned by {d, b, l} is a 3x3x3 (Complex or Normal) let me ask you to do the following. Perform the operation F followed by the operation d on this subset. Do you still think they are a 3x3x3?
I will work on this.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 9:11 pm 
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bmenrigh wrote:
No I'm not talking about purely 3x3x3 pieces here. The UD piece is an imaginary piece on the complex 3x3x3.
Yes, my bad. On first reading I wasn't thinking the U and D faces were opposite so in my head I was seeing a normal edge piece. I caught that on my second reading and edited the above post with a P.S. Read that and see if that helps. I had to go hunting for a 5x5x5 to help me visualize things and the best I could find was one of my Uniaxial 3x3x3s so I couldn't actually make the turns I wanted to on the puzzle and still ended up doing them in my head. So I could easily have something backwards there. But I firmly believe the approach is correct.

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 9:23 pm 
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bmenrigh wrote:
Brain has been pretty much incapable of higher level thought on anything other than this problem for about two days now. Even if I turn out to be wrong in the end I'm really enjoying trying to think deeply about these abstract puzzles. I think debate and disagreement push creativity and logical boundaries a lot faster consensus.
Oh it really wears me down. I feel like I see something so clearly and I simply can't get anyone else to 'see' it. That coupled with the fact that in many other ways people like you and Andreas really make me look like an amateur. You're programming skills just being one example. So I have nothing but the most respect for you guys. And now I find myself in the uncomfortable position of proving you wrong [which I really do believe that you are. ;)] Yes I know its just good debate but I guess I have 'debated' too many topics with woman and those NEVER fail to ALWAYS turn into arguments as they (yes I'm being overly general and I'll probably get stepped on as a result) can't seem to separate themselves/their emotions from the topic. And I really really hate to argue.

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 9:42 pm 
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wwwmwww wrote:
bmenrigh wrote:
Brain has been pretty much incapable of higher level thought on anything other than this problem for about two days now. Even if I turn out to be wrong in the end I'm really enjoying trying to think deeply about these abstract puzzles. I think debate and disagreement push creativity and logical boundaries a lot faster consensus.
Oh it really wears me down. I feel like I see something so clearly and I simply can't get anyone else to 'see' it. That coupled with the fact that in many other ways people like you and Andreas really make me look like an amateur. You're programming skills just being one example. So I have nothing but the most respect for you guys. And now I find myself in the uncomfortable position of proving you wrong [which I really do believe that you are. ;)] Yes I know its just good debate but I guess I have 'debated' too many topics with woman and those NEVER fail to ALWAYS turn into arguments as they (yes I'm being overly general and I'll probably get stepped on as a result) can't seem to separate themselves/their emotions from the topic. And I really really hate to argue.

Carl

Those are kind words, thank you. I have a huge amount of respect for you too. Last night I read the entire thread by you and Andreas about holding point pieces and your discovery of new virtual pieces on various puzzles. Then I looked at the dates... 2009... A full year before I'd even begun thinking about these things. You've made huge contributions to the twisty puzzle theory and I've made none (or close to none).

I've been trying my hardest to read (without emotion) and re-read until I understand, every point you've made and then come up with a question / example / counter-example / rebuttal.

Eventually I think we're going to come to understand the branch in the logical tree where we diverge. I'd really like to understand if that branch is the cause of a mistake or of a valid alternative view on something. I started out thinking we differed because of a valid alternative view but I'm leaning towards one of us being wrong. I'm actually feeling pretty 50/50 on if it's you or me.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 9:53 pm 
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wwwmwww wrote:
P.S. Ok... reading this again you want to hold the D face center as your holding point. Correct? Now you are asking me how you can turn the D face and the UD piece (which is imaginary so ignore what I just said above) . Well the answer should be to turn U and 2 in sequence... both in the opposite direction that you wanted to turn D and then to follow this with a global rotation of the entire puzzle. The U turn moves the layer containing the piece UD by 90 degrees. The 2 turn which is a slice turn moves the UD piece by -90 degrees. Note there are other pieces in this 2 turn that are moving by +90 degrees. You have to remember how the 2 turn is defined. On the normal face turn Complex 3x3x3 is a linear combination of a of a U turn and a D turn followed by a global rotation of the entire puzzle. So after these two turns the D piece and the UD piece are aligned and the global rotation gives you what you want.
I had not considered the UD piece twisting the opposite way as the rest of the pieces in the 2 slice. I do think this is the correct definition though.

