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 Post subject: Re: An incomplete picture... a theory threadPosted: Fri Nov 30, 2012 5:26 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
You've created the special theory of relativity for twisty puzzles.
Looks like I need to edit the picture in my sig. I LOVE that quote.
bmenrigh wrote:
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.
We can start there. We still need the tables for a face center and an edge serving as a holding point.
bmenrigh wrote:
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.
I beieve I know how to enumerate them and then how to group them into types. Not sure I have the easilest picture in my head but it basically entails listing all the parts and creating the signaure of each part relative to each of the parts of the type chosen as the holding part type. Then the 'complete' signature of each part becomes the list of all these seperate signatures. From these 'complete' signatures it should be possible to tell which parts are of the same type.
bmenrigh wrote:
And thank you! It probably took a lot of patience to put up with my broken arguments .
Oh it was well worth it.

Carl

P.S. Once we start making tables and naming the pieces in the Complex 4x4x4 here is a question I'd be interested in trying to answer. Does the Clockwork Complex 4x4x4 reduce to the Real Clockwork 4x4x4? Both are Face-turning Hexahedrons with only a single independent layer per axis. So I'm curious if this would give us a similar situation as the Real 2x2x2 and the Complex 2x2x2 being the same puzzle. Or does the Clockworks Complex 4x4x4 contain new pieces? If so I bet they'd be very interesting.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Fri Nov 30, 2012 7:20 pm

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
Carl, you seem to be suggesting that nobody agreed with your view of the complex 4x4x4. I've never had a problem with it, indeed I think it's the most mathematically elegant version, I just preferred an alternative. Part of the reason I think is that I wasn't comfortable with grouping the pieces into types.

However, I've had some breakthroughs now that this thread has made me reconsider the problem, and I'm almost ready to prefer your version of the complex 4x4x4 (I'll never disregard my old version, 2 puzzles are always better than 1, right? ). I've had my own ideas about listing all the pieces and an algorithm to do so, which doesn't go through every piece of a given (real) piece type. I'll try and clarify my thoughts a little more on the issue, and I'll have a go of independently looking at the pieces types of the complex 4x4x4. If they match with what you think is correct, then it means I'll have understood what's going on, and I'll be fully converted. I've got some code from a recent project which I should be able to recycle when I get the time.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Sat Dec 01, 2012 12:56 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bobthegiraffemonkey wrote:
Carl, you seem to be suggesting that nobody agreed with your view of the complex 4x4x4. I've never had a problem with it, indeed I think it's the most mathematically elegant version, I just preferred an alternative. Part of the reason I think is that I wasn't comfortable with grouping the pieces into types.
Please accept my apologies. But if you don't "prefer" the "most mathematically elegant version" a part of me still feels like I've done something wrong. I was hoping to get away from calling "the alternatives" THE Complex 4x4x4 and to gain some consensus as to what the Complex 4x4x4 exactly was.
bobthegiraffemonkey wrote:
However, I've had some breakthroughs now that this thread has made me reconsider the problem, and I'm almost ready to prefer your version of the complex 4x4x4 (I'll never disregard my old version, 2 puzzles are always better than 1, right? ).
Oh my picture doesn't destroy these puzzles. It just calls them subsets of the Complex 5x5x5. You are certainly still welcome to play with and discuss them. But I don't view them as THE Complex 4x4x4 and if they are still referred to as that I fear its just going to create confusion and that new comers that might be interested in joining us in some of these deeper theory threads could be discouraged if those of us who have been debating these topics for years can't come to some consensus.
bobthegiraffemonkey wrote:
I've had my own ideas about listing all the pieces and an algorithm to do so, which doesn't go through every piece of a given (real) piece type. I'll try and clarify my thoughts a little more on the issue, and I'll have a go of independently looking at the pieces types of the complex 4x4x4. If they match with what you think is correct, then it means I'll have understood what's going on, and I'll be fully converted. I've got some code from a recent project which I should be able to recycle when I get the time.
NICE!!! However I haven't fully counted all the piece types in the Complex 4x4x4 yet myself. I have a method for doing so and I've applied that method to the Complex 2x2x2 using a corner as a holding point. I've also applied it to the Complex 3x3x3 using a core and a corner as a holding point. I still need to generate the tables for using an edge and a face center as holding points but I don't expect any surprises. And the approach is general enough that it is easy to apply to the Complex 4x4x4 as well but to date I've just done everything by hand and the Complex 4x4x4 just has so many pieces that it would be a real head ache to do all that by hand. Brandon is a much better programmer then I so I'm very happy I got him converted over, so I think we will see a detailed list of the Complex 4x4x4 pieces types sometime soon. Of the symmetry types I expect to find are these:

C => Corners [4 types / 8 copies of each]
X => X-Faces [6 types / 24 copies of each]
W => Wings [6 types / 24 copies of each]
L => Obliques [4 types / 48 copes of each]

So this would be a total of 20 types and give us the 512 pieces we expect. Care to guess how I did that in my head without even making any tables?

