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 Post subject: How to visualize pieces of a complex puzzlePosted: Sat Oct 22, 2011 4:05 pm

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
Ever since I first heard about complex pieces (see any one of a dozen threads by Carl or Andreas), I've been constantly confused on how you can possibly imagine how a piece moves if it doesn't have any volume. Carl found a way to simulate a complex 3x3 on a multi 5x5 with some bandaging, but I find that slightly unsatisfying (compared to my new found idea) because it has duplicate pieces (125 cubies to represent 64 pieces), the bandaging makes it nigh-impossible to build, and circle puzzles are always somewhat confusing even if you get use to them.
A few weeks ago, I was chatting with Brandon in IRC about building a generic 3-cycle finder when I realize how to see these pieces and how they move. I'm not sure if Carl or Andreas think about it this way, but it works for me and wanted to share it.

The key to this new idea is that every piece on the complex 3x3x3 is a cube. I know what you're thinking, "duh, of course all the cubies are cubes." I'd say you're right, but not for the right reason. Under this model all the pieces on the complex tetrahedron are tetrahedrons, and all the pieces on the complex dodecahedron are dodecahedrons. In short, every piece is the size and shape of the whole puzzle. The pieces you see on a physical puzzle are like looking through a small view port to the whole piece. The stickers not only show you where the piece's solve state location is, it tells you when the piece can move (ignoring deeper cut puzzles for now and only on face turning puzzles). Like on the cube, the stickers on the red-white edge (when in the solved state) tell you that the piece moves on the when the red or white faces move. When the piece is in the green-orange position, the piece will move when green or orange face turns. Probably didn't explain this 100% clear but hopefully it will make sense soon.
Attachment:

dodeca.png [ 17.27 KiB | Viewed 8027 times ]

So how many pieces are there and how can we identify them? I like to imagine each piece as a 6 bit number (for the cube which has 6 turning cutting planes) that tells us which faces can turn it. Each bit would tell us if it's affected by the respective face in UFRBLD. The core is not affected by any turns so it's signature would be 000000. The U face center is only affected by turns in U so it'd be 100000. The RB edge would be 001100 and the UFL corner would be 110010. There's 6 bits with 2 possible values each so there are 2^6 possible signatures so there's 64 pieces in the complex 3x3x3. This is much more than the standard 27 you find on a normal Rubik's cube.
Now how do we use these signatures and know how they move? For each piece, we need a fully painted cubie with your color scheme of choice. For each piece, we put a black dot on the the sides with a 1 in the signature for that face and we don't do anything for the sides with a 0.
Attachment:

solved.png [ 33.42 KiB | Viewed 8027 times ]

When we want to turn the U face, we examine each cubie and see if it has a top on the U face. If it does, we turn the cube. If we want to turn the R face, we look at the right side of all the pieces and if there's a dot, we turn the cube like we would do a right turn. Here's a pic after a U turn.
Attachment:

u.png [ 42.27 KiB | Viewed 8027 times ]

One of the strange quirks of this model is that pieces have no permutation, only orientation.

This system works for any of the platonic solids. A complex tetrahedron has only 16 pieces, the complex octahedron has 256, the complex dodecahedron has 4,096, and the complex icosahedron has 1,048,576.

You can also apply this to hybrid puzzles like a Super-X. Your pieces would be in the shape of a cuboctahedron and there would be 16,384 pieces. I can't think of a way to make this work for jumbling yet, but the non-jumbling complex rhombic dodecahedron (edge turning cube) would have 4,096 pieces and the non-jumbling rhombic triacontahedron (edge turning dodecahedron) would have 1,073,741,824 pieces.

You can even apply this to puzzles with multiple cuts like the 5x5x5. Rather than just having 1 dot on each face, you need 2 dots, or a circle and a dot to represent each slice that affects the piece. This monster would also have 4,096 pieces.

You can try doing this at home with 64 dice + a pen, but I made an app to play with the complex 3x3x3 if you want. I tried to emulate the gelatin brain interface a bit because it's quite familiar to many solves and using macros would be absolutely necessary. You can rotate the entire cube using ctrl+click. As you might guess, it is incredibly hard to visualize even a simple 3x3x3 so I added a normal viewable 3x3. If anyone solves this without any of the 3x3 views, let me know. I hope everything is pretty straight forward, but if it's confusing, I'll try to explain how to use it.
I originally wanted write this for the complex dodecahedron since it's a step up from when Brandon solved the multi-dodecahedron. But if I did, the pieces on the screen would be smaller than your mouse cursor so that's impractical unless I change the filters to completely hide all but the pieces you're interested in.

http://landonkryger.com/rubik/complex/Complex.jar

You can play with some of these pieces on gelatin brain. Here's a small table showing which one the pieces are in. There could be more puzzles that have these extra pieces.
Code:
Core   Face   Edge         Corner   Inverted Edge      Inverted Face   Inverted Core
[]   [U]   [UF]   [UD]   [UDF]   [UFR]   [UDFR]   [UDLR]   [UDLRF]   [UDLRFB]
3.1.1   0   0   0   0   0   1   0   0   0   0
3.1.2   0   1   1   0   0   1   0   0   0   0
3.1.7   1   1   1   0   0   1   0   0   0   0
3.1.31   0   0   1   0   0   1   1   0   1   0
3.1.32   0   0   1   0   1   1   0   1   0   0
3.1.33   1   1   1   0   0   1   1   0   0   0
3.1.36   0   0   1   0   0   1   0   0   1   0

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Sun Oct 23, 2011 4:11 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
The program is an amazing achievement.
Sadly it demonstrates very well how diffcult it is to visualize the NHP's aka Non-Holding-Point pieces aka imaginary pieces.

BTW: Some of the NHPs have been implemented in a physical puzzle although one which uses bandaging:
viewtopic.php?f=15&t=15003
This puzzle consists of
[] (the core; this time visible in form of the 8 connected corners)
[UD] (represented by 4 edges which behave interconnected)
[UDRL] (represented by 2 opposing faces)
[UDRLFB] (the inverted core; visible because connected with the faces)

Sorry for contributing nothing more helpful.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Oct 25, 2011 6:45 pm

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
I like this representation. Brandon also revealed your idea to me a couple of days before.

I think this is a very good way to define a complex puzzle. I wonder how general it can go. I try to find the most general and minimal math definition of it. And I'm not sure how to do it.

For example, I try to imagine the 2x2x2 as a complex puzzle with three moves U, F, and R. Since there are three possible moves, there should be 2^3=8 pieces and there really are 8 pieces on the 2x2x2. They are perfectly distinguished by whether it can be moved by U, F, R, respectively. So the UFR corner is [UFR], the UFL corner is [UF] etc. the LBD corner is [] (the core). To complete the model, we need to answer the question: after the U move, where is [UFR] going? Well, it's going to [UF], but what's a concise way to define it? I don't know.

What's a universal model that includes both the complex platonic solids and this 2x2x2?

Eventually I would like to see a somewhat abstract definition of a complex puzzle. Like, for operations 1,2,...,k, the state is represented by an array of 2^k entries. After defining some transition rules for the array, a complex puzzle is defined. I would like to see something like this, is it possible?

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Oct 25, 2011 7:02 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
schuma wrote:
[...]To complete the model, we need to answer the question: after the U move, where is [UFR] going? Well, it's going to [UF], but what's a concise way to define it? I don't know.

What's a universal model that includes both the complex platonic solids and this 2x2x2?

Eventually I would like to see a somewhat abstract definition of a complex puzzle. Like, for operations 1,2,...,k, the state is represented by an array of 2^k entries. After defining some transition rules for the array, a complex puzzle is defined. I would like to see something like this, is it possible?

Hi Schuma, I think this is actually already hidden in the model.

Lets use the 2x2x2 cube as an example. If you define a cube as UFRBLD and the U operation does UFRBLD -> URBLFD then when you look at the UFR piece as the "UFR..." piece and you apply the same U operation, it turns into "UR..F." -- that is -- the UFR piece is now in the spot that was occupied by the UFL piece. Of course, the UFL piece is now "U..LF." and in the spot that was occupied by the "U..BL." piece.

So to populate this puzzle with pieces you'd have the "UFR...", "U.RB..", "U..BL.", "UF..L.", ".FR..D" and so on.

As long as you define the "handle" or "grip" permutations for each of the types of twists you can keep track of the permutation and orientation of each piece and you will automatically learn about where they went.

What the model doesn't tell you is where these pieces are physically located. When you look at the motion of the pieces you only know that a piece went into another piece's spot, not where that spot is located physically. I don't believe there is any permutation or orientation ambiguity though.

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Oct 25, 2011 7:18 pm

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
bmenrigh wrote:
Lets use the 2x2x2 cube as an example. If you define a cube as UFRBLD and the U operation does UFRBLD -> URBLFD then when you look at the UFR piece as the "UFR..." piece and you apply the same U operation, it turns into "UR..F." -- that is -- the UFR piece is now in the spot that was occupied by the UFL piece. Of course, the UFL piece is now "U..LF." and in the spot that was occupied by the "U..BL." piece.

Hi Brandon,

Once you are talking about a cube as UFRBLD, you are embedding the 2x2x2 pieces in a complex 3x3x3 cube. That's a valid way to talk about it and I understand it. But I don't think this is the minimal way to define the 2x2x2.

Since I look at the 2x2x2 as a complex puzzle of only three moves U,F, and R, there is no B, L, or D in my dictionary. I don't know what "L" is. I even don't know what a "cube" is. I only know there are three operations U F R (maybe someone calls them as operations #1, #2, #3) and there should be eight pieces in the puzzle. Is there a way to define a puzzle that makes sense?

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Oct 26, 2011 5:40 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
There were some discussion between me and Carl Hoff about how to define a Complex 2x2x2. We found nothing useful. In the meantime I had the chance to rethink this problem:

The 2x2x2 can be viewed as a subset of the 3x3x3. Just "delete" the edges, faces and the core.
The Skewb can bew viewed as a subset of the Master Skewb. Just "delete" the additional pieces.
This rule even works, when ZHP's (aka virtual) pieces are introduced.

I defined for myself: To make up a useful definition of the complex 2x2x2 one must be able to view it as a subset of the complex 3x3x3.
I looked up the 10 NHP (aka imaginary pieces) again and recognized that there are only 4 piece types where no pairs of opposite sides are present: [], [U], [UL] and [ULF] which are the four piece types of the 3x3x3.
That would mean the complex 2x2x2 is the 3x3x3 or maybe just the Fused Cube.

And I have another problem with the concept of a Complex 2x2x2:
One piece type of the Complex 3x3x3 is [UDL]. There are 12 samples of this piece. I can create all these 12 samples by applying all hexaheral symmetries to [UDL]. The same is true for the P's and ZHP's in any other puzzle.
When I am not allowed to introduce (for example) D, R and B I can't apply the symmetries to the pieces of the complex 2x2x2 you aim at.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Oct 26, 2011 11:56 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Nan I thought about your UFL 2x2x2 last night and I think the trouble with it and why it doesn't fit well in this model is that it has stored cuts.

In this model you describe each piece with a binary mask. Each bit is a turnable axis -- 0 means it doesn't turn when that axis is turned and 1 means it does. With your 3-bit Complex 2x2x2 the pieces with only a single bit set ("U..", ".F.", and "..L") become turnable by the other faces due to the stored cuts.

I think there are two options for making it fit into this model. The first is to add turnable axis until there are no stored cuts. You get the complex 3x3x3.