If you either define slices in this way or just define slices as doing the U, D', re-orient then the holding piece is just re-orientation games and all of the complex pieces will stay consistent with each other.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 10:00 pm 
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wwwmwww wrote:
As for the 4x4x4 (Complex or Normal) defined by {U, d, D}, {F, b, B}, {R, l, L}.... if you still believe the subset of pieces left after you remove the pieces turned by {d, b, l} is a 3x3x3 (Complex or Normal) let me ask you to do the following. Perform the operation F followed by the operation d on this subset. Do you still think they are a 3x3x3?
For this group to be isomorphic with the Complex 3x3x3 there must be a function that maps each state of one group to the other while preserving all properties / structure of the group.

In the case of doing F, then d on the 4x4x4 subgroup, in order to perform the d in a way that is consistent with the 3x3x3 you have to tread d as though it were the middle slice on the 3x3x3 between the U and D faces. If you do U, D', re-orient the 4x4x4 subgroup stays 100% consistent with a 3x3x3.

I started making images to demo this but I think you're issue with this answer, if you have one, will be defining the d slice as U, D', re-orient, not whether images look good or not.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 10:11 pm 
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bmenrigh wrote:
If you either define slices in this way or just define slices as doing the U, D', re-orient then the holding piece is just re-orientation games and all of the complex pieces will stay consistent with each other.
AGREED!! One small step in the right direction.

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 10:22 pm 
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bmenrigh wrote:
I started making images to demo this but I think you're issue with this answer, if you have one, will be defining the d slice as U, D', re-orient, not whether images look good or not.
Granted I'm not very happy with that definition. d is an independent layer. That slice layer is not. But that actually wasn't the point I was trying to make. Maybe it was F' then d that I wanted you to do. Its hard for me to see all this in my head without being able to hold a puzzle in my hands. Anyways what I was wanting you to do was to rotate the F face until the only remaining face center on that side was in the d layer. Now if you rotated the d layer then that one face center moves as we've already removed all the other face centers from the d layer. Nothing else on the puzzle should move. How can this picture remain consistent with the 3x3x3 even if I do let you get away with that definition of the d slice on a 3x3x3? As you say...
bmenrigh wrote:
For this group to be isomorphic with the Complex 3x3x3 there must be a function that maps each state of one group to the other while preserving all properties / structure of the group.
To keep things simple let's just look at the Normal 3x3x3 and the 27 pieces that remain of the Normal 4x4x4. I would have to believe there are many more states the 27 pieces on the Normal 4x4x4 can attain that can't be mapped to 3x3x3 states. You are MORE then welcome to post pictures. I was half tempted to make a few POV-Ray animations myself... just too lazy at the moment.

Carl

P.S. Thinking a bit more... if I did move the F layer the wrong direction the first time then there shouldn't have been ANY pieces in the d layer. So I would have thought the F turn followed by nothing on the 3x3x3 would have mapped very nicely so I'm not sure you would have needed to even define the d layer. I do think some pictures would be nice

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Thu Nov 29, 2012 10:55 pm 
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bmenrigh wrote:
Eventually I think we're going to come to understand the branch in the logical tree where we diverge. I'd really like to understand if that branch is the cause of a mistake or of a valid alternative view on something. I started out thinking we differed because of a valid alternative view but I'm leaning towards one of us being wrong. I'm actually feeling pretty 50/50 on if it's you or me.
If you ask me... this is rooted in your 'face' approach to these puzzles and not the 'axis' approach. To you a 2x2x2 is a 3x3x3 where the two cuts overlap in the center on each axis. Seeing as you still view these as two cuts you aren't giving up your hold on the 3x3x3 pieces that now no longer have any volume. So you bring these pieces back when you make the Complex 2x2x2. To me that is boring as you are just remaking the Complex 3x3x3. In my picture there are no longer two cuts on that axis... its just one. So I don't give the 3x3x3 pieces a chance to come back into play and force the Complex 2x2x2 to be something new. Using your picture I think one could argue that the 2x2x2 wasn't a deep cut puzzle. The pieces of one face no longer have an isomorphic mapping to the other pieces in the puzzle if you keep the zero volume 3x3x3 pieces. So its hard for me to say that view is wrong... my biggest disagreement with it is that it doesn't put the 2x2x2 and the 4x4x4 on equal footing with the 3x3x3 and the 5x5x5. They just become subsets. As a collector it hits me this way... assume I wanted to collect all the NxNxN puzzles from N=2 to N=7 and that I somehow missed a 2x2x2 and they were no longer available. Using your picture I could just go buy a 3x3x3 and take the stickers off the face centers and the edges and then I could proudly display it with my complete collection. Personally I'd never be happy with that... but can I really say its wrong?

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Fri Nov 30, 2012 2:10 pm 
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I am afraid I can only add two things:
1. B12C111 is indee the slices-only-cube.
2. Carl got me. HC2 is NOT a subset of HC3. The same is true for every axis system.

Besides that:
I was absent for one day.
After my last post there were 36 postings.
There is a reason I post only one time per thread in a single thread.
This time it is me who is to occupied with his job to follow all this.