Carl

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 Post subject: Re: An incomplete picture... a theory threadPosted: Sat Dec 01, 2012 2:44 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
[...]I have a method for doing so and I've applied that method to the Complex 2x2x2 using a corner as a holding point. I've also applied it to the Complex 3x3x3 using a core and a corner as a holding point. I still need to generate the tables for using an edge and a face center as holding points but I don't expect any surprises. And the approach is general enough that it is easy to apply to the Complex 4x4x4 as well but to date I've just done everything by hand and the Complex 4x4x4 just has so many pieces that it would be a real head ache to do all that by hand. Brandon is a much better programmer then I so I'm very happy I got him converted over, so I think we will see a detailed list of the Complex 4x4x4 pieces types sometime soon. Of the symmetry types I expect to find are these:

C => Corners [4 types / 8 copies of each]
X => X-Faces [6 types / 24 copies of each]
W => Wings [6 types / 24 copies of each]
L => Obliques [4 types / 48 copes of each]

So this would be a total of 20 types and give us the 512 pieces we expect. Care to guess how I did that in my head without even making any tables?

Carl

Nice table, it's probably right or pretty close to right. I think I see the 4 corner-like sets. I'm not sure what the difference between an X-face and an Oblique is.

I'm still working through how to generate and represent all of the pieces for the Complex 4x4x4. I'm trying to start at the Complex 3x3x3 using something other than the core as the holding point but I'm having trouble. I see how to do it by starting with the core as the holding point and then I could generate your table by choosing another holding point and adjusting the grip pattern for each piece. It works because defining the slice layers as the dependent layers is nicely symmetrical and when you rotate the puzzle the dependent layers stay in the same location. That doesn't happen with the Complex 4x4x4 so I'm going to have to figure out a way that doesn't rely on this symmetry.

Also, in the process of thinking a bunch about the Complex 3x3x3 as part of our debate over the Complex NxNxN, I think I've realized it's much easier to solve than I thought at first. Somewhere that I can't find now, Andreas wrote that the only redundant pieces are the three UD pieces. I'm pretty sure they are redundant but I also think the Core, Inverted Core, inverted UD pieces are all redundant. Andreas posted some GAP code to demonstrate this so I'd be very interested in understanding where I'm going wrong or where the GAP code is wrong. I'll post more about this after I do more thinking and testing. I'm probably going to have to solve the Complex 3x3x3 again

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 Post subject: Re: An incomplete picture... a theory threadPosted: Sat Dec 01, 2012 2:50 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
bobthegiraffemonkey wrote:
[...]I'll never disregard my old version, 2 puzzles are always better than 1, right?

Matt, are these your alternative versions?
http://www.twistypuzzles.com/forum/view ... 11#p227011

I hadn't seen that post before but after reading it, it is identical to what I was trying to do here:
viewtopic.php?p=292900#p292900

I think we both agree that those start out as the Complex 5x5x5 and then we eliminate pieces that behave as though they are in the middle layer. The 216-piece puzzle might be a subset of the Complex 4x4x4, I'm not sure.

They are neat subsets of the Complex 5x5x5 but neither match up with how I think the Complex 4x4x4 should be defined.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Sat Dec 01, 2012 6:40 pm

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
You made me waste one of my valued free days.
Anyway. Here we go.

viewtopic.php?f=1&t=18470

I succeeded in making up a table for the Complex5x5x5 with the core as HoldingPoint.
I did it by hand!
To be able to do this, I had to use the core of the 5x5x5 as holding point.

I came up with the following:
There are 4096 different subset for pieces of the Complex 5x5x5
There are 220 different types of pieces.
10 of them are NonZeroVolumeHoldingPoints = VHP aka "real"
210 are NonHoldingPoints = NHPs aka "imaginary"
As it was in the case for the Complex 3x3x3 all piece types have an inverse one.
In this case there are 12 types which are self-inverse. All 12 are NHPs.

4 out of 220 piece types have the symmetry of the core. One of them is the VHP-Core.
24 out of 220 piece types have the symmetry of the faces. Two of them are VHPs.
6 out of 220 piece types have the symmetry of the corners. Two of them are VHPs.
24 out of 220 piece types have the symmetry of the edges. Two of them are VHPs.
59 out of 220 piece types have the symmetry of the T-Faces. One of them is a VHP.
31 out of 220 piece types have the symmetry of the W or X. Two of them are VHPs.
20 out of 220 piece types have the symmetry of the Obliques.
12 out of 220 piece types have the symmetry of the UD-piece of the Complex 3x3x3.
36 out of 220 piece types have the symmetry of the UDL-piece of the Complex 3x3x3.
4 out of 220 piece types have a symmetry never been found before. {U,D,f,b} is an example.

Sadly I wasn't able to filter the Complex 4x4x4 out of this mess.
I deleted all variants with a symmetry on at least one axis: {U,D} or {u,d} or {U,D,u,d} or {}
I deleted all variants with both slices on at least a single axis: {U,u,d} and others.
The remaining list has 35 types and 1000 pieces which can't be true.
What additional conditions could be applied for filtering.