The other is to tweak the model so instead of using a binary mask you use a higher-base mask and then define the U, F, and L operations to do more than permute digits -- they must also modify the digits to make the new twist handles active.

The other trouble with the UFL 2x2x2 is that this model for viewing complex pieces likes symmetry. The UFL 2x2x2 doesn't have all of the symmetry that a cube has.

As for the difference between a "Complex 2x2x2" and a "Complex 3x3x3", I think both of these are misnomers. What we have is a complex face-turning cube with only two states per axis (turns with axis or does not turn with axis). The Complex 2x2x2 should be the same puzzle as the Complex 3x3x3.

Also, if you extend the model to support 2 cuts per face or the "Complex 4x4x4" that should be the same puzzle as the "Complex 5x5x5". The middle slice of the 3x3x3 and 5x5x5 is not a turnable slice in the puzzle but the part that doesn't turn when you turn two opposite faces.

Edit: I think "stored cuts" is the wrong terminology here. More like "stored axes".

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Oct 27, 2011 9:57 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Sorry for the slow reply. Andreas just PM'ed me about this thread. I'd missed it till now.
GuiltyBystander wrote:
Carl found a way to simulate a complex 3x3 on a multi 5x5 with some bandaging, but I find that slightly unsatisfying (compared to my new found idea) because it has duplicate pieces (125 cubies to represent 64 pieces), the bandaging makes it nigh-impossible to build, and circle puzzles are always somewhat confusing even if you get use to them.
I don't really consider them duplicate pieces. They are just discontinuous parts of the same piece. But if you consider them duplicate, you could also just drop all but 64 cubies of a Multi-5x5x5 and also have another model where each piece is a cube in a 5x5x5 array with 61 voids in it. Note the voids would also move.

As this stage you don't even need to talk about circle cubes as we are just talking about a model, as you are. Its when you try to actually build a working Multi-5x5x5 that you start talking about circle cubes and there you even start to run into discontinuous parts just trying to capture all the real piece types. Take the 3x3x3 Face Center of my Real5x5x5 design for example.

When I get that design into Solidworks and have it tested I hope to take a shot at turning it into a Complex3x3x3... but that is a ways down the road.
GuiltyBystander wrote:
I'm not sure if Carl or Andreas think about it this way, but it works for me and wanted to share it.
Yes I think this is very similiar to the way Matt Galla initially defined the set of Complex pieces. Andreas and I had been talking about a subset of these we had called virtual pieces.
GuiltyBystander wrote:
One of the strange quirks of this model is that pieces have no permutation, only orientation.
I think I see why. Each piece can only be in the correct orientation if its also in the correct position, looking at my Multi-5x5x5 model for example.
GuiltyBystander wrote:
You can try doing this at home with 64 dice + a pen, but I made an app to play with the complex 3x3x3 if you want.
I don't see many taking you up on the 64 dice idea but the app sounds very nice. I downloaded it and got a bunch of class files I have no idea how to use. Can you walk me though setting this up so I can use it?

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Oct 27, 2011 10:22 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
As for the difference between a "Complex 2x2x2" and a "Complex 3x3x3", I think both of these are misnomers. What we have is a complex face-turning cube with only two states per axis (turns with axis or does not turn with axis). The Complex 2x2x2 should be the same puzzle as the Complex 3x3x3.

Also, if you extend the model to support 2 cuts per face or the "Complex 4x4x4" that should be the same puzzle as the "Complex 5x5x5". The middle slice of the 3x3x3 and 5x5x5 is not a turnable slice in the puzzle but the part that doesn't turn when you turn two opposite faces.
Take a look at my thread here: The Complex NxNxN Puzzles.

Andreas and I agree the Complex NxNxN puzzles are pretty well defined for N=odd. I wanted to take a shot at coming up with a definition which worked for both N=odd and N=even and didn't change the existing understanding of the N=odd Complex puzzles. And I think what I came up with works rather well. The Complex 2x2x2 is just your basic 2x2x2. The Complex 4x4x4 however is another beast altogether. In that thread I explain how it could be modeled. It will have 64 real pieces, 448 imaginary pieces, and look like an 8x8x8 MultiCube with just 3 independant layers per axis of rotation. And you would like that model too as each piece is just one cubie. It doesn't have any discontinuous parts.

Note the real pieces of the Complex 4x4x4 are the same real piece present in a Multi-4x4x4. If you say the Complex 5x5x5 is the same puzzle as the Complex 4x4x4 then you have a problem as the Complex 4x4x4 has MORE real pieces then are present on the Multi-4x4x4 yet the Multi-4x4x4 should contain ALL the real pieces. If you go that far you are in effect saying the Multi-4x4x4 and the Multi-5x5x5 are the same puzzle and I feel strongly that isn't the case.

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Oct 27, 2011 12:45 pm

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
Guys,

Thanks for pointing out the difficulty to consider the 2x2x2 as a complex puzzle of three operators. I think that's just too hard to fit into a unified definition. So forget about it.

What I was thinking is to generalize this puzzle to non-cube, non-platonic solids type things, because I think this is an elegant way to define puzzles without considering geometry, but only with group operations.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Oct 27, 2011 12:53 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
wwwmwww wrote:
Take a look at my thread here: The Complex NxNxN Puzzles.
Just took a closer look back at that thread myself. In the very last post Matt aka bobthegiraffemonkey (not Matt Galla aka Allagem) points out a few problems/concerns he has with my Complex N=even model. His first concern is he states its hard to "realise how many pieces there are". Granted its not easy but I believe I showed how it can be done.

His second concern is he presents 2 other models for defining the Complex NxNxN puzzles that he considered equally valid but different from mine. I disagree with his first model as its built from the Complex (n+1)x(n+1)x(n+1) and you shouldn't have to go to the higher order to explain things. Yes, most 2x2x2's are built on top of a bandaged 3x3x3 but the model of just 8 cubies works just as well to explain the 2x2x2. You don't need to make the picture more complicated then necessary. And I'm not sure I understand his second model or that he understood mine. He gives a few examples in this format:

3x3x3: 10 distinct pieces, 0 chiral, 64 total

2x2x2:
Matt's model 1:
4, 0, 27 (3^3)
My Model:
4, 0, 8, (2^3)
Matt's model 2:
1, 0, 8 (2^3)

In my model I show all the corners are the same type of piece so my entry should be:
1, 0, 8 (2^3)

4x4x4:
Matt's model 1:
165, 20, 3375 (15^3)
My Model:
120, 56, 512 (2^9)
Matt's Model 2:
10, 1, 216 (6^3)

And I'm not getting his numbers for my model of the Complex 4x4x4. I view it as having:

4 types of pieces each with 8 copies (corners interior and exterior)
12 types of pieces each with 24 copies (wings and x-centers both interior and exterior)
4 types of pieces each with 48 copies (obliques interior and exterior)

So yes, it has 512 pieces total but only 20 types. I don't get where his 120 and 56 figures come from. Not sure about how to check for chirality. I'm guessing its the 4 oblique piece types.

And I'm really lost on his second model.

Carl

EDITED: To correct count of Complex 4x4x4 piece types.

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Last edited by wwwmwww on Thu Nov 10, 2011 11:31 pm, edited 4 times in total.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Oct 27, 2011 1:15 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Hi Carl, I have not had the time to fully read and digest that other thread but I will try to this evening.

The trouble I'm having with the complex 2x2x2 versus the complex 3x3x3 and the complex 4x4x4 versus the complex 5x5x5 is that on a Complex puzzle you don't really get to choose the depths of the cuts.

That is, if the pieces of a complex puzzle are defined by what faces move that piece then the cut depth is already somewhat pre-defined for that piece.

So for the complex 2x2x2, there is a piece that doesn't turn with the U or D faces but do turn with the F and L face. That's the same thing as a 3x3x3 edge on the complex 3x3x3 which doesn't turn with UD but does turn with FL.

Put another way, the 3x3x3 has one cut per face. So does the 2x2x2 but the depth has been lowered to the point where the two meet and share one cut. The number of faces hasn't changed though and so the 2^6 pieces for the complex 3x3x3 should be the same 2^6 pieces on the complex 2x2x2.

To extend this to the complex 4x4x4 versus complex 5x5x5 the same reasoning for the middle layer applies.

the only difference is that there are now 2 cuts per face. For each face there are now four choices:

Using the U face and u slice as an example:
• Piece does not turn with U face or u slice
• Piece turns with U face but not with u slice
• Piece does not turn with U face but does turn with u slice
• Piece turns with U face and u slice

Therefor the complex 4x4x4 should have 4^6 == 2^12 == 4096 pieces.

The complex 5x5x5 is the same puzzle as the complex 4x4x4 because the complex has the middle slice layer due to the definition of of the pieces.

Put another way, there are no even complex puzzles because they are forced to have a middle slice layer by definition of the pieces.

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Oct 27, 2011 2:52 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
The trouble I'm having with the complex 2x2x2 versus the complex 3x3x3 and the complex 4x4x4 versus the complex 5x5x5 is that on a Complex puzzle you don't really get to choose the depths of the cuts.
In a way you do. Its more in how you set up the puzzle. After you read the other thread I hope my idea makes more sense.
bmenrigh wrote:
That is, if the pieces of a complex puzzle are defined by what faces move that piece then the cut depth is already somewhat pre-defined for that piece.
So how do you define what faces move? A 3x3x3 has 2 independent faces per axis of rotation. These are typically taken to be LRUDFB. But with that choice you have already picked your holding point, the core of the puzzle. The core is unique so its perfectly fine to look at just this one choice for your analysis and you get all 10 types and all 64 pieces. Your 10 types are:

( 1) [UDLRFB]
( 2) [UDLRF]
( 3) [UDLR]
( 4) [UDLF]
( 5) [ULF]
( 6) [UDL]
( 7) [UD]
( 8) [UL]
( 9) [U]
(10) []

BUT a 2x2x2 only has 1 independent face per axis. So you don't get to pick 6 faces (or layers) which move. There are only 3. So which 3 do you pick? Any one choice of 3 means you've picked one of the 8 corners as a holding point. But here is where its different than a 3x3x3, you've now picked a holding point which isn't a unique piece type. So you MUST look at all 8 posibilities at the same time. If you do this for a 2x2x2 you will see there are 8 types of pieces each of this type:

(1) {[],[A],[B],[C],[AB],[AC],[BC],[ABC]}

Look at the table I have posted in the other thread. To prove that this is a valid way of looking at things I did the same type of analysis on the 3x3x3 where I picked a corner as a holding point. It also requires looking at all 8 corners at the same time and your defining patterns become much more complex but you again get the total of 64 pieces and 10 types. Also see that table in the other thread. By defining patern I mean this:

[ULF] = corner if the core is chosen as a holding point
{[],[A],[B],[C],[AB],[AC],[BC],[ABC]} = corner if a corner is chosen as a holding point.
bmenrigh wrote:
So for the complex 2x2x2, there is a piece that doesn't turn with the U or D faces but do turn with the F and L face. That's the same thing as a 3x3x3 edge on the complex 3x3x3 which doesn't turn with UD but does turn with FL.
The complex2x2x2 simply doesn't have a U and a D face that are independant. You can only have one of these in the picture at a time... not both. Note if you turn the U face it means you MUST be holding the puzzle by one of the cubies on the D face. So the D face can't be turned.
bmenrigh wrote:
Put another way, the 3x3x3 has one cut per face. So does the 2x2x2 but the depth has been lowered to the point where the two meet and share one cut. The number of faces hasn't changed though and so the 2^6 pieces for the complex 3x3x3 should be the same 2^6 pieces on the complex 2x2x2.
Where you say "The number of faces hasn't changed"... I'd say the number of independent turnable layers HAS changed. Its changed from 6 to 3 and the number of pieces has changed from 2^6 to 2^3.
bmenrigh wrote:
To extend this to the complex 4x4x4 versus complex 5x5x5 the same reasoning for the middle layer applies.

