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 Post subject: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Tue Jul 27, 2010 9:07 pm 
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This topic got started here:

http://twistypuzzles.com/forum/viewtopic.php?f=15&t=18117

But I didn't want to get too side-tracked in that thread so I'm starting this one.

I'm trying to figure out what THE Order=3 Corner-Turn MultiCube should look like. It will have all the pieces of the Compy Skewb on its surface plus 4 new pieces found on the Elite Skewb. I'll try to update this thread with an animation or two and some better drawings in the future but for now I took a quick attempt at making what I think a face of this puzzle MIGHT look like.

Attachment:
O3CTMC.png
O3CTMC.png [ 26.42 KiB | Viewed 5304 times ]


The blue pieces are the Skewb. The red pieces are the two order=3 pieces of the Tetrahedral Twins. The green pieces are from the Dino Skewb. And the purple pieces are an odd bunch. All 4 pieces of the elite skewb are there and numbered. The corner becomes 3 edges pieces that are always together. The face center is now 4 pieces that are always together. And the other two are pairs of pieces that are always together. Now its getting late and my brain is fried... but I seem to be stuck with 3 other pieces. The yellow pair is stuck together. The gray pair is stuck together. And there is that brown piece. The top pic is intented to show 2 of the 4 turnable layers of the puzzle.

So what the heck are these other 3 pieces? Andreas... anyone? I'm tempted to think they may be virtual pieces...

http://twistypuzzles.com/forum/viewtopic.php?f=1&t=15667

but in the current state of my mind I could have easily made some stupid mistakes in the above picture. Can a physical copy of this puzzle be made? I have no idea but I'll certainly try to make a POV-Ray pic of what it could look like.

Carl

P.S. I meant for this to be in general puzzle topics. Can someone move it for me? Thanks...

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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Wed Jul 28, 2010 12:12 am 
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I have a hard time just visualizing how the master skewb pieces move and interact. For reference so we can see how you're counting, can you tell/show us what you're counting as the order-1 and order-2 Corner-Turn MultiCube?

Would it help to identify the pieces on a higher order Face-turning Octahedron first? For some reason, I want to call face turning puzzles the "dominate" puzzle and all corner/edge turning puzzles shape mods.

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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Wed Jul 28, 2010 8:27 am 
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GuiltyBystander wrote:
I have a hard time just visualizing how the master skewb pieces move and interact. For reference so we can see how you're counting, can you tell/show us what you're counting as the order-1 and order-2 Corner-Turn MultiCube?

The Order=1 Corner-Turn MultiCube is your basic Skewb. It doesn't have any pieces on the inside.

The Order=2 Corner-Turn MultiCube has the 6 types of pieces seen in this animation (Counted as 0 through 5).

Image

Though I guess I did sort of jump the gun as the Order=2 Corner-Turn MultiCube with all its pieces viewable on the surface hasn't been worked out yet. I'll try to get to that one too.
GuiltyBystander wrote:
Would it help to identify the pieces on a higher order Face-turning Octahedron first?

I'll try to find the time to make an animation showing the pieces of an Order=3 Corner-Turn MultiCube (Face-turning Octahedron). They are actually the same puzzle. You could make it in the shape of a sphere if you wanted to and the cuts will all be the same. We are solving the volume of the puzzle so the shape of the surface doesn't come into play. The puzzle is simply defined by the location of the cut planes.
GuiltyBystander wrote:
For some reason, I want to call face turning puzzles the "dominate" puzzle and all corner/edge turning puzzles shape mods.

To me ALL puzzle are shape mods of their spherical form. Well almost... a spherical helicopter cube is actually a different puzzle then a cubic helicopter cube due to external bandaging.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Thu Jul 29, 2010 11:34 am 
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I don't think that this can be done for the simple reason the layers can't be interconnected like that.

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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Thu Jul 29, 2010 12:56 pm 
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Urgh! Carl is back again!

Right now I can't identify the pieces "left over" but maybe you can identify them for yourself.
I don't think that you have to think about "virtual" pieces because:
1. You use equidistant planar cuts where virtual pieces have never appeared so far.
2. The CornerTurning MultiCube with 3 cuts per axis has 10 kinds of "real" pieces which is your number of pieces.