It would help me very much if I saw only a single posting from each of you when I look here the next time.

Last edited by Andreas Nortmann on Sun Dec 02, 2012 3:23 am, edited 1 time in total.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Sat Dec 01, 2012 6:41 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
I'm not sure what the difference between an X-face and an Oblique is.
There are X-Faces: (there are two types of X-Faces highlighted here)

These are Obliques: (I'm calling this a single piece type that is highlighted... though I guess you could also consider it two chiral pairs of piece types)

What I did to get these numbers is to assume this mapping of the Complex 4x4x4 to a Multi-8x8x8 is valid:
So what I did to create the above table was simply count the piece types in the Multi-8x8x8. Like the 10 piece types in the Complex 3x3x3 map to the 10 piece types in the Multi-5x5x5, I'm expecting the same thing here. And unlike the Multi-5x5x5 picture of the Complex 3x3x3 I don't think any cubies will have to be joined to form a single piece. In other words, I'm not expecting any unique new symmetry types to appear as does happen with the N=odd puzzles.
Andreas Nortmann wrote:
It would help me very much if I saw only a single posting from each of you when I look here the next time.
Ok... one post it is.
Andreas Nortmann wrote:
You made me waste one of my valued free days.
Something tells me you enjoyed at least a part of it.
Andreas Nortmann wrote:
I succeeded in making up a table for the Complex5x5x5 with the core as HoldingPoint.
I did it by hand!
You did this today. WOW!!! I figured it was easier to convert the programmers and to let them do the dirty work. And converting the programmers wasn't easy... this is SOMETHING. Thanks for all that hard work.
Andreas Nortmann wrote:
I came up with the following:
There are 4096 different subset for pieces of the Complex 5x5x5
There are 220 different types of pieces.
10 of them are NonZeroVolumeHoldingPoints = VHP aka "real"
210 are NonHoldingPoints = NHPs aka "imaginary"
As it was in the case for the Complex 3x3x3 all piece types have an inverse one.
In this case there are 12 types which are self-inverse. All 12 are NHPs.

4 out of 220 piece types have the symmetry of the core. One of them is the VHP-Core.
24 out of 220 piece types have the symmetry of the faces. Two of them are VHPs.
6 out of 220 piece types have the symmetry of the corners. Two of them are VHPs.
24 out of 220 piece types have the symmetry of the edges. Two of them are VHPs.
59 out of 220 piece types have the symmetry of the T-Faces. One of them is a VHP.
31 out of 220 piece types have the symmetry of the W or X. Two of them are VHPs.
20 out of 220 piece types have the symmetry of the Obliques.
12 out of 220 piece types have the symmetry of the UD-piece of the Complex 3x3x3.
36 out of 220 piece types have the symmetry of the UDL-piece of the Complex 3x3x3.
4 out of 220 piece types have a symmetry never been found before. {U,D,f,b} is an example.
Very Very interesting... For some of these symmetries I know how many pieces to expect of each type:

core = 1 piece
faces = 6 pieces
corners = 8 pieces
edges = 12 pieces
T-Faces = 24 pieces
W or X = 24 pieces (but can you not distinguish W from X?)
Obliques = 48 (unless you are counting chiral pairs as two piece types... not sure of your view on this)
UD type = 3 pieces
UDL type = 12 pieces
UDfb type = I was going to say I have no idea... but some math tells me the answer is 6 pieces AND you consider chiral pairs as a single piece type.
Andreas Nortmann wrote:
Sadly I wasn't able to filter the Complex 4x4x4 out of this mess.
I deleted all variants with a symmetry on at least one axis: {U,D} or {u,d} or {U,D,u,d} or {}
I deleted all variants with both slices on at least a single axis: {U,u,d} and others.
The remaining list has 35 types and 1000 pieces which can't be true.
What additional conditions could be applied for filtering.
Great question... I'm honestly not sure if you use the core as a holding point. An approach I think I'd take would be to use the outer most real corner as the holding point. When this was done for the Complex 2x2x2 and the Complex 3x3x3 as seen here:

and here:

You will note that the signature of the 2x2x2 corners is IDENTICAL for both puzzles.

So I believe if one were to do that for the Complex 4x4x4 and the Complex 5x5x5 it would easily allow one to see the other puzzles inside the Complex 5x5x5. I'm confident the picture of the Complex 5x5x5 produced this way is equivalent to the Complex 5x5x5 were the core is the holding point but since the Complex 4x4x4 doesn't have a core I think its going to make it difficult to pull those pieces out without generating the intermediate picture of the Complex 5x5x5 where the outer real corner is the piece type picked to serve as the holding point. After this is done it may be possible to come up with filtering conditions that can be applied to allow this step to be skipped... but at the moment at least I can't see an easier way. I'll just say I'm confident that the Complex 4x4x4 is a subset of the Complex 5x5x5 just as the Complex 2x2x2 is a subset of the Complex 3x3x3... but pulling it out using a core holding point picture I don't think is a trivial task.