the only difference is that there are now 2 cuts per face. For each face there are now four choices:

Using the U face and u slice as an example:
• Piece does not turn with U face or u slice
• Piece turns with U face but not with u slice
• Piece does not turn with U face but does turn with u slice
• Piece turns with U face and u slice

Therefor the complex 4x4x4 should have 4^6 == 2^12 == 4096 pieces.
Again I can use the same logic I used above and say they are different. The Complex5x5x5 has 4 independantly turnable layers per axis and 3 axis and 2^(4*3) or 4096 pieces. The Complex4x4x4 has only 3 independantly turnable layers per axis and 3 axis and 2^(3*3) or 512 pieces.
bmenrigh wrote:
The complex 5x5x5 is the same puzzle as the complex 4x4x4 because the complex has the middle slice layer due to the definition of of the pieces.
And I'd say the initial definition of the turnable faces you built your model on wasn't appropriate for the complex 4x4x4, as you threw in an extra degree of freedom per axis which isn't available on the 4x4x4.
bmenrigh wrote:
Put another way, there are no even complex puzzles because they are forced to have a middle slice layer by definition of the pieces.
This strikes me as the many math problems that prove 1=2 (or something similiar) and resolving the conflict by saying therefore 2 doesn't exist. I understand what you are doing and it certainly is an approach that can be taken. The math is simplier as all the odd NxNxN's have a unique cubie that can be chosen as the holding point, the core, where the resulting defining paterns of the piece types are much simplier. But if this approach does work to generalize to N=even (and it still works for N=odd too) why not accept it?

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Oct 27, 2011 3:09 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
schuma wrote:
Thanks for pointing out the difficulty to consider the 2x2x2 as a complex puzzle of three operators. I think that's just too hard to fit into a unified definition. So forget about it.
It's too hard... so forget about it. Oh that pains my ears. NEVER!!!! The problem that others have given up on because its too hard is exactly the type of problem I LOVE to find.

Carl

P.S. In other words... I choose to tackle problems like this not because they are easy, but because they are hard. Now where have I head that before? That is of course a rhetorical question.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Nov 10, 2011 3:26 pm

Joined: Sun Aug 29, 2010 1:56 pm
Location: Berlin, Germany
fantastic!!
Well done, GuiltyBystander!
bmenrigh wrote:
As long as you define the "handle" or "grip" permutations for each of the types of twists you can keep track of the permutation and orientation of each piece and you will automatically learn about where they went.
It makes the inner representation of such a lot of puzzles!

wwwmwww wrote:
I don't see many taking you up on the 64 dice idea but the app sounds very nice. I downloaded it and got a bunch of class files I have no idea how to use. Can you walk me though setting this up so I can use it?
You have to rename the downloaded *.zip into a *.jar file. If you have installed the java jre on your computer, you can now double-click the jar to run it. It's like opening a *.txt file with the notepad. The java.exe opens the jar file, loads the class-files reads the manifest file and starts the main class.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Nov 10, 2011 5:48 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Stefan Schwalbe wrote:
You have to rename the downloaded *.zip into a *.jar file. If you have installed the java jre on your computer, you can now double-click the jar to run it. It's like opening a *.txt file with the notepad. The java.exe opens the jar file, loads the class-files reads the manifest file and starts the main class.
Thanks. I'll give that a try. I don't believe I have installed java jre but I should be able to find that I believe. Also thanks for bringing this thread back up again.
bmenrigh wrote:
Hi Carl, I have not had the time to fully read and digest that other thread but I will try to this evening.
bmenring did you ever get a chance to digest that other thread? I'm quite curious to hear your take on the idea.

Carl

P.S. bmenrigh what is the puzzle in your avitar picture? I don't believe I've seen it before.

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Last edited by wwwmwww on Thu Nov 10, 2011 11:40 pm, edited 1 time in total.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Nov 10, 2011 7:23 pm

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
Look what you made me do, I've went and read through my old posts to try and understand them again!

First of all, I certainly do understand the thinking behind your model, it's just that I have difficulty working with it. That's probably more due to laziness on my part than anything else. Don't worry about my weird numbers for your model.

The thoughts behind my second model are as follows. Since normal even order cubes have cuts running through the middle of the puzzle, you can divide it up into 8 sections. Let the cube have 2n layers, then the outer n layers of 3 adjacent faces will have an intersection equal to one of these 8 sections. All the pieces in a section can therefore be defined in terms of those n layers of the 3 faces, and each piece is affected by at least one of these layers. Extending to a multicube and using a multi 4x4x4 as an example, each piece is affected by either the inner layer or outer layer of each axis. Corners are affected by 3 outer layers, wings by 2 outer layers and 1 inner layer, centers by 1 outer layer and 2 inner layers, and internal corners by 3 inner layers. One way to define a complex 4x4x4 then (perhaps not the best way, or widely agreed on, but certainly one way) is to allow other possibilities, so long as every piece in the section is affected by at least one layer of each face. One such possibility is a piece affected by the outer layer of 2 axes, and both layers of the other axis. The only chiral piece type (I think) in this model, is affected by the outer layer of one axis, the inner layer of another, and both layers of the third. Total piece numbers are calculated for a section first, then multiplying by 8.

For the record, I don't see why my first idea was so bad. I think one valid way of thinking about even order cubes is as odd order cubes with one more layer, without the middle layer.

This was probably poorly explained, but I'm tired and trying to figure out what went though my head over a year ago. Hopefully it will give a better explanation of my idea. It was only ever really intended to give a different perspective. Any questions, it will probably be several days before I can reply. I hope to look through this thread better soon, but I'm pretty busy these days.

Matt

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Fri Nov 11, 2011 11:42 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bobthegiraffemonkey wrote:
Look what you made me do, I've went and read through my old posts to try and understand them again!
Thanks... the first time I read it when you posted it I didn't get it and I had meant to get back to it to try harder and never did until now myself.
bobthegiraffemonkey wrote:
The thoughts behind my second model are as follows. Since normal even order cubes have cuts running through the middle of the puzzle, you can divide it up into 8 sections. Let the cube have 2n layers, then the outer n layers of 3 adjacent faces will have an intersection equal to one of these 8 sections. All the pieces in a section can therefore be defined in terms of those n layers of the 3 faces, and each piece is affected by at least one of these layers. Extending to a multicube and using a multi 4x4x4 as an example, each piece is affected by either the inner layer or outer layer of each axis. Corners are affected by 3 outer layers, wings by 2 outer layers and 1 inner layer, centers by 1 outer layer and 2 inner layers, and internal corners by 3 inner layers. One way to define a complex 4x4x4 then (perhaps not the best way, or widely agreed on, but certainly one way) is to allow other possibilities, so long as every piece in the section is affected by at least one layer of each face. One such possibility is a piece affected by the outer layer of 2 axes, and both layers of the other axis. The only chiral piece type (I think) in this model, is affected by the outer layer of one axis, the inner layer of another, and both layers of the third. Total piece numbers are calculated for a section first, then multiplying by 8.
Ok... I think I see what you are doing. But let's look at your 4x4x4 example. A 4x4x4 has 3 independant layers per axis of rotation. You are in effect talking only an 8th of the puzzle and only looking at 2 independant layers per axis of rotation. Actually that's isn't quite correct because if you did do that exactly you would get each 8th of the puzzle to look like a Complex 3x3x3 with 64 pieces and when multiplied by 8 you would have 512. 512 pieces is what I consider the correct answer and I'm not sure why this approach gives you that as its not my approach. Your actual approach produces a smaller number [you say 216 (6^3) but I haven't tried to verify that] and this is due to your requirement that each piece MUST be moved by at least one of the layers releated to each face. The complex 3x3x3 has pieces in slice layers which don't move with either face layer on a given axis.

My approach doesn't break the puzzle up into 8th but just deals with the 3 independant layers on each axis of rotation. Looking at each piece by itself you can ask yourself does this piece move with independant face (pick one "A", "B", or "C") on axis (pick one "X", "Y", or "Z")? This answer is either Yes or No. So its 2 possible states you are looking at. To cover all possibilities and thus count all the pieces you simply do 2^(3*3) or 512 pieces and this equation should hold for all Complex NxNxN puzzles (N=even or odd).

2 ^ (Number of independant faces per axis * Number of axes)

I suspect this can be generalized even further to cover Complex Cuboids but as I'm still trying to convince most that we can have Complex NxNxN puzzles with N even I'm not sure I want to open that can of worms yet.

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Nov 16, 2011 7:50 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
bmenring did you ever get a chance to digest that other thread? I'm quite curious to hear your take on the idea.

Yes I have now but it took me a while to figure out why I don't like the existing models. I think where I differ is in the way to count axis and cuts. I've hinted at this in prior posts by saying things like "on a Complex puzzle you don't really get to choose the depths of the cuts".

Ultimately I think it comes down to us talking about slightly different things. That is, there is the "Complex Face Turning Cube With One Cut Per Face" which is a mathematical object and then there are physical realizations of this puzzle such as the 3x3x3, 2x2x2, Gelatinbrain's 3.1.31, etc.

When you're talking about the mathematical object -- the Complex puzzle -- the geometric details of the shape of the cuts or depth or cuts per axis don't matter. When you are talking about a physical puzzle they do.

So lets look at some physical realizations of this face-turning one-cut-per-axis cube. The way I see it is that you take a blank cube, you apply a single cut, and then you apply the symmetries of the cube (24) to get the final puzzle. Here is the cut progression as you apply the symmetries for the 3x3x3:
Attachment:

3x3x3_cut_progression.png [ 17.07 KiB | Viewed 7301 times ]
Note that this puzzle is a subset of the Complex 3x3x3.

If you adjust the cut down a bit and then apply the symmetries you get:
Attachment:

2x2x2_cut_progression.png [ 15.5 KiB | Viewed 7301 times ]
And again, this is a subset of the Complex 3x3x3.

And of course you can choose other shapes for the cut:
Attachment:

curvy_3x3x3_cut_progression.png [ 21.94 KiB | Viewed 7301 times ]
And again, you'll get some subset of the Complex 3x3x3.

I argue that you can add any crazy curve or zig-zag cut you want and then apply symmetries and the puzzle you will get will still be a subset of the complex 3x3x3. The 2x2x2 and 3x3x3 and 3.1.31 are all subsets of the Complex 3x3x3 so I think calling it the "Complex 3x3x3" is a bit of a misnomer. Calling it the "Complex Face Turning Cube With One Cut Per Face" seems a bit more technically accurate.

When you're talking about "cuts per axis" I think you're really talking about the implementation details of a single realization and not something fundamental to the Complex 3x3x3. I think it is a mistake to say that the 2x2x2 only has one cut per axis and therefor the "Complex 2x2x2" is somehow different than the "Complex 3x3x3" because the number of cuts per axis is really just a function of one of the subsets of the complex puzzle that it is derived from.