The pieces are:
F03, C04, C13, X39, T66, W66, X93a, X93b, F30, C40
where the first digit is the number of pieces of this kind which are twisted while twisting an outer layer. The second digit refers to the number of moved pieces within a "slice turn".
I could immediately identify two pieces:
C13 is identical to the "subcorners" of the tetrahedral twins
X39 is identical to the "edges" of the tetrahedral twins
Sadly there are two kinds of pieces with identical siganture.

For those who are interested:
I came up with the list above by analyzing Carl's animation which resulted in this list of pieces: O, C1, E3, X9, F3, C4
Then I splitted up all these pieces by adding the central third cutting plane.
I haven't done this right now. I performed this analysis months ago and had the results rotting on my HDD.

To confuse everybody I haven't lost so far:
The class of cornerturning hexahedrons with 3 cuts per axis has 9 kinds of virtual pieces but I would be surprised if they were needed here.


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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Thu Jul 29, 2010 3:40 pm 
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Andreas Nortmann wrote:
I don't think that you have to think about "virtual" pieces because:
1. You use equidistant planar cuts where virtual pieces have never appeared so far.
2. The CornerTurning MultiCube with 3 cuts per axis has 10 kinds of "real" pieces which is your number of pieces.


Not sure point 1 is true as I have layers that are disconnected... so the cuts are more complicated that single planes. One cut is actually several planes.

And point 2 is off... True I found the 10 "real" piece types but I also found two other piece types. The gray and yellow sets are two examples of one type and the brown piece appears to be another type. So we have 10 real types but I think I've found 12 types. I should be able to figure out what these two other pieces are but I don't have the time to dig into it now.

Thanks,
Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Sun Aug 01, 2010 7:05 am 
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Here we go with the 2nd analysis:
Attachment:
O3CTMC3.png
O3CTMC3.png [ 112.32 KiB | Viewed 4878 times ]

These are my results:

1 is equivalent to C04
3 is equivalent to C40 (three pieces act connected)
4 is equivalent to C13 (three pieces act connected)

12 is equivalent to F03
11 is equivalent to F30 (four pieces act connected)

9 is equivalent to T66
7 is equivalent to X39
5 is equivalent to X93b (two pieces act connected)
8 is equivalent to X93a (two pieces act connected)
I haven't found a good method to distinguish between this pieces yet.

2 and 6 (both!) are equivalent to W66
One piece of 2 and one of 6 act connected to each other. See the image above.

10 is equivalent to G and is indeed a virtual piece. (two pieces act connected)
These twisting possibilities are blocked by fixing this piece in space:
SideDRB, SideULB, SliceURB, SliceDLB


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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Sun Aug 01, 2010 3:48 pm 
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Ok... I've been away from this long enough to get rusty. As I recall here is the basis of the letters.

http://www.speedsolving.com/wiki/index.php/Center
http://www.speedsolving.com/wiki/index.php/Edge

0 - core ()
C - corner
E - edge (precisely: midge)
F - face (precisely: central center)
W - Wings
T - T-center
X - X-center
L - Obliques

Here are the 10 real pieces...
Andreas Nortmann wrote:
F03, C04, C13, X39, T66, W66, X93a, X93b, F30, C40
where the first digit is the number of pieces of this kind which are twisted while twisting an outer layer. The second digit refers to the number of moved pieces within a "slice turn".

And in the image above we have the piece types numbered from 1 to 12. And to define which is the outer layer and which is the slice turn... remember the inner skewb of The Order=3 Corner-Turn MultiCube is only changed with slice turns. So we have...

Piece 1 is equivalent to C04, aka the Skewb's corners.
Piece 12 is equivalent to F03, aka the Skewb's face centers.

Continuing to move from the inside out we move on to the Tetrahedral Twin's pieces.

Piece 4 is equivalent to C13 (three pieces act connected) is the nontrivial Tetrahedral Twin's corner.
Piece 7 is equivalent to X39 is the Tetrahedral Twin's edge (X-center when in the form of a cube).

Continuing outward the next puzzle is the Dino Skewb.

Piece 6 is equivalent to W66 and this is the Dino Skewb Edge or Wing, as its not a middle edge piece.
Piece 9 is equivalent to T66 and this is the Dino Skewn Face or T-Center piece.

If I see what you are saying about Piece 2 above its not really another piece at all but part of the piece labeled Piece 6 in the picture. They are connected... correct? There are only a total of 24 of these pieces.... not 48.
Andreas Nortmann wrote:
One piece of 2 and one of 6 act connected to each other. See the image above.

Yes that appears to be exactly what you are saying.