Carl

P.S. Note the phrase "I'm confident..." is just a nice way of saying I've been too lazy to prove it. I certainly believe it to be true but I'm open to being proven wrong.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Sun Dec 02, 2012 9:03 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Hi,
here comes my posting for this day.
The next one will be busy. Maybe I won't be able to answer.
wwwmwww wrote:
UDfb type = I was going to say I have no idea... but some math tells me the answer is 6 pieces AND you consider chiral pairs as a single piece type.
The symmetry of UDfb indeed produces 6 pieces per type.
And in my book there are 48 pieces for the obliques.
wwwmwww wrote:
(but can you not distinguish W from X?
I came to the conclusion that distinguishing between W and X is rather arbitrary from the mathematical point of view because they share the same symmetry.
Look again at the piece W66 in this example:
viewtopic.php?p=223243#p223243
There you see that there are two seemingly different pieces. One would be X and one would be W in the known notation. But these two behave connected and are therefore a single piece type. Is it X? Or is it W?

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 Post subject: Re: An incomplete picture... a theory threadPosted: Mon Dec 03, 2012 11:27 pm

Joined: Sat Nov 17, 2012 12:36 am
Location: Melbourne, Australia
Hi All, This is a repost of something that I wrote on a more recent thread as I wasn't sure whether to bump this thread or not but I have been told that this is the correct place for my thoughts. Please forgive me if I have misunderstood the nature of complex nxnxn puzzles.

I'd like particularly to deal with Complex 2x2x2 that Carl was mentioning earlier in this thread. My issue with his arguments regarded the orbits of the pieces (the orbit of a piece is the number of places that that piece can end up in after a series of moves). A complex 2x2x2 should have three pieces in one orbit and all of the rest of the pieces in their own orbit, the regular 2x2x2 treating one corner as fixed has the fixed corner in its own orbit and only one other orbit with the 7 remaining corners, this is not what the complex 2x2x2 should look like.

To represent a Super Complex 2x2x2 one only needs a super 3x3x3 cube and then to restrict moves to the slice moves E, M and S. This combines the 8 corners as the "fixed centre", the edges in the M slice form one piece which is equivelent to the M face and so on, The faces are in pairs and orbit the three positions having 8 posible orientations in each position, and the centre is the piece that changes orientation with every move.

If this isn't clear please let me know and I will explain in greater detail, but there is an upshot of this, clearly even though the 2x2x2 and the complex 2x2x2 each have 8 pieces they are different puzzles, which means that the 2x2x2 is not contained in the complex 2x2x2 and the smallest complex puzzle that it is contained in is the complex 3x3x3.

I haven't put in enough thought to see what happens with higher order complex nxnxn for even ns, but I would speculate that they exist but don't contain the nxnxn puzzle that they relate to.

Joe

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 Post subject: Re: An incomplete picture... a theory threadPosted: Tue Dec 04, 2012 1:41 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Knaves wrote:
Hi All, This is a repost of something that I wrote on a more recent thread as I wasn't sure whether to bump this thread or not but I have been told that this is the correct place for my thoughts. Please forgive me if I have misunderstood the nature of complex nxnxn puzzles.
Saw the original post. Your post here is only 1 day after the one above it so I certainly wouldn't call it a bump. And the other thread looks like its turning into one about the solving... so I'll post my reply here.
Knaves wrote:
I'd like particularly to deal with Complex 2x2x2 that Carl was mentioning earlier in this thread. My issue with his arguments regarded the orbits of the pieces (the orbit of a piece is the number of places that that piece can end up in after a series of moves).
I view the definition of an orbit as the places themselves... not just the number of places. But that doesn't appear to be the problem here.
Knaves wrote:
A complex 2x2x2 should have three pieces in one orbit and all of the rest of the pieces in their own orbit,
You just lost me. How many pieces are you saying are in the Complex 2x2x2? You appear to be saying the number of orbits is 3 less then the number of pieces and I don't see why you say that.
Knaves wrote:
the regular 2x2x2 treating one corner as fixed has the fixed corner in its own orbit and only one other orbit with the 7 remaining corners, this is not what the complex 2x2x2 should look like.
I could see why you say the Regular 2x2x2 has two orbits if held by a corner but I believe I'd argue it just has one. Any of the 8 corners COULD serve as the holding point and in all 8 pictures you have the same puzzle. And the way I perform my analysis is to look at all 8 pictures at once. This way the "signature" of all 8 corners is identical and I'd be inclined to say all 8 corners exist in the same orbit. Granted my background is Physics and not math so I may be a bit fuzzy on the exact mathematical definition of orbit. All that said... you describe the regular 2x2x2 and then say that is NOT what the Complex 2x2x2 SHOULD look like. Why? Basically in my model the Complex 2x2x2 IS the regular 2x2x2.
Knaves wrote:
To represent a Super Complex 2x2x2 one only needs a super 3x3x3 cube and then to restrict moves to the slice moves E, M and S. This combines the 8 corners as the "fixed centre", the edges in the M slice form one piece which is equivelent to the M face and so on, The faces are in pairs and orbit the three positions having 8 posible orientations in each position, and the centre is the piece that changes orientation with every move.
This is the Slice-Turn Only 3x3x3. I agree its a Face-Turn Hexahedron with only 1 independent layer per axis but I don't follow what you believe this should be considered the Complex 2x2x2. To me the Complex NxNxN puzzles are a superset of their regular NxNxN puzzle with all the imaginary pieces added in. The puzzle you just described doesn't even contain the 2x2x2 so its NOT a superset of the 2x2x2. Not that I don't consider the Slice-Turn Only 3x3x3 an interesting puzzle... I do. But I view it as a symmetrically bandaged subset of the Complex 3x3x3. They way you have it described it contains both the real and imaginary cores. It contains imaginary face centers. And it contains the 3 UD pieces (not sure we have a better names for this type of piece). And actually I just realized something rather interesting... this puzzle exists within the Complex 3x3x3 with NO bandaging at all. I guess that shouldn't surprise me as the 2x2x2 exists within the Complex 3x3x3 as well and that doesn't require the Complex 3x3x3 to be bandaged in any way either.
Knaves wrote:
If this isn't clear please let me know and I will explain in greater detail, but there is an upshot of this, clearly even though the 2x2x2 and the complex 2x2x2 each have 8 pieces they are different puzzles, which means that the 2x2x2 is not contained in the complex 2x2x2 and the smallest complex puzzle that it is contained in is the complex 3x3x3.
I agree the 2x2x2 and the Slice-Turn Only 3x3x3 are different puzzles. I don't follow why the Slice-Turn Only 3x3x3 should be considered the Complex 2x2x2.