So to address things like "the 3x3x3 has a middle layer and the 2x2x2 doesn't so that means the complex 3x3x3 is different than the complex 2x2x2", I'd say that the middle layer on the 3x3x3 is just a physical realization of some of the pieces of the Complex 3x3x3 and than the 2x2x2 just doesn't physically have those pieces. That doesn't mean that just because they aren't there physically they aren't there mathematically.

If you want to identify a piece on a physical puzzle you have to look at the interaction and depths of the cuts but if you want to define a piece mathematically it's just "turns with this face" or "does not turn with this face" and hence the binary definition.

So what about the Complex 4x4x4 / Complex 5x5x5? Well I'd say that it's just the "Complex Face Turning Cube With Two Cuts Per Face" and to make physical realizations of it you just add two cuts and then apply symmetries:
Attachment:

5x5x5_cut_progression.png [ 23.82 KiB | Viewed 7301 times ]

Just like with the 2x2x2 versus 3x3x3, the 4x4x4 and 5x5x5 are just subsets of the "Complex Face Turning Cube With Two Cuts Per Face". The difference mathematically is that you can turn with cut 1, turn with cut 2, turn with both, or not turn with either so there are 4 states rather than 2. The strange pieces that turn with both cuts can be seen on Gelatinbrain's 3.1.35 and they are just a subset of the pieces on the Complex 4x4x4 / 5x5x5.

Again I argue that no mater how crazy you make the cuts curve and zig-zag, you will always make a puzzle that is just a subset of the Complex 4x4x4 / 5x5x5 (whether or not the puzzle is physically realizable).

To put everything I've said in a slightly different way, I think the cuts per axis, number of layers, and the pieces of the puzzle are all a symptom of a choice of the shape and depth of the cut but there are other choices that lead to other subsets of the complex puzzle and they are not from fundamentally different complex puzzles but just different choices to the subset of pieces selected from the same complex puzzle.

I hope this makes sense . It seems like a more abstract and flexible way to look at it. Perhaps I'm being shortsighted in not seeing it differently?

Brandon

wwwmwww wrote:
P.S. bmenrigh what is the puzzle in your avitar picture? I don't believe I've seen it before.
I'm not sure if you still haven't seen this post but it was shown off here Dreidel Skewb

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Nov 16, 2011 8:51 pm

Joined: Sun Oct 08, 2006 1:47 pm
Location: Houston/San Antonio, Texas
Brandon,

I completely agree (and always have, if you remember ever reading my posts on the subject from long ago ). Having a system like this also points out the ambiguity in what people want to call a 4x4x4 megaminx. It really does seem to view things from a much more generalized perspective. It also makes more sense for puzzles that do not have opposite faces. I would say Oscar's Meteor Madness "feels" like the same order puzzle as a standard 3x3x3 even though it only has one cut per axis.

I wish I had more time to catch up on all the latest stuff in twistypuzzles but I'm so busy completing my senior year of undergrad (earning a BS in engineering and a BA in mathematics - no easy feat) that I barely have time to sleep let alone play with puzzles

I will reveal that my senior thesis in mathematics will be over something some people here might find VERY interesting

Peace,
Matt Galla

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Thu Nov 17, 2011 1:45 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Brandon and Matt,

We may have to agree to disagree. There are parts of this I agree with and parts of this I disagree with. But I view the core difference between my view and your view is you look at cuts per face and I look at cuts per axis. I view the axis approach as more fundamental as they are independant of the geometry. You can have a spherical 3x3x3 for example and I could agrue where are the faces. That is a weak argument I know as it still has cubic symmetry but I still prefer to deal with the axis system over the faces of a puzzle. Allow me to try and explain...
bmenrigh wrote:
I think it is a mistake to say that the 2x2x2 only has one cut per axis and therefor the "Complex 2x2x2" is somehow different than the "Complex 3x3x3" because the number of cuts per axis is really just a function of one of the subsets of the complex puzzle that it is derived from.
Here is why I don't like this argument. Let's look at the "Complex 5x5x5". Also let's say we have a cube in space with one corner at X=Y=Z=1 and the opposite corner at X=Y=Z=-1. Let's take your face approach. We can name the 6 faces X, Y, Z, -X, -Y, and -Z. So let's just pick the 2 cuts for the X face. We can call the two cut depths x=a and x=b with the following restrictions: 0â‰¤a<1 and 0â‰¤b<1. If we look at the case where a=0 and 0<b<1 then from the symmetry of the cube you can see that the "a" cut of the X and the -X face are degenerate such that there are only 3 cut planes on the x-axis. So YES, I agree that the Complex 4x4x4 is a degenerate example of the Complex 5x5x5. However let's look at another degenerate example. Let's say a=b and 0<a<1. We now have only 2 cut planes on the x-axis so the Complex 3x3x3 is ALSO a degenerate example of the Complex 5x5x5. So why is it not an equal mistake to say that the Complex 3x3x3 is somehow different then the Complex 5x5x5, because the Complex 3x3x3 is really just a function of one of the subsets of the Complex 5x5x5? To go one step further I can show the Complex 5x5x5 is just a subset of the Complex 7x7x7, etc and carry this all the way to the conclusion that there is only one complex puzzle... the Complex infinite^3.

If we can acknowledge that the Complex 3x3x3 is a subset worth looking at in its own light despite the fact that its also part of the Complex 5x5x5 why can't we also assume the the Complex 4x4x4 is a subset worth looking at in its own light?
bmenrigh wrote:
If you want to identify a piece on a physical puzzle you have to look at the interaction and depths of the cuts but if you want to define a piece mathematically it's just "turns with this face" or "does not turn with this face" and hence the binary definition.
With this I agree 100%. I just look at the independant layers per axis and NOT the layers or cuts per face. The Complex 4x4x4 has 3 independant layers per axis. Each piece mathematically can be defined as either turning with a given layer or not turning with a given layer. Since there are 3 axes, there are a total of 9 independant layers in the puzzle and therefore the Complex 4x4x4 has 2^9 or 512 pieces. This is totally general and gives the Complex 3x3x3 a total of 2^6 or 64 pieces and it gives the Complex 5x5x5 a total of 2^12 or 4096 pieces. In fact the very same model that GuiltyBystander used to make his app for the Complex 3x3x3 I think could be used to make a great Complex 4x4x4.

I just see nothing in this model which requires you to jump from 2 independant layers per axis to 4 independant layers per axis. Why can't we stop and expore the case where there are only 3 independant layers per axis?
bmenrigh wrote:
It seems like a more abstract and flexible way to look at it. Perhaps I'm being shortsighted in not seeing it differently?
Don't get me wrong. I understand the way you are looking at it and I don't really feel there is a right or a wrong way. Its just a different point of view. But I consider the concept of an axis of rotation just as abstract as the concept of a face. And I'd argue my view is more flexible. It allows for an odd number of independant layers per axis. You model only allows for an even number.
bmenrigh wrote:
I'm not sure if you still haven't seen this post but it was shown off here Dreidel Skewb
Ahh thanks!!! No I hadn't seen that till you pointed it out. If you ask me that puzzle deserves its own thread in the new puzzle section.
Allagem wrote:
I would say Oscar's Meteor Madness "feels" like the same order puzzle as a standard 3x3x3 even though it only has one cut per axis.
Of course it "feels" very much like a 3x3x3. And I believe there is a very good reason for that. The 3x3x3 has 3 axes of which each has 2 independant layers so the total puzzle has 6 independant layers. The Meteor Madness puzzle has 6 axes and each has 1 independant layer so again you are looking at a puzzle with a total of 6 independant layers. I'd expect it to "feel" alot like a 3x3x3.

In this case the Meteor Madness cuts aren't deep cut so you are looking at a puzzle which in someways is similiar to a Pyraminx minus the trivial tips. If you mirrored the cuts such that you had 2 per axis and not one it would give you the Master Skewb I believe. This is a play on the geometic relationship between the tetrahedron and the octahedron. Its this interplay which has produced puzzles like Shim's F-Skewb. I suspect you could mirror the cuts of a Meteor Madness about the origin and create a similiar interplay in geometry if you'd like. Its much more complex though due to there being more axes.
Allagem wrote:
I will reveal that my senior thesis in mathematics will be over something some people here might find VERY interesting
I think I can confirm that. How did things turn out with the Goldwater Scholarship?

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Sat Nov 19, 2011 6:09 pm

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
Guys, I've had a bit of a crazy idea which was inspired by some of this thread. It's a little weird, but it could lead to some pretty interesting puzzles.

So, going back to the alternative way to visualise pieces of a complex puzzle, with the cubes and dots and stuff. Now, say we wish to make a twist of the U face on a complex 3x3x3. We find all the pieces which currently have a dot on the U face, and rotate the cubes representing those pieces in the same direction about the UD axis. Alternatively, this is a 4-cycle of the FRBL faces of these cubes. What if we change that a little? Say we do some other cycle of the faces on all the cubies with a dot on the U face when we do a U move? Perhaps we could switch the U and D faces of these pieces, which would still be symmetric about the U face and would still lead to one 'twist' per face, and with everything else the way it was, we get a different puzzle, with the same number of pieces as a complex 3x3x3.

Or, what if we go a little crazier. What if, instead if a U 'twist' doing a 4-cycle of FRBL, it did FRLB? It works the same way, you find all the pieces with a dot on the U face, and do the cycle. For example, say we have a solved state, and a piece which is affect by that particular twist and has its faces labelled as UDRLFB. After that twist, it would be UDBFRL, and the dot(s) would move accordingly. Define an equivalent cycle for each other face, and again you would get a new puzzle. This time though, it doesn't work so well, as there are 2 cycles possible per face. A U twist would do either FRLB or FBRL. Well, it would sort of be like going from a 3x3x3 to a 5x5x5, where we have 2 possible twists per face instead of 1. We pick one of the possibilities for each face and give it one colour of dot, then give the other twist a different colour of dot. Then we have 2 'twists' per face which behave in a similar manner to the 2 twists per face of a 5x5x5, and a puzzle with the same number of pieces as a complex 5x5x5. Also, there are various different ways of picking which twist from each face to be a particular colour of dot, and different choices would lead to different puzzles.

It is easy to see that the number of pieces would get very large very quickly. However, we can just choose a subset of them to work with, just like we can view a 3x3x3 as picking a subset of the pieces from a complex 3x3x3 and using the same rules as it does (I realise this seems a little backwards since people chronologically went from a normal 3x3x3 and derived a complex 3x3x3, but it works). We could, for example, take the pieces which have 3 dots of one colour around one corner, and three dots of the other colour around the other corner. Then, we only have 8 pieces to solve. We can also choose any other subset we like.

It turns out I lied a little earlier. I said there were 2 possibilities for picking the 4-cycle for the U face. Clearly this is wrong, it just happens that there are 2 which are more symmetrical about the U face. There are 6 possibilities altogether (2 per axis), and we can choose any combination of them to define the 'twists' of a puzzle.

There are obviously many more cycles of faces you can choose to perform to each piece affected when doing a particular 'twist' , from 6-cycles of the faces to having a simultaneous 3-cycle and 2-cycle. You can also clearly extend this to other shapes and make things even crazier.

Plenty of food for thought here, and it certainly confused me a lot. However, it provides a new perspective for designing puzzles, and might be interesting for some of you to analyze. There may even be more possibilities this can be extended to, and there probably are. I only wish I had more time so I could study this more and show how to derive many common puzzles from it, as I'm sure it will cover a lot of well-known puzzles.