Continuing outward the next puzzle is the Elite Skewb.

Piece 3 is equivalent to C40 (three pieces act connected). This is the Elite Skewb Corner.
Piece 5 is equivalent to X93b (two pieces act connected). This is the outer Elite Skewb X-Center.
Piece 8 is equivalent to X93a (two pieces act connected). This is the inner Elite Skewb X-Center.
Piece 11 is equivalent to F30 (four pieces act connected). This is the Elite Skewb Face Center.

The other pieces seen on the surface of the Elite Skewb are the Dino Skewb Pieces.

So this just leaves one unexplained piece... (on the surface)
Andreas Nortmann wrote:
10 is equivalent to G and is indeed a virtual piece. (two pieces act connected)
These twisting possibilities are blocked by fixing this piece in space:
SideDRB, SideULB, SliceURB, SliceDLB

What is "G" again? Looks like there are 12 pieces of this type, two on each face.

And you also say...
Andreas Nortmann wrote:
The class of cornerturning hexahedrons with 3 cuts per axis has 9 kinds of virtual pieces...

I wonder if any of the other 8 virtual pieces have any physical volume to them inside this puzzle.

What we've done here really makes me think of the idea of connecting isolated layers of a 5x5x5 MultiCube to make the Complex 3x3x3 talked about here:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=15667
Where by we gave the imaginary pieces real physical volume.

Could even more apparent layers be added to this... (There must only be 4 turnable layers but each could consist any any number of connected or isolated apparent layers)... to bring more of the virtual pieces to the surface? Can they all even be given a physical volume inside this puzzle with this technique? And dare I even mention the imaginary pieces? Has there number even been calculated for the Order=3 Corner-Turn MultiCube?

And I guess I should rename this thread. The puzzle its about is actually MORE then The Order=3 Corner-Turn MultiCube. Its the 10 pieces of The Order=3 Corner-Turn MultiCube PLUS one virtual piece.

Cool stuff,
Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Sun Aug 01, 2010 6:09 pm 
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Andreas Nortmann wrote:
10 is equivalent to G and is indeed a virtual piece. (two pieces act connected)
These twisting possibilities are blocked by fixing this piece in space:
SideDRB, SideULB, SliceURB, SliceDLB


Attachment:
O3CTMC4.png
O3CTMC4.png [ 95.59 KiB | Viewed 4803 times ]


Just to be sure... I've labeled the sides correctly to match your blocked twists notation... haven't I? Still wondering about "G".

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Sun Aug 01, 2010 9:25 pm 
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Carl, I'm alittle confused by what you're asking here.... I'm sure I could help if I just understood your question.

First off, remembering that I count orders differently than most people, can you make sure this is correct: You are looking for a puzzle with 3 parallel cutting planes: one that is exactly through the center (deepcut), and then one offset on each side. So that's what has been called an elite skewb correct?

Second, you are looking for the multicube. My understanding of your definition of multicube is as follows: using dodecahedral cored puzzle as an example, as you make the cuts deeper and deeper through the puzzle (which is equivalent to keeping the cut planes stationary and "filling in" the space farther and farther away from the center, making the puzzle larger and larger) you run into different types of pieces that behave differently. This can be seen going from megaminx to pyraminx crystal to starminx to "master pentutlimate" as Scott I believe called it (a name I actually don't like for 2 reasons. Oh well...) Now then as per your discussion here a multicube includes all of these pieces simultaneously. IMO the most direct way to make this is with a virtual program that allows an exploded view of the puzzle, including all internal pieces. I think this definition also implies all orientations are visible right?

Now then, it sounds like you want a puzzle that is equivalent to a (3 parallel cuts per axis)-order vertex turning cube (which will forever and always be equivalent to a (3 parallel cuts per axis)-order face turning octahedron assuming all orientations are visible on both). However you want a puzzle that has all necessary information visible from the surface only... Correct? And in an attempt to pursue this, certain layers on this puzzle are forced to spin together? This is where I get confused. If I am understanding this right so far, can you better indicate which layers spin together and which are separate? It sounds to me like you've started with a complicated idea and are now trying to shove it into basic terms you can understand when you should be doing it the other way around :wink:

As soon as I can figure out what you're getting at here I'll be happy to help :)