Carl

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 Post subject: Re: An incomplete picture... a theory threadPosted: Tue Dec 04, 2012 12:06 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
With regard to bandaging of the (Complex) 3x3x3 as a representation of the (Complex) 2x2x2, we've already tackled another puzzle like that in Oskar's Enabler Cube which can be thought of as a "one independent layer per axis" where that layer is the outer and slice layer of a 3x3x3: http://twistypuzzles.com/forum/viewtopic.php?p=292124#p292124

Even though this puzzle has a similar definition to the Complex 2x2x2, it creates a puzzle which is basically a 3x3x3 with an additional parity restriction.

I'm not sure what exactly it is that makes this such a different puzzle than the Complex 2x2x2 but my guess would be that the choice of which two layers are bandaged together per axis must be arbitrary (the halves must be isomorphic) and the all possible variations of the definition have to be equal to re-orientation of one. For the Complex 2x2x2, the choice of the dependent layers is arbitrary but for the Enabler cube it is not.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Tue Dec 04, 2012 5:11 pm

Joined: Sat Nov 17, 2012 12:36 am
Location: Melbourne, Australia
Ok, so there are a few things I need to tackle here, first I need to describe the complex 2x2x2 in abstract.
I will call the three operators X, Y, and Z so that we won't have any preconceptions about what they do.
We need one piece that is invariant to X,Y, and Z (the core) and one piece that is always effected by turns X, Y and Z(the anti-core).
We need a piece that is only effected by X, one only effected by Y and one only effected by Z (the faces).
And Finally we need a piece affected by <X,Y>, one affected by <X,Z> and one affected by <Y,Z> (the edges).

These define what orbits the pieces should have:
1. The core (can't move so it must be in it's own orbit.)
2. The anti-core (must remain affected by all moves so must be in its own orbit.)
3,4,5. The three faces (each face is only affected by itself so nothing can move it into a different plane, they must each be in their own orbit, see complex 3x3x3 for more insight into this.)
6. The three edges share an orbit,(because they have two operators, ie. after X the <X,Y> edge has moved into the <X,Z> position)

So the complex 2x2x2 should have 6 orbits for its 8 pieces, the regular 2x2x2 has 1 orbit, each piece can be in any position, although if we treat one corner as fixed we have two orbits one with our "core" and one with the remaining 7 pieces, these are not the same puzzle.

The slice super 3x3x3 is a representation of the complex 2x2x2 in the same way that a restricted turn super 5x5x5 with inner pieces was a representation of a complex 3x3x3.

If you prefer you can treat the cube as having the centre fixed and the corners turn as a block with every move. Then your moves are RL', UD' and FB'. Your core is your core your faces are your faces although your F and B centres are the same piece. Your corners are the anti core and your edges are the edges but all the edges in the M layer move together etc.
alternatively you can black out most of the prices of the puzzle until you are left with only a 2x2x2 corner of the 3x3x3. It's way as you move the pieces around the puzzle each piece has only one physical representation.