Have fun

Matt S

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Nov 22, 2011 5:08 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bobthegiraffemonkey wrote:
Or, what if we go a little crazier. What if, instead if a U 'twist' doing a 4-cycle of FRBL, it did FRLB?
Its very hard for me to visualize this. Are you sure this is a new puzzle? How do you know you haven't just renamed the B and L faces and in a much more abstract way are actually dealing with the same puzzle?

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Nov 22, 2011 7:21 pm

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
wwwmwww wrote:
bobthegiraffemonkey wrote:
Or, what if we go a little crazier. What if, instead if a U 'twist' doing a 4-cycle of FRBL, it did FRLB?
Its very hard for me to visualize this. Are you sure this is a new puzzle? How do you know you haven't just renamed the B and L faces and in a much more abstract way are actually dealing with the same puzzle?

Carl

The simplest way to realise that this 'twist' is different is by creating a puzzle with both of them being possible. It is clear that for such a puzzle they will have a different effect by definition. Imagine a piece which could be affected by both 'twists', and it is clear that they affect the piece differently as they permute the 'faces' of that piece differently. Does that make any sense Carl or do I need to explain better?

I keep putting 'twist' in inverted commas for all this, as I'm not sure it really fits the standard definition of a twist, it's too abstract just now to really say.

Just to add some more info as I didn't get time to say everything in my last post. I've had a look at all the possible different cycles you can define relative to a face. By my calculation, there are 537 in total, as follows (indexed by cycle size(s)):

6 60
5 72
4 45
3 20
2 15
4,2 45
3,3 40
3,2 60
2,2 90
2,2,2 90

1-cycles and 0-cycles are ignored as they don't do anything. Now, let's take this to extremes. Let's say we have one of each 'twist' for every face. How many pieces do we have in total if we take the complex version of the puzzle? Well, with some thought the number of pieces will be (2^(number of types of twist per face))^6 which is
(2^537)^6 = 2^(537*6) = 2^3222
unless I have made some mistake.

Clearly, this extreme puzzle contains more pieces than a normal 3x3x3 has states! I think it will be difficult to calculate how many possible states there are for this puzzle. It is also an interesting coincidence with the number as it contains (2^3) and (222) which can be different notations for a 2x2x2 puzzle. In practice, it would be essentially impossible to solve, but it is interesting in theory, as it does not contain more than one of a certain 'twist' per face so it is not higher order in the conventional sense (ie. going from 3x3x3 to 5x5x5), yet we can have a very large number of possible twists defined relative to just the 6 faces of a cube, and it contains all the different pieces which can exist in puzzles derived from this with at most one of each possible 'twist' per face.

I want a good name for this hypothetical puzzle. I'm thinking of maybe extending the 'complex' name, and maybe referencing quaternions/Hamiltonian numbers. How about Quaternion Cube? Any better ideas? I personally don't think it feels right but I can't think of anything better. I'm not sure we can really get many new physical puzzles out of this, but it could lead to some interesting simulated puzzles. Does anyone want to try making a simulator for some of them? Does anyone even understand the strangeness happening inside my head or am I just crazy?

Matt S

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Sun Nov 27, 2011 11:51 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bobthegiraffemonkey wrote:
The simplest way to realise that this 'twist' is different is by creating a puzzle with both of them being possible. It is clear that for such a puzzle they will have a different effect by definition. Imagine a piece which could be affected by both 'twists', and it is clear that they affect the piece differently as they permute the 'faces' of that piece differently. Does that make any sense Carl or do I need to explain better?
Sorry for the slow reply. I've been a bit under the weather the last few days. Anyways, I think I follow the general idea but its so abstract I'm having a hard time seeing the specifics. Any chance you could make a picture or two that might help.

For example lets look at a normal 90 degree face twist of a 3x3x3. Would one of your other 'twists' be a 180 degree turn of the same face? If so are these actually new pieces as its just a subset of the possible 3x3x3 states? Say you combine a normal 3x3x3 and a 3x3x3 which just allowed 180 degree turns. You are back to having just your normal 3x3x3 again as you haven't really added anything new.
bobthegiraffemonkey wrote:
I've had a look at all the possible different cycles you can define relative to a face. By my calculation, there are 537 in total, as follows (indexed by cycle size(s)):

6 60
5 72
4 45
3 20
2 15
4,2 45
3,3 40
3,2 60
2,2 90
2,2,2 90

1-cycles and 0-cycles are ignored as they don't do anything.
I'm not sure I understand this calculation. Any chance you could make a picture which shows what some of these 'twists' do to the cube. I assume they can be viewed as a normal 3x3x3 but that one 'twist' of a face changes the location and/or orientation of the face center, edges, and corners on that face. So again if that state is reachable on a normal 3x3x3 via a sequence of 90 degree face turns I think you are still looking at a normal 3x3x3. You've just taken the sequence and replaced it with a new 'twist'. You should be able to find a sequence of these new 'twists' whch would map back to a normal 90 degree turn.

That said I can see cases where a new 'twist' wouldn't fall into this catagory. Say a 'twist' was define such that it just flipped a single edge piece on that face. You are now in a state which isn't reachable on a normal 3x3x3. But if you had a puzzle with 3 types of twists; a normal 90 degree face turn, a flip of a single edge piece, and a 120 degree rotation of a single corner piece; couldn't you make all possible 537 cycles as some combination of these?
bobthegiraffemonkey wrote:
Does anyone even understand the strangeness happening inside my head or am I just crazy?
Not sure yet. You may be on to something that is just beyond my understanding at the moment. Not that hard to do as I still have to really struggle to understand the 4D and higher puzzles out there. But I think you may be making a pictue where most of your 'twists' are linear combinations of the other 'twists' and I'm not really sure if they add anything new. I have to confess I don't understand your counting method above so its just as likely I'm the one that's out in left field and none of what I just said applies.

Carl

_________________
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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Sun Nov 27, 2011 5:36 pm

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
wwwmwww wrote:
Any chance you could make a picture or two that might help.

Sure, as soon as I get time.

wwwmwww wrote:
For example lets look at a normal 90 degree face twist of a 3x3x3. Would one of your other 'twists' be a 180 degree turn of the same face? If so are these actually new pieces as its just a subset of the possible 3x3x3 states? Say you combine a normal 3x3x3 and a 3x3x3 which just allowed 180 degree turns. You are back to having just your normal 3x3x3 again as you haven't really added anything new.

There certainly is some redundancy in this system, but I'm not sure of the best way to fix that.

wwwmwww wrote:
bobthegiraffemonkey wrote:
I've had a look at all the possible different cycles you can define relative to a face. By my calculation, there are 537 in total, as follows (indexed by cycle size(s)):

6 60
5 72
4 45
3 20
2 15
4,2 45
3,3 40
3,2 60
2,2 90
2,2,2 90

1-cycles and 0-cycles are ignored as they don't do anything.
I'm not sure I understand this calculation. Any chance you could make a picture which shows what some of these 'twists' do to the cube. I assume they can be viewed as a normal 3x3x3 but that one 'twist' of a face changes the location and/or orientation of the face center, edges, and corners on that face. So again if that state is reachable on a normal 3x3x3 via a sequence of 90 degree face turns I think you are still looking at a normal 3x3x3. You've just taken the sequence and replaced it with a new 'twist'. You should be able to find a sequence of these new 'twists' whch would map back to a normal 90 degree turn.

First point to make here: I am not thinking of a 3x3x3 with centers, edges and corners. I am thinking of abstract pieces which are each represented by a cube with coloured faces, an idea which was presented earlier in this thread (which I hope I picked up correctly). There will be dots on some faces which correspond to certain twist types. When you do a twist assigned to a certain face, you look at that face of all the cubes representing the pieces. Any of them which have the correct type of dot on the relevant face will be affected by that twist. Before, we were implicitly assuming that the only valid possibility was a 4-cycle of the faces adjacent to the face the twist was assigned to, since this corresponds to what physically happens on nxnxn cubes. I'm looking at what happens if you perform a different permutation of the faces of the cubes representing the pieces. Just to look at that in a little more detail: instead of a U turn being thought of as rotating the pieces about an axis going through the U face on a physical puzzle, think of it as a 4-cycle of the faces which lie adjacent to the U face (F, R, B and L) on the cubes which represent the pieces. For a normal 3x3x3, these are equivalent, but the alternative approach gives a way to turn the construction on its head and say, "well this is one way to think of a 3x3x3, but what other (possibly very abstract) puzzles could this construction describe?"

Quick sample calculation which I will do for the 5-cycles. There are 5! ways to order 5 faces, and (6choose5=x) ways of picking 5 faces to order. but 5!*x gives too large a number for the cycles, as you can start at any of 5 points in the cycle, and the inverse of each of these will be essentially the same twist. So we get (5!*6)/5/2=72 distinct cycles. A similar calculation applies to the other cycles. Here there is some easy to find redundancy, as doing any 5-cycle twice gives a different 5-cycle, so the number of useful 5-cycles is half of the given number. I honestly have no idea how many of the different 'twists' can be removed without losing anything non-trivial.

wwwmwww wrote:
That said I can see cases where a new 'twist' wouldn't fall into this catagory. Say a 'twist' was define such that it just flipped a single edge piece on that face. You are now in a state which isn't reachable on a normal 3x3x3. But if you had a puzzle with 3 types of twists; a normal 90 degree face turn, a flip of a single edge piece, and a 120 degree rotation of a single corner piece; couldn't you make all possible 537 cycles as some combination of these?
bobthegiraffemonkey wrote:
Does anyone even understand the strangeness happening inside my head or am I just crazy?
Not sure yet. You may be on to something that is just beyond my understanding at the moment. Not that hard to do as I still have to really struggle to understand the 4D and higher puzzles out there. But I think you may be making a pictue where most of your 'twists' are linear combinations of the other 'twists' and I'm not really sure if they add anything new. I have to confess I don't understand your counting method above so its just as likely I'm the one that's out in left field and none of what I just said applies.

Carl

A nice idea, but as I've hopefully managed to explain, its not what I'm thinking of.

Here's a little extra thing which I just thought of tonight, which develops things a little more. Since I'm not longer just rotating the cubes which represent the pieces, but permuting their faces, we can give each of these faces an independent orientation. First, let's give this in terms of a normal supercube. We do a U move, which for all the pieces affected by the twist does the 4-cycle F->L->B->R, rotates the face in the U position CW, and the face in the D position CCW. However, we don't have to be so tame. What if, instead, we leave the U and D faces in the orientation they had, and after we do the 4-cycle, we rotate whichever face is in the F position CW? Obviously, there is a lot of freedom for combining these rotations, and combined with several kinds of 'twists', could lead to some pretty confusing stuff. Intuition tells me that this approach will make it very easy to make puzzles with very few pieces which are extremely difficult to solve, and which cannot be physically implemented in general. Obviously, this is only useful if I manage to explain this well enough eventually for someone to understand it .

I have some plans to learn some Java programming when I have the time (which I don't have much of these days), which was originally intended for other puzzle-based projects for simulated 4D puzzles, but I will probably make a playable version of some of what I have described here, unless someone figures out what I' describing and beats me to it. If anyone wants to, go ahead. while I would enjoy it, it would be one less thing on my to-do list. Then again, this construction applies equally well in 4 dimensions, so I can combine the two into the one super-powerful weapon of mass confusion. Oh, the possibilities for chaos ...

Matthew S

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Jan 23, 2012 2:20 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
I am pleased to report after dozens of hours of work, I have finally solved the Super-Complex 3x3x3 (the complex 3x3x3 where the orientations of all pieces are visible).