Edit: Being more polite, hopefully.


Last edited by Andreas Nortmann on Fri Nov 30, 2012 4:13 pm, edited 1 time in total.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Fri Nov 30, 2012 3:00 pm 
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Okay Carl you've convinced me :D

Your table of holding points on the Complex 3x3x3 showing everything works out no mater how you choose the dependent layer per axis unravels all of my arguments and opposition to your view.

In general your idea of a holding point is a very powerful one and quite a bit more generic than my previous view. The reason I was introducing a middle layer for even-layers-per-axis puzzles is that the holding point I chose to use was always some abstract core at the center of the puzzle. If the puzzle had a middle layer then turning the middle layer would turn the core. If it didn't have a middle layer nothing would turn the core.

If instead you force yourself to pick some finite, non-imaginary volume inside of the puzzle as the fixed holding point that all pieces move around then you don't get to pick some new piece that forces a middle layer to spring out of the nether.

I don't get to have my [r, f] piece on the Complex 4x4x4 because that piece would be forced to turn by the dependent layer on the axis between U and D. 3.1.21 has this piece because it introduces a new dependent layer on each axis and is actually a subset of the Complex 5x5x5, not Complex 4x4x4.

My experience with "grip analysis" (what Andreas calls "twistability analysis") is really born out of experience solving. It is much more convenient mentally to define the dependent layer as the middle layer and if there isn't a middle layer, introduce an imaginary one. I can't introduce a new layer and then use that to prove there is another layer.


EDIT: I'll also mention that there is nothing special about a middle layer. My choice of calling the middle layer the set of pieces that don't turn with the others is but one of N choices where N is the number of layers per axis. By elevating the middle layer to something special I was forcing the 4x4x4 to have one when instead the right approach it to pick a different layer as the dependent layer.

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Fri Nov 30, 2012 4:23 pm 
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Andreas Nortmann wrote:
I am afraid I can only add two things:
1. B12C111 is indee the slices-only-cube.
2. Carl got me. HC2 is NOT a subset of HC3. The same is true for every axis system.
Thanks.
Andreas Nortmann wrote:
Besides that:
I was absent for one day.
After my last post there were 36 postings.
There is a reason I post only one time per thread in a single thread.
in a single day?
Andreas Nortmann wrote:
This time it is me who is to occupied with his job to follow all this.
No problem. It will still be here if you ever want to catch up. Your contribution is ALWAYS welcome. That and I know where you are coming from... its taken me 3 months to get back to thinking about your question for pretty much the exact same reason so we all understand.

Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Fri Nov 30, 2012 4:40 pm 
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bmenrigh wrote:
Okay Carl you've convinced me :D
WOW!!! I really feel like I've acomplished something. Thank you so much. I must say you aren't an easy man to convince and I do agree that certainly helped flush out all the details that I'd probably glossed over too much before.

If you are still interested in making some pieces tables for the Complex 4x4x4 I'd be interested in seeing them but there is no rush. In other words let's NOT do that today. I feel like I need break after finally pulling someone over to my side. That said I CERTAINLY enjoy the company.

Thanks again
Carl

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 Post subject: Re: An incomplete picture... a theory thread
PostPosted: Fri Nov 30, 2012 4:54 pm 
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wwwmwww wrote:
bmenrigh wrote:
Okay Carl you've convinced me :D
WOW!!! I really feel like I've acomplished something. Thank you so much. I must say you aren't an easy man to convince and I do agree that certainly helped flush out all the details that I'd probably glossed over too much before.
It's never easy to convince somebody when they are sure they are right :lol: . Showing the middle layer is not special for the Complex 3x3x3 is what won me over. You've created the special theory of relativity for twisty puzzles. I wanted an absolute fixed reference frame and you've showed we have to choose a reference frame from a piece that exists and define everything relative to that.

wwwmwww wrote:
If you are still interested in making some pieces tables for the Complex 4x4x4 I'd be interested in seeing them but there is no rush. In other words let's NOT do that today. I feel like I need break after finally pulling someone over to my side. That said I CERTAINLY enjoy the company.
So making the table for the Complex 3x3x3 wouldn't be that hard or take that much time or code. Part of the reason though is that the Complex 3x3x3 is nicely symmetrical around the central core piece and it's easy for me to count the number of unique piece types via Burnside's lemma. I'm not yet sure how to go about counting the unique piece types for the Complex 4x4x4, much less enumerate them all. Actually I don't know how to do it for the Complex 2x2x2 either. I understand how it works conceptually but I haven't made the leap from conceptual -> algorithm yet.

wwwmwww wrote:
Thanks again
And thank you! It probably took a lot of patience to put up with my broken arguments :oops: .

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