I hope that that explains things better but I will explain things further when I have more time.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Tue Dec 04, 2012 7:08 pm

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
That's not quite correct, but a forgivable mistake (I feel) and one related to the current problems of listing piece types on the complex 4x4x4.

Take a 2x2x2 and fix the DLB corner in place. The allowed twists are U, R and F. If you look at each corner, you will see that you get all of the pieces required of a complex 2x2x2: -, U, R, F, UR, UF, RF, URF. The reason it doesn't match your idea is that you are taking a piece which is not affected by a twist of a particular axis to lie in a middle slice somehow, but instead it lies in the slice which we cannot turn freely (which is decided by which piece we fix in place).

So the UR piece on the complex 2x2x2 is NOT the same as the UR piece on a 3x3x3: on the 3x3x3 with the core fixed in place the layer we cannot move freely on the FB axis is the middle layer, but on the 2x2x2 there is no middle layer on the FB axis, instead the RU piece is in the layer on the FB axis we cannot turn freely, i.e. the B layer.

The confusion is that for the 2x2x2 the UR piece (in terms of which twists move this piece) is in the URB position (in the usual sense of a location on the 2x2x2).

Does this help you understand the construction that everyone else is using?

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 Post subject: Re: An incomplete picture... a theory threadPosted: Tue Dec 04, 2012 10:13 pm

Joined: Sat Nov 17, 2012 12:36 am
Location: Melbourne, Australia
I don't feel that I have made a mistake, it is a difference of opinion as to what the complex 2x2x2 is, and if it is consistent with other complex nxnxns.

The 2x2x2 is not consistent with other complex nxnxns, it does not have an anticore, it does not have faces pieces and it does not have antiface/edge pieces, however Carl who predicted that the complex 2x2x2 would have 8 pieces predicted that it would have that combination.

The Slice cube does have those properties.

The argument that it doesn't matter which piece you choose to be the core seems a bit fallacious to me. It should be a piece that has the following property: X could be the core if for any piece Y that you chose to be stationary X would be in an orbit of one piece.

There is no such piece in an even nxnxn there is an all odd nxnxn's and that's where the problem comes from. It is easy to define an odd complex nxnxn. Even ones require more thought. I am still trying to get my head around whether an even complex nxnxn even makes sense but if it does then t he complex 2x2x2 can be emulated by the slice 3x3x3 and is not equal to the regular 2x2x2.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 12:27 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Knaves wrote:
I don't feel that I have made a mistake, it is a difference of opinion as to what the complex 2x2x2 is, and if it is consistent with other complex nxnxns.
Arg... Why does this have to be so hard? I feel like getting someone to see this picture the way I see it is like pulling teeth.

I personally feel there in ONE Complex 2x2x2 and it is what it is. But let's take a few steps backwards.

What you are calling X and Y are operations. For example the turning of the X-face layer or the turning of a slice layer named 'X'. I call the sequences of letters [XY] for example a signature. And these signatures apply to a given piece. They are found by examining that piece and identifying it as moving with the operation X and the operation Y and no other operation or it would be listed in the pieces signature.

So... lets apply this to a puzzle where there is agreement. He can leave the imaginary pieces out of the picture for now to make it even easier to see. So let's look at the good old normal 3x3x3. If we pick the core as the holding point we have this list of operations R, F, U, D, B, and L. We can now look at the 27 pieces in the puzzles and look at the signature of each one. To make things even simpler we can limit ourselves to just looking at the corners. We have 8 corners and here are their signatures:

[RUF]
[RUB]
[RDF]
[RDB]
[LUF]
[LUB]
[LDF]
[LDB]

These signatures all list 3 adjacent faces and as the naming is arbitrary we can say these signatures are all the same format and describe a given piece type, THE CORNER. The signature of a face center would have the format [F] for example. There would be 6 of those. Does this abstract picture tell us much about the orbits? The corners are all in the same orbit. But as each face center never changes places with any others are they all in the same orbit? I don't think so. I guess it depends on how one defines orbit but that was never what this type of analysis was intended to answer.

So now we know how to apply the theory... lets apply it again to the 3x3x3. Its the same puzzle. I'm just now asking you to hold it by a corner. Nothing fundamental about the puzzle changes... we are just picking the reference frame from which we are going to look at the puzzle. So in this new picture of a normal Rubik's cube our operations are going to become R, F, U, 1, 2, and 3 where the numbers represent turns of a slice layer. Using the EXACT same analysis method we used above we now see the CORNERS have what appear to be different signatures. The 8 corners would now appear listed like this:

[]
[B]
[D]
[DB]
[L]
[LB]
[LD]
[LDB]

[] - in this case JUST means holding point. This isn't a CORE. We know that because we defined the holding point AND the operations in order to make this picture.