I started working on sequences to solve it a few months ago and got stuck. I couldn't figure out how to handle the orientation of the face centers and UD pieces, or the inverted face centers and UD^-1 pieces.

Realizing that some of the concerns I had regarding their orientation could be avoided early on in the solve or tackled when needed, I decided to take up the task today.

For this challenge I used GuiltyBystander's great simulator in concert with Gelatinbrain's 3.1.7, 3.1.31, 3.1.32, and 3.1.33 with a whole lot of copy-and-paste back and forth. Because I want this solution to be reproducible I've generated my own scramble in a "nothing up my sleeve" way using this program:
Code:
#!/usr/bin/perl

use strict;
use warnings;

my \$rc4_state = pack('C' x 256, (0 .. 255));
my \$rc4_i = 0;
my \$rc4_j = 0;

sub rc4_init {
my \$key = shift;
my \$keylen = length \$key;

for (\$rc4_i = 0; \$rc4_i <= 255; \$rc4_i++) {
\$rc4_j = (\$rc4_j +
unpack('C', substr(\$rc4_state, \$rc4_i, 1)) +
unpack('C', substr(\$key, \$rc4_i % \$keylen, 1))) & 0xFF;

# Swap
(substr(\$rc4_state, \$rc4_i, 1), substr(\$rc4_state, \$rc4_j, 1)) =
(substr(\$rc4_state, \$rc4_j, 1), substr(\$rc4_state, \$rc4_i, 1));

}

\$rc4_i = 0;
\$rc4_j = 0;
}

sub rc4_byte {
\$rc4_i = (\$rc4_i + 1) & 0xFF;
\$rc4_i = (\$rc4_j + unpack('C', substr(\$rc4_state, \$rc4_i, 1))) & 0xFF;

# Swap
(substr(\$rc4_state, \$rc4_i, 1), substr(\$rc4_state, \$rc4_j, 1)) =
(substr(\$rc4_state, \$rc4_j, 1), substr(\$rc4_state, \$rc4_i, 1));

return substr(\$rc4_state, ((unpack('C', substr(\$rc4_state, \$rc4_i, 1)) +
unpack('C', substr(\$rc4_state, \$rc4_j, 1))) &
0xFF), 1);
}

sub rc4_int {

return unpack('C', rc4_byte());
}

my @face_names = ('U', 'F', 'R', 'B', 'L', 'D');
my @amt_names = ('', '', '2', '\'');

rc4_init('Super-Complex 3x3x3');

my @moves;
my \$last_move = -1;
# a scramble of 128 moves should be enough
for (my \$i = 0; \$i < 128; \$i++) {

my \$rand_face = (rc4_int() & 0x07);
while ((\$rand_face > 5) || (\$rand_face == \$last_move)) {
\$rand_face = (rc4_int() & 0x07);
}
\$last_move = \$rand_face;

my \$rand_amt = (rc4_int() & 0x03);
while ((\$rand_amt > 3) || (\$rand_amt == 0)) {
\$rand_amt = (rc4_int() & 0x03);
}

# push this move onto our list
push @moves, \$face_names[\$rand_face] . \$amt_names[\$rand_amt];

}

print '/* Scramble */', "\n";
print join(', ', @moves), "\n";

The scramble it generates (128 moves) is:
Code:
F2, B2, L', B', D', F, B2, L', B', F, B2, L', B', F, B2, L', B2, L', B, F2, B2, L', R', L', U, F2, B2, L', R', B', F, B2, L', B', U, F2, B2, L', B', D, F2, B2, L', B', D, F2, B2, L', B', D, F2, B2, L', B', F, R2, B2, L', B', F, B2, L', B', D', F, B2, L', B', F, B2, L', B', F, B2, L', B2, L', B, F2, B2, L', R', L', U, F2, B2, L', R', B', F, B2, L', B', U, F2, B2, L', B', D, F2, B2, L', B', D, F2, B2, L', B', D, F2, B2, L', B', F, R2, B2, L', B', F, B2, L', B', D', F, B2, L', B', F

I then solved in phases, copy and pasting between programs as needed. The hardest part was the UDF pieces which on Gelatinbrain's 3.1.32 come in 3 groups of 4 identical pieces. I first solved them so that they "looked" solved and then I went about permuting indistinguishable pieces using GulityBystander's program to tell them apart. I avoided slice moves because I feel like a slice is poorly defined on the Complex 3x3x3.

Here is my 839 move solution:
Code:
/* Super 3x3x3 */
B', L, F', D', F, D', B', D', B', R'2, D', R, D, R', D, F', D', F, D, R', D', R, D', L', D', L, D'2, L, D', L', D', F', D, F, D'2, F, D', F', D', R', D, R, D, R, D', R', D', B', D, B, D, B, D', B', D', L', D, L, D', B, D', B', D', L', D, L, R', B', D', B, D, R, D, F, D', B', D, F', D', B, D'2, L, D, L', D, L, D'2, L', D'2, L, R, D'2, R', L', B', F', D'2, F, B'2, F', B', F, B', L', R, U', D, F, B', L, F', B, U, D', L, R', F', B, F, B', R, L', R', L, R, L', R', L, U, D', F, B', R', L, D', R, L', F', B, U', D, R'2, L', R', L, U, D', F, B', R', L, D', R, L', F', B, U', D, R,
/* Inverted centers permutation */
L, R', U', D, F', B, L', B', F, D', U, R, L', B,
/* Inverted edges */
D', L', B, U, L, U', L', B', U, R, B, U, B', U', R', U', L, D, F'2, R'2, L, B, L, D, L', D', B', L', D, F, L, F', L', D', R'2, F'2, B, D'2, U, F, U, R, U', R', F', U', R, B, U, B', U', R', D'2, B', F'2, B, R, B, D, B', D', R', B', D, L, B, L', B', D', F'2, R', D, F, R, F', R', D', F, L, D, F, D', F', L', F', R, L, U, B, U, L, U', L', B', U', L, F, U, F', U', L', F, L, U, L', U', F', L', U, B, L, B', L', U', R, B', L, F, L, U, L', U', F', L', U, B, L, B', L', U', B, R', F, D, F, R, F', R', D', F', R, U, F, U', F', R', B, U', F, D, F, R, F', R', D', F', R, U, F, U', F', R', U, B',
/* UDF pieces (visible permutation) */
F, R, D, R', U, R, D', R', U', B'2, D, L, U, L', D', L, U', L', B'2, F', R, B'2, R, U, R', D, R, U', R', D', F'2, U, L, D, L', U', L, D', L', F'2, B'2, R', F', D'2, R, D', L, D, R', D', L', F'2, R, U, L, U', R', U, L', U', F'2, D', F, U, D'2, R, D', L, D, R', D', L', F'2, R, U, L, U', R', U, L', U', F'2, D', U', L'2, B, R'2, D, R', U, R, D', R', U', B'2, D, L, U, L', D', L, U', L', B'2, R', B', L'2, R', L, D', R, D, L', D', R', B'2, L, U, R, U', L', U, R', U', B'2, D'2, B'2, U, R, U', L, U, R', U', L', B'2, R, D, L, D', R', D, L', D'2, R,
/* UDF pieces (invisible permutation) */
B, L', R, L', U, L', D, L, U', L', D', B'2, U, R, D, R', U', R, D', R', B'2, L', R', B', D, L', R, L', B, L', F, L, B', L', F', D'2, B, R, F, R', B', R, F', R', D'2, L', R', D', R', L, R, U', D, U', F, U', B, U, F', U', B', R'2, F, D, B, D', F', D, B', D', R'2, U', D', R', B, U', D, U', R, U', L, U, R', U', L', B'2, R, D, L, D', R', D, L', D', B'2, U', D', B', L', R'2, B', F, B', U, B', D, B, U', B', D', R'2, U, F, D, F', U', F, D', F', R'2, B', F', R', L, U, D, L'2, U, B, U', F, U, B', U', F', L'2, B, D, F, D', B', D, F', D, U', D, L',
/* Face and UD orientations */
U, R, D, B', L', B', L', B, L, B, D', R', U', B, U, R, D, B', L', B', L', B, L, B, D', R', U', B, U, R, D, B', L', B', L', B, L, B, D', R', U', B,
/* Inverted face and UD^-1 orientations */
L, U, L, B, L', B', U', L', B, D, L, D', L', B', L, U, L, B, L', B', U', L', B, D, L, D', L', B', L, U, L, B, L', B', U', L', B, D, L, D', L', B', U, L, B, L', B', U', L, F, U, L, U', L', F', L', U, L, B, L', B', U', L, F, U, L, U', L', F', L', U, L, B, L', B', U', L, F, U, L, U', L', F', L', U, R, U, B, U', B', R', U', B, L, U, L', U', B', U, R, U, B, U', B', R', U', B, L, U, L', U', B', U, R, U, B, U', B', R', U', B, L, U, L', U', B', R, B, R, U, R', U', B', R', U, F, R, F', R', U', R, B, R, U, R', U', B', R', U, F, R, F', R', U', R, B, R, U, R', U', B', R', U, F, R, F', R', U'

You can paste the scramble and then the solution into 3.1.7 or 3.1.31 or 3.1.32 or 3.1.33 and see that it solves the puzzle. To paste into GuiltyBystander's program you first have to strip out the /* ... */ comments.

This was the hardest puzzle I have ever solved. The puzzle is "deep-cut" in that half the pieces (32) move in every twist and the permutations of some of the pieces are linked to the orientations of others. Isolating pieces to make a 3-cycle was very challenging. Unfortunately, I wasn't able to figure out a pure way to orient some of the pieces so I ended up apply 3-cycles that had a side effect 3-times to cancel out the 3-cycle and leave a twist in some other pieces. I hate doing this but I couldn't come up with any alternative. Applying sequences 3 times coupled with solving one piece at a time in each cycle made for a very lengthy solve. This move count could easily be cut in half if I was willing to double or triple the time I spent on the solve.

Notice that I solve the orientations of the face centers on the Super 3x3x3 and allow them to get broken later before solving them again. I believe this is required to link their orientations up so that they can be solved in pairs later using my non-pure sequences.

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Jan 23, 2012 10:39 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
I am pleased to report after dozens of hours of work, I have finally solved the Super-Complex 3x3x3 (the complex 3x3x3 where the orientations of all pieces are visible).
VERY VERY IMPRESSIVE!!!! Congratulations!
bmenrigh wrote:
I avoided slice moves because I feel like a slice is poorly defined on the Complex 3x3x3.)
A slice on the Complex 3x3x3 is defined just as it is on the 3x3x3. It is a linear combination of the two opposite faces being turned in the same direction about their axis of rotation followed by a global rotation of the entire puzzle in the opposite direction. It is sort of ugly but it is there.
bmenrigh wrote:
This was the hardest puzzle I have ever solved.
Very interesting.... I'm not sure I would have expected that. I guess for your next challenge you'll take on the Super-Complex 3x3x3x3.
bmenrigh wrote:
The puzzle is "deep-cut" in that half the pieces (32) move in every twist and the permutations of some of the pieces are linked to the orientations of others.
Does it fall within the definition of deep cut that states the puzzle is cut into two isomorphic groups? Look at the UD pieces, there are only 3 in the whole puzzle and each twist only moves 1 of the 3. I'm having a hard time seeing how the two seperate groups of 32 pieces could be isomorphic to each other.