So do the corners magically now become different piece types. Are they now in different orbits. Do I really need all new algorithms to solve this 3x3x3 as its just morphed into a totally different puzzle. NO!!!! Good old common sense tells us these are the EXACT same corners we have come to know and love. So how do we reconcile this with these apparently DIFFERENT piece signatures? What I did was to go back and look at the definition of a signature... at the point I first did this the ONLY type of piece this method had ever been applied to was the CORE. Its a unique piece type in this picture and the UNIQUENESS helps keep the picture simpler. But what I wanted to do was to KNOWINGLY pick a corner piece. There are 8 copies of a corner piece in a 3x3x3. So what if what I'm looking at is actually only 1/8 of the actual true signature. Can I again get all the corners to appear to have the same signature if I look at the full signature? One that looks at all 8 pieces of the signature where each of the 8 corners takes it's turn serving as a holding point. The answer is YES. And that is EXACTLY what I did here:

Look at rows 28, 29, 30, 31, 34, 35, 36, and 37. The full signature where one uses a corner as a holding point is what is between the {} symbols. What is between the [] is just a part of the picture. With the full signature in view it is again clear that the corners are all the same (as they should be). So why is all this necessary? Because the 2x2x2 doesn't have a core that can serve as a holding point... it only has the 8 corners. And remember only REAL and VIRTUAL pieces can serve as holding points. And NONE of the NxNxN puzzles (N odd or even) have any Virtual pieces. So there is NO need to go looking in the Complex (Real + Imaginary) 2x2x2 to see if one can find an Imaginary core.

So while you have given me some insight into the Slice-Turn Only 3x3x3. Your analysis is based on an incomplete application of this method. You are only seeing PART of the picture and trying to work things backwards. It doesn't work that way... a signature is built up from the operations. And those operations need to be well defined for the signature to mean anything. If your holding point is a 2x2x2 corner and your operation is a 2x2x2 face turn... you can't change to saying you are now holding the core of a Slice-Turn Only 3x3x3 and that your operations are slice turns. And if you start an analysis of the Slice-Turn Only 3x3x3 don't call it the Complex 2x2x2. It breaks the very meaning of the name... Complex = Real+Imaginary and your puzzle doesn't even contain the Real 2x2x2.

Follow?
Carl

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 12:53 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Knaves wrote:
The 2x2x2 is not consistent with other complex nxnxns, it does not have an anticore, it does not have faces pieces and it does not have antiface/edge pieces, however Carl who predicted that the complex 2x2x2 would have 8 pieces predicted that it would have that combination.
I never said the Complex 2x2x2 would have an anticore. Can you point me to were you think I said that?

You can count the pieces by asking if the pieces are IN or OUT of the independent layers. And sure one piece must be in all 3 and one piece must not be in any. But these pieces aren't cores. And when that counting method was used I was just trying to count pieces. I wasn't trying to identify piece types. You have 8 choices of which layers to pick as the independent ones so its only a partial picture and can't be used to classify the pieces. Note: I know going in that my choice of independent layers picks a corner as a holding point which is a NON-UNIQUE piece type.

Carl

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 7:55 am

Joined: Sat Nov 17, 2012 12:36 am
Location: Melbourne, Australia
I am very sorry it seems that I am talking about something completely different. I understand that holding a given piece still changes how you think about some of the orbits but mostly changes what moves are available for each turn. However I don't see what this signature gains us, orbits seem a lot more useful... But that isn't important.

My issue with calling the 2x2x2 the complex 2x2x2 is that one can immediately see that there is a missing core and anti core. ( ie. there needs to be a piece that doesn't react to any operation and one that reacts to all) these pieces would stay in their own orbit regardless of your holding piece. Once you expand that you get the complex 3x3x3. I don't see another resolution for this. I was trying to find a complex puzzle that was the order of your proposed complex 2x2x2 and the slice 3x3x3 was the only one I could find.

I see that this doesn't fit with your model but if the 2x2x2 is the complex 2x2x2 then I don't understand how you can say that there are pieces missing from the 3x3x3. Everything that makes the complex 3x3x3 make sense tells me that the smallest complex cube that contains the 2x2x2 is the complex 3x3x3.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 9:54 am

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
Knaves wrote:
My issue with calling the 2x2x2 the complex 2x2x2 is that one can immediately see that there is a missing core and anti core. ( ie. there needs to be a piece that doesn't react to any operation and one that reacts to all) these pieces would stay in their own orbit regardless of your holding piece.

You seems to be basing your way of building the complex 2x2x2 on assumtions about what you should get at the end, rather than starting with a 2x2x2, turning it into a complex puzzle, and looking at what you get. You need to be careful about what you assume; there is no reason to assume that a complex 2x2x2 has to have a core and anti-core. You should always start with basic definitions and build from there without other assumption.

An nxnxn has n layers per axis, and we get a complex nxnxn by picking n-1 layers on each axis (with the nth layer being dependent on the others, thus we have a real piece unaffected by one layer on each axis which can serve as a reference or holding point) and defining pieces with every possible combination of those. This is enough to define complex puzzles and it is minimal and unambiguous. If you do something similar with assumptions about what types of pieces you get at the end, then you will end up with those pieces, not because they necessarily end up being there but because you chose to have them there.