Very very nice,
Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Jan 23, 2012 1:30 pm

Joined: Sun Aug 29, 2010 1:56 pm
Location: Berlin, Germany
bmenrigh wrote:
I am pleased to report after dozens of hours of work, I have finally solved the Super-Complex 3x3x3 (the complex 3x3x3 where the orientations of all pieces are visible).
Hi Brandon, astonishing, I'm surely impressed too. I played with the thought, to do that, but actually never came to it. Thank you, that you did it, even harder with all orientations visible.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Jan 23, 2012 2:15 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
VERY VERY IMPRESSIVE!!!! Congratulations!
Thanks, I've been wanting to solve this puzzle for a very long time but due to the complexity of it, much more effort was required than a typical puzzle.
wwwmwww wrote:
bmenrigh wrote:
I avoided slice moves because I feel like a slice is poorly defined on the Complex 3x3x3.)
A slice on the Complex 3x3x3 is defined just as it is on the 3x3x3. It is a linear combination of the two opposite faces being turned in the same direction about their axis of rotation followed by a global rotation of the entire puzzle in the opposite direction. It is sort of ugly but it is there.
Of course you're right. For some reason I assumed that in order to do the middle slice between R and L it mattered if you turned up R and then L or L and then R (that they wouldn't commute) but of course they do. I should have thought about that more.
wwwmwww wrote:
bmenrigh wrote:
This was the hardest puzzle I have ever solved.
Very interesting.... I'm not sure I would have expected that.
Me either. The trick to solving any puzzle though is to find ways to isolate one piece type from another to create a cycle or a way to change the orientations of pieces of one type without affecting pieces of another type. Because there are 10 piece types in such a compact puzzle it is quite hard to separate out the target piece from all of the others to create a cycle.
I spent a long time worrying about what would happen if I had a single UD piece twisted or a single inverted face center twisted. I spend so much time trying to find sequences for these cases before I realized that some of the cases can't happen or that I can avoid them early on by solving the face center orientations before working on the inverted face centers, inverted edges or UDF pieces.
wwwmwww wrote:
I guess for your next challenge you'll take on the Super-Complex 3x3x3x3.
A tesseract has 8 cells so that should have 2^8 == 256 pieces. It sure seems like it would have more. I think the Complex FTO is a much more realistic challenge for a human though.
wwwmwww wrote:
bmenrigh wrote:
The puzzle is "deep-cut" in that half the pieces (32) move in every twist and the permutations of some of the pieces are linked to the orientations of others.
Does it fall within the definition of deep cut that states the puzzle is cut into two isomorphic groups? Look at the UD pieces, there are only 3 in the whole puzzle and each twist only moves 1 of the 3. I'm having a hard time seeing how the two seperate groups of 32 pieces could be isomorphic to each other.
Yeah you're right again. The UD pieces are not the only pieces that break the isomorphism. I should have known this because I had to make heavy use of "anti-moves" -- that is, turn the opposite face and then re-orient the puzzle. If this were a real deep-cut puzzle with isomorphic halves that wouldn't have helped anything. In fact, most of the difficulty of the puzzle stemmed from the need to look at what doesn't move when I turn a face, rather than what does. Most of my commutators made use of a mix of regular moves and anti-moves and it was the need to do this that took me so many hours to find all of the sequences I needed.

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Dec 03, 2012 1:06 am

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
Sorry for bumping. I've finally solved this complex 3x3x3 puzzle. This solve is long overdue. I was at first intimidated by the UI. Recently I made up my mind to solve it, and then I spent time understanding how everything works. After that I have to say everything makes perfect sense. I especially like "1 layer" "2 layer"... views. They helped a lot.

This is a very elegant puzzle. I won't say it's the hardest, but it's complicated in a very neat way. I certainly want to solve more puzzles like this. I think I'm going to scramble and solve this puzzle again by 180 degree turns only. I guess this is equivalent to the "edge-turning complex tetrahedron", isn't it? I hope this is another interesting puzzle.

Brandon: You mentioned that you've solved the "Super-Complex 3x3x3". The default puzzle in Complex.jar is such a puzzle, isn't it? All the [UDF] pieces are actually distinguished in the 64-dice view.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Dec 03, 2012 7:19 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Hey Nan, congrats and welcome to the club! It sounds like you solved everything using the interface provided by Landon's program? I used the set of 3.1.x provided by Gelatinbrain for the solve and I used Landon's Complex.jar visualization as a way to distinguish orientation and identical pieces.

Here are my "notes":
Code:
Solve the complex 3x3x3:

* Solve super-rubik's cube (3.1.7)
* Solve inverted face centers on 3.1.31
* Solve inverted edges on 3.1.31
* Solve UDF pieces on 3.1.32
* Solve face/UD and inverted face / UD^-1 orientations

Face and UD half-turn (makes a mess of other pieces, only do on super rubik's cube):
[U, R, L, U'2, L', R', U, R, L, U'2, L', R']

Inverted edges 3-cycle (affects UDF edges too) cw:
[B, U, L, U', L', B', U, R, B, U, B', U', R', U']
Inverse ccw:
[U, R, U, B, U', B', R', U', B, L, U, L', U', B']

Flip two inverted edges (affects UDF edges too):
[D, F, D, L, D', L', F', D', L, B, D, B', D', L', U, B', D, F, D, L, D', L', F', D', L, B, D, B', D', L', B, U']

UDF 3-cycle (affects inverted UD and inverted face orientations):
[U, F, U', B, U, F', U', B', R'2, F, D, B, D', F', D, B', D', R'2]

UDF pieces two flip:
[U, F, U', B, U, F', U', B', R'2, F, D, B, D', F', D, B', D', R'2, U'2, R'2, D, B, D', F, D, B', D', F', R'2, B, U, F, U', B', U, F', U]

UDF pieces on same face (R) cw:
[B, L', R, L', U, L', D, L, U', L', D', B'2, U, R, D, R', U', R, D', R', B'2, L', R', B']

UDF pieces on same face (R) 2-2 swap accross:
[B, L', R, L', U, L', D, L, U', L', D', B'2, U, R, D, R', U', R, D', R', B'2, L', R', B', D, L', R, L', B, L', F, L, B', L', F', D'2, B, R, F, R', B', R, F', R', D'2, L', R', D']

UDF pieces on same face (R) 2-2 swap diagonal:
[B, L', R, L', U, L', D, L, U', L', D', B'2, U, R, D, R', U', R, D', R', B'2, L', R', B', F, R, L, F'2, R, U, R', D, R, U', R', D', F'2, U, L, D, L', U', L, D', L, R', L, F']

Inverted UD and inverted face pair (ONLY) half turn:
[U, F, U', B, U, F', U', B', R'2, F, D, B, D', F', D, B', D', R'2]x3

Inverted UD and inverted face pair (ONLY) quarter turn (U and R with U going CW):
[L, U, L, B, L', B', U', L', B, D, L, D', L', B']x3

(inverted):
[B, L, D, L', D', B', L, U, B, L, B', L', U', L']x3

Face pair and UD (ONLY) half turn (U and R):
[F, D, B, R', U', R', U'2, R, U, R, B', D', F', R'2]x3

Face pair and UD (ONLY) quarter-turn (U and R with U going cw):
[F, D, B, R', U', R', U', R, U, R, B', D', F', R]x3

I spent a huge amount of time worrying about and searching for routine to twist the orientations of the inverted face centers, inverted UD pieces, inverted core, face centers, and UD pieces. I was worried that it would be possible to decouple the inverted face centers from the inverted UD pieces in a way that would make the puzzle unsolvable. I was worried about the same for the UD pieces and face centers.

Because of this, I made sure everything was synced up after the regular 3x3x3 solve and then I only used routines that would preserve this synchronization the whole way through.

I now think all of this concern was a waste of time and that the UD, inverted UD, core, and inverted core are all completely redundant. Was that your experience?

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Dec 03, 2012 12:54 pm

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
bmenrigh wrote:
I now think all of this concern was a waste of time and that the UD, inverted UD, core, and inverted core are all completely redundant. Was that your experience?
Does that mean your statement for the redundancy was born out of experience?
I ask this because you asked for my GAP-Code of the Complex3x3x3 and I posted a link to it.
I considers the core (trivial!) and UD as redundant but argue the other two.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Dec 03, 2012 2:08 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Andreas Nortmann wrote:
bmenrigh wrote:
I now think all of this concern was a waste of time and that the UD, inverted UD, core, and inverted core are all completely redundant. Was that your experience?
Does that mean your statement for the redundancy was born out of experience?
I ask this because you asked for my GAP-Code of the Complex3x3x3 and I posted a link to it.
I considers the core (trivial!) and UD as redundant but argue the other two.

This suspected redundancy is part experience, part thinking about it.

The core is redundant because 1) it never moves and 2) the face centers show the orientation of the core (they're "attached" to it).

The UD pieces are redundant because they only twist when the face centers they're attached twist. If the face center orientations are solved then the UD pieces are solved.

I think the inverted core is redundant because the inverted face centers show its orientation and are basically attached to it. If all of the inverted centers permuted correctly then the inverted core is oriented correctly.

I think the inverted UD pieces are tied to the orientation+permutation of the inverted face centers. If you completely solve the inverted face centers I think the inverted UD pieces must also be solved.

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Dec 03, 2012 9:32 pm

Joined: Sat Nov 17, 2012 12:36 am
Location: Melbourne, Australia
Hi All, I'm new to the forums and thought that I'd jump into this quagmire with some insights from my mathematical background. 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: How to visualize pieces of a complex puzzlePosted: Mon Dec 03, 2012 9:40 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Hi Joe,
Your comments are welcome but most of the theory aspect of this thread has been moved to viewtopic.php?f=1&t=24706 where a more recent discussion on these things is happening.

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Mon Dec 03, 2012 10:44 pm

Joined: Sat Nov 17, 2012 12:36 am
Location: Melbourne, Australia
Thanks, I wasn't sure whether to bump an old thread, I will repost it there.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Dec 04, 2012 12:56 am

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
bmenrigh wrote:
I spent a huge amount of time worrying about and searching for routine to twist the orientations of the inverted face centers, inverted UD pieces, inverted core, face centers, and UD pieces. I was worried that it would be possible to decouple the inverted face centers from the inverted UD pieces in a way that would make the puzzle unsolvable. I was worried about the same for the UD pieces and face centers.

My plan was as follows:
1. Solve super-stickered 3x3x3. [UD] is automatically solved when [U] and [D] are solved.
2. Solve inverted core.
3. Solve the orientation of inverted faces. [UDLR] pieces are automatically solved.
4. Permute [UDF] pieces
5. Flip [UDF] pieces
6. Solve inverted edges
7. Flip inverted edges

My algorithms are borrowed from GB 3.1.31 and .32 and are pretty long, and are originally based on face-center-rotation algorithms of the super 3x3x3. I started from there because those algorithms preserve the common edge and corner pieces. Your algorithms looks mystic for me. How did you come up with these short algorithms? Aren't they commutators?

bmenrigh wrote:
I think the inverted UD pieces are tied to the orientation+permutation of the inverted face centers. If you completely solve the inverted face centers I think the inverted UD pieces must also be solved.

I think of the inverted core, face, and [UD] pieces as follows. Imagine a person stand on the inverted core. In his reference system, the inverted core never rotates by definition, which behaves just like the core for us. So all the results for the common core, face, [UD] are valid. Does this simple argument work?

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Dec 04, 2012 2:01 am

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
I've just solved the 180-degree only complex 3x3x3. It's a relatively easy puzzle to solve. Having said that, there are more "redundant" pieces that I don't understand the reason why they are redundant.

For example, my first step is to solve all the type-[U] faces, which are trivial. When this trivial step is finished, not only [UD] are solved, but the inverted core, [UDLR] pieces are also solved. These types seem to be far apart.

Furthermore, when I solved [UDF], the inverted faces were also automatically solved, always. I wonder if anyone can explain this.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Dec 04, 2012 6:50 am

Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
schuma wrote:
I've just solved the 180-degree only complex 3x3x3. It's a relatively easy puzzle to solve. Having said that, there are more "redundant" pieces that I don't understand the reason why they are redundant.

For example, my first step is to solve all the type-[U] faces, which are trivial. When this trivial step is finished, not only [UD] are solved, but the inverted core, [UDLR] pieces are also solved. These types seem to be far apart.

Furthermore, when I solved [UDF], the inverted faces were also automatically solved, always. I wonder if anyone can explain this.

Ok, I'll take a shot at this. I'll look at the first example first. It seems to be related to commutativity within the half-turn slice group. For example, when performing half turns of slices to make the checkerboard pattern, it doesn't matter what order we do the moves in.

I perfer to consider this within the context of the emulation on the multi-5x5x5 cube as it follows on from this point, and I'll be turning layers 2 and 3.

Each of the [U] pieces are independent and will be solved when the total numberof twists of each face is even. The [UDRLFB] piece is located at the core of the 5x5x5 and behaves like the centres of the 3x3x3 when doing half turns of slices, which commute. Then, when an even number of each twist has been performed, all the moves affecting this piece will cancel each other out. The [UDRL] pieces are the face centres of the 5x5x5 and essentially follow the same argument, but note that turns on the different axes that affect one of these pieces permute it the same but affect its orientation differently, but it is not too hard to see that these commute so again everything cancels and they end up solved along with the [U] pieces.

The second example came as somewhat of a surprise to me, I've not quite figured it out yet but I've ran out of time for now. I'll try and answer it later if someone doesn't work it out in the meantime, and hopefully my answer so far makes sense.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Dec 04, 2012 11:51 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Andreas Nortmann wrote:
bmenrigh wrote:
I now think all of this concern was a waste of time and that the UD, inverted UD, core, and inverted core are all completely redundant. Was that your experience?
Does that mean your statement for the redundancy was born out of experience?
I ask this because you asked for my GAP-Code of the Complex3x3x3 and I posted a link to it.
I considers the core (trivial!) and UD as redundant but argue the other two.
I'm not much of a solver but I would think GAP should certainly be able to answer this question.

Looking at the picture Knaves just painted of the Slice-Turn Only 3x3x3 you can view this puzzle as simply a subset of the Complex 3x3x3 that contains the core, the inverted core, the UD pieces, and the inverted UD pieces. So if by redundant you mean this puzzle should be trivial (i.e. a giant 1x1x1) then I think you are wrong bmenrigh and Andreas is correct. The Slice-Turn Only 3x3x3 I would consider a valid puzzle... its not a very hard one as there is alot of interrelatedness and has rather few states in comparison to a full 3x3x3 but I don't think I'd buy into calling them all redundant.

Not being much of a solver maybe I'm misunderstanding something here.

Carl

P.S. Corrected this post based on the error Andreas pointed out. This topic is comfusing enough without me leaving stupid errors around.

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Last edited by wwwmwww on Wed Dec 05, 2012 11:34 am, edited 1 time in total.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Tue Dec 04, 2012 12:30 pm

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
After bmenrigh wrote more precisely in which way he understands redundancy here I used GAP and found he is right.
Adding UDLR or UDLRFB to the Super 3x3x3 adds something.
Adding UDLR or UDLRFB or to the (Super 3x3x3 + UDLRF) adds nothing.
After solving U, UL, ULF and UDLRF the two piece types bmenrigh mentioned are solved too.

The slices-only-cube is the subset [{}, UD, UDLR, UDLRFB] of the Complex 3x3x3.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Dec 05, 2012 11:29 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Andreas Nortmann wrote:
After bmenrigh wrote more precisely in which way he understands redundancy here I used GAP and found he is right.
Adding UDLR or UDLRFB to the Super 3x3x3 adds something.
Adding UDLR or UDLRFB or to the (Super 3x3x3 + UDLRF) adds nothing.
After solving U, UL, ULF and UDLRF the two piece types bmenrigh mentioned are solved too.
Ok I believe I follow. Thanks. Let's see if I can summarize this. There are 10 piece types in the Complex 3x3x3:

The 10 defining patterns:
( 1) UDLRFB => inverted core; 1 moved in every turn
( 2) UDLRF => inverted faces; 5 moved per turn
( 3) UDLR => inverted version of UD
( 4) UDLF => inverted edges; 8 moved per turn
( 5) ULF => normal corners; 4 moved per turn; impossible to be inverted
( 6) UDL => new symmetry type not found in the 3x3x3; impossible to be inverted
( 7) UD => another new symmetry type not found in the 3x3x3;
( 8) UL => normal edges; 4 moved per turn
( 9) U => normal faces; 1 moved per turn
(10) [] => normal core; 0 moved per turn

So I can drop piece types 1, 3, 7, and 10 and as long as I have super copies of all the others then I have a puzzle with the same number of permutations as the Complex 3x3x3. Correct?

Or are you saying the Super 3x3x3 + UDLRF contains the same number of permutations as the Complex 3x3x3? The UDLF and UDL both add something don't they? Or do you only need one or the other?
Andreas Nortmann wrote:
The slices-only-cube is the subset [{}, UD, UDLR, UDLRFB] of the Complex 3x3x3.
You are absolutly correct. I was being very sloppy above. And this picture is true regardless of rather you pick the central core or the outer 8 corners to act as the holding point.

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Dec 05, 2012 1:27 pm

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
wwwmwww wrote:
So I can drop piece types 1, 3, 7, and 10 and as long as I have super copies of all the others then I have a puzzle with the same number of permutations as the Complex 3x3x3. Correct?
That is correct.
wwwmwww wrote:
Or are you saying the Super 3x3x3 + UDLRF contains the same number of permutations as the Complex 3x3x3?
I didn't want to say something like that. Nonetheless I checked it and saw: Super3x3x3+UDLRF has less permutations than Complex3x3x3.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Dec 05, 2012 1:53 pm

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
wwwmwww wrote:
( 5) ULF => normal corners; 4 moved per turn; impossible to be inverted
( 6) UDL => new symmetry type not found in the 3x3x3; impossible to be inverted

I wouldn't call them "impossible to be inverted". I'd say they are "self-inverted", just like a tetrahedron is self-dual.

wwwmwww wrote:
So I can drop piece types 1, 3, 7, and 10 and as long as I have super copies of all the others then I have a puzzle with the same number of permutations as the Complex 3x3x3. Correct?

Agree.

wwwmwww wrote:
Or are you saying the Super 3x3x3 + UDLRF contains the same number of permutations as the Complex 3x3x3? The UDLF and UDL both add something don't they?

[UDLF] and [UDL] add a LOT to super 3x3x3 + [UDLRF]. Most of the moves in my solution are dealing with these two types. I think of them as the major difficulty of solving the Complex 3x3x3.

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Dec 05, 2012 7:01 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Andreas Nortmann wrote:
wwwmwww wrote:
So I can drop piece types 1, 3, 7, and 10 and as long as I have super copies of all the others then I have a puzzle with the same number of permutations as the Complex 3x3x3. Correct?
That is correct.
NICE!!!! This has some VERY interesting ramifications. Most notability this gives the Complex 3x3x3 a property that I didn't know it had. Previously I believed the Complex 3x3x3 needed a very complex mechanism to be built on top of a Multi-5x5x5. I've worked out the details of a mechansim for a Multi-5x5x5 which I called the REAL 5x5x5 and I mostly have the details worked out how to bandaged this into the Complex 3x3x3 in my head. Its high on my Solidworks to do list. However this development greatly simplifies things. Since the inner 3x3x3 contains these pieces which we've just called reduntant I don't actually need them anymore to build a physical Complex 3x3x3 that has all the same states. So I now don't need to compine two very complex mechanisms in one puzzle. I just need a 5x5x5 which uses a Slice-Turn Only 3x3x3 as a core (core used in the sense that a Geared Mixup is the core of a Clockwork 4x4x4 and not core the piece type). And I believe this will be MUCH easier to make then if I really needed to make the full Multi-5x5x5. This really needs to be my next Solidworks project. And yes I do still plan to port the Real 5x5x5 model over to Solidworks as well and this will make that easier as well as I now don't have to worry about how to pull all the Complex 3x3x3 ideas I have into that design.
schuma wrote:
I wouldn't call them "impossible to be inverted". I'd say they are "self-inverted", just like a tetrahedron is self-dual.
Agreed... that is a much nicer way to look at it. Thanks for this insight.

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Dec 05, 2012 7:08 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
For the UD and UD^-1 to be redundant the face centers and inverted face centers must show orientation (have super stickers). I'm not sure if that'll mean you need super stickers on the 5x5x5-based realization of the puzzle.

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

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Wed Dec 05, 2012 7:25 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
For the UD and UD^-1 to be redundant the face centers and inverted face centers must show orientation (have super stickers). I'm not sure if that'll mean you need super stickers on the 5x5x5-based realization of the puzzle.
Yes... that is EXACTLY what that means. But trust me making a set of super stickers is FAR easiler then actually working all those pieces into the design. I still believe the full design is possible but it could easily require a cube with about a 100mm edge length and even then with the complexity of the design and the number of pieces needed (FAR FAR more then the 64 you'd really want) I suspect that it'd turn like crap. So any way to simplify it will make designs more afordable to test and should result in something that is more likely to be a playable puzzle.

Carl

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Sat Dec 08, 2012 4:19 am

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
Hi guys,

After discussing about the redundant pieces and reading this post:

viewtopic.php?f=1&t=18470

I find the number of permutations very useful. So I decide to go ahead and compute this number for some other complex puzzles. Here's what I've got:

https://github.com/nanma80/complex

I use Ruby + GAP to compute it. Please find out what it is and how to use in the readme section of the page. Also, some computation results in notes.txt:

Complex edge-turning tetrahedron:
Number of permutations of pieces (ignoring orientation):
42467328
Number of permutations of stickers (considering orientation):
2717908992

Complex vertex-turning cube:

Number of permutations of pieces (ignoring orientation):
256173997827463780521415556667434027160193098673921602898863225475501536067567951337866994\
5809209917440000000000000000000000000000000000

Number of permutations of stickers (considering orientation):
776864058028093956806901137727727231682263264853083948548575098290738974207281162073295584\
401118072441719234341086020239360000000000000000000000000000000000

For larger puzzles like edge-turning cube and face-turning dodecahedron, GAP cannot quickly do it (I gave up on edge-turning cube after waiting for an hour). Does anyone know of some access of powerful machine on which I may have permission to install GAP?

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 Post subject: Re: How to visualize pieces of a complex puzzlePosted: Sat Dec 08, 2012 9:53 pm

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
I used my program to analyzed the vertex turning cube (or face turning octahedron). The only redundant piece types are the counterpart of those on the complex 3x3x3:

- core
- four pieces moved by antipodal corners, e.g., [ UFR, DBL ]
- inverted antipodal pieces
- inverted core.

After removing these pieces, removing any other piece type will cause a reduction of the number of permutations, assuming all pieces are orientation sensitive.

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