When you decided that the complex 2x2x2 should have a core, you got a complex 2x2x2 with a core since that's what you wanted to make, but it doesn't make sense to define it that way.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 11:57 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Knaves wrote:
I am very sorry it seems that I am talking about something completely different. I understand that holding a given piece still changes how you think about some of the orbits but mostly changes what moves are available for each turn. However I don't see what this signature gains us, orbits seem a lot more useful... But that isn't important.
You missed the main point I was trying to make about your model. Here:
Knaves wrote:
I don't feel that I have made a mistake, it is a difference of opinion as to what the complex 2x2x2 is, and if it is consistent with other complex nxnxns.
You argue that you model is just as valid as it produces a consistant picture with the other Complex NxNxN models. I was trying to point out that your model isn't even consistent with itself when just describing the Normal 3x3x3. Once you understand why... which I don't think you do... you'll see the error that you are making.
Knaves wrote:
My issue with calling the 2x2x2 the complex 2x2x2 is that one can immediately see that there is a missing core and anti core. ( ie. there needs to be a piece that doesn't react to any operation and one that reacts to all) these pieces would stay in their own orbit regardless of your holding piece.
Why should these pieces exist in the Complex 2x2x2? You are hunting for them instead of letting the theory tell you what is there.
Knaves wrote:
Once you expand that you get the complex 3x3x3. I don't see another resolution for this. I was trying to find a complex puzzle that was the order of your proposed complex 2x2x2 and the slice 3x3x3 was the only one I could find.
See... trying to find = hunting. You are twisting the theory to meet your expectations.
Knaves wrote:
I see that this doesn't fit with your model but if the 2x2x2 is the complex 2x2x2 then I don't understand how you can say that there are pieces missing from the 3x3x3. Everything that makes the complex 3x3x3 make sense tells me that the smallest complex cube that contains the 2x2x2 is the complex 3x3x3.
The Complex 2x2x2 has 2^3 pieces. The Complex 3x3x3 has 2^6 pieces. This is why the 3x3x3 I claim is missing some imaginary pieces. The 2x2x2 doesn't have any imaginary pieces as they are all accounted for by the real pieces.

Carl

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 5:04 pm

Joined: Sat Nov 17, 2012 12:36 am
Location: Melbourne, Australia
I think my theory is very different to yours I have some very distinct issues with yours which have been dismissed. When I have time I. Will start a new thread with the basis of my theory, with pictures etc. it will probably be some time over Christmas break.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 5:29 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Knaves wrote:
I think my theory is very different to yours I have some very distinct issues with yours which have been dismissed. When I have time I. Will start a new thread with the basis of my theory, with pictures etc. it will probably be some time over Christmas break.
What you've created is interesting but I'm hesitant to call it a valid description of the Complex 2x2x2.

The Complex 3x3x3 introduces new pieces due to combinatorics without altering the geometry or layers or bandaging of the puzzle.

Carl's mode for the Complex 2x2x2 does the same. It doesn't have to introduce a new layer and then bandage layers or any other sort of modification. The same with the Complex 4x4x4.

The slice-3x3x3 of yours is neat and yes it has 8 pieces and yes it has a core and anti-core which is nice but it's something else entirely. If only because it has achieved the two layers by bandaging.

I certainly would like to hear how your theory relates to the orbits of pieces though. There is probably a lot of cool things yet-to-be-discovered there.

_________________
Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 7:20 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Knaves wrote:
I think my theory is very different to yours I have some very distinct issues with yours which have been dismissed. When I have time I. Will start a new thread with the basis of my theory, with pictures etc. it will probably be some time over Christmas break.
I look forward to it. Granted the friendly "debating" does tend to wear me down so if I've appeared frustrated know that I'm NOT trying to discourage you. This is the best way to learn and theories (yes even mine ) could always have holes in them. So the best way for them to gain weight is to stand the test of everyone (Oh why does it really need to be everyone? LOL!!) trying to break them.

The discussion here and the one in the "How to visualize pieces of a complex puzzle" thread about the Slice-Turn Only 3x3x3 has certainly sparked a few ideas in my head and I now believe a physical version of the Complex 3x3x3 could actually be much closer then I was thinking just a week ago. If I didn't have a day job (no I'm NOT hoping it goes away... I actually really like this one) I think I could bang out a design to test by Christmas. As is... maybe I can aim for the next IPP.

Carl

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 Post subject: Re: An incomplete picture... a theory threadPosted: Wed Dec 05, 2012 7:34 pm

Joined: Sat Nov 17, 2012 12:36 am
Location: Melbourne, Australia
Yes I agree complex 2x2x2 is not a good description for the slice 3x3x3, I'm thinking of a different set of terminology to talk about my theory, it will have areas that will overlap with complex theory and I will definitely mention this is where my ideas were influenced but it will have a different focus. I will not add further to this thread until I get my head around how a complex 2x2x2 could be a 2x2x2 because that is what my stumbling block is.

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