Peace,
Matt Galla


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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Mon Aug 02, 2010 10:38 am 
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Carl. The image you posted is correct, although you should have written F instead of T. :wink:
wwwmwww wrote:
What is "G" again? Looks like there are 12 pieces of this type, two on each face.
Stupid me. "G" is not another classification symbol, like E,F,T etc. It is just the internal name it got in my humongous Excel-table. You know which is meant.
wwwmwww wrote:
I wonder if any of the other 8 virtual pieces have any physical volume to them inside this puzzle.
I don't dare to answer this without a exploded view. You have to help me (and yourself) here.
wwwmwww wrote:
Could even more apparent layers be added to this... (There must only be 4 turnable layers but each could consist any any number of connected or isolated apparent layers)... to bring more of the virtual pieces to the surface?
I would guess: YES.
If you want to bring ALL virtual pieces to the surface you should start with the most complicated ones, two types of virtual SkewbCores:
In the first the slices URF, DLF, DRB, ULB stay fixed.
In the second the sides URF, DLF, DRB, ULB stay fixed.
wwwmwww wrote:
And dare I even mention the imaginary pieces? Has there number even been calculated for the Order=3 Corner-Turn MultiCube?
Matt Galla: Can you explain how to generalize your concept for puzzles with more than two cuts per axis?

Allagem wrote:
First off, remembering that I count orders differently than most people, can you make sure this is correct: You are looking for a puzzle with 3 parallel cutting planes: one that is exactly through the center (deepcut), and then one offset on each side. So that's what has been called an elite skewb correct?
One EliteSkewb is shown here:
http://www.twistypuzzles.com/forum/view ... 15&t=17292
Although the puzzle Carl hinted to would have more than three cuts it has 4 independently movable entities per axis like the Elite Skewb. That also means my twistability analysis can treat it as if it had just three cuts per axis.
In the picture Carl reposted the 4 twistable entities are these:
Red: Side ULB
Blue: Slice ULB
Green: Slice DRF
White: Side DRF


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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Mon Aug 02, 2010 12:09 pm 
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I'm kinda' confused now. Isn't a skewb a order 2 puzzle?? :?

A 3x3x3 is a order 3 puzzle, huh?
then a 2x2x2 is a order 2, that means a order 2 puzzle is deepcut, and the skewb is. then a dino cube, or master skewb is a order 3 puzzle, because it got a middle slice, and one slice on both sides of it.

or am I wrong???

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Mon Aug 02, 2010 7:03 pm 
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blueShinyApple wrote:
I'm kinda' confused now. Isn't a skewb a order 2 puzzle?? :?

All depends on how you define order. I'm aware of at least 3 definitions that have been used on this site. The one I use is the number of cuts per axis of rotation (or the number of rotatable layers minus one). With that definition the skewb and the 2x2x2 are order 1. The 3x3x3 is then order 2.

Some say order = the number of rotatable layers. This makes the 3x3x3 order 3 which some are happy with and then yes the skewb and 2x2x2 are order 2.

Still others say order = the number of cuts per rotatable face/corner/edge/etc. This makes the 2x2x2 AND the 3x3x3 both order 1 puzzles. The 4x4x4 and 5x5x5 are then order 2.

So to use that term around here one really does need to define it first... my bad.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube
PostPosted: Mon Aug 02, 2010 7:23 pm 
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Andreas Nortmann wrote:
Carl. The image you posted is correct, although you should have written F instead of T. :wink:

My bad. I keep forgetting if its Top/Bottom or Front/Back.
Andreas Nortmann wrote:
I don't dare to answer this without a exploded view. You have to help me (and yourself) here.

Will do. It will have to wait till I get some free time so it may be a week or two but I'll get there.
Andreas Nortmann wrote:
I would guess: YES.

Me too but I want to do more then guess. I suspect there are some more of the virtual piece that can be made real this way but I'm really not sure they all can.
Andreas Nortmann wrote:
If you want to bring ALL virtual pieces to the surface you should start with the most complicated ones, two types of virtual SkewbCores:
In the first the slices URF, DLF, DRB, ULB stay fixed.
In the second the sides URF, DLF, DRB, ULB stay fixed.

Which reminds me that I started making an animation while I was unemployed to show a way to make those two pieces visable and still haven't finiahed it. I need to do that. It might help here.
Andreas Nortmann wrote:
Matt Galla: Can you explain how to generalize your concept for puzzles with more than two cuts per axis?

Even just looking at order=2 puzzles I still have a question. Are the virtual pieces a subset of the imaginary ones? Or are there pieces which are virtual but NOT imaginary. I know the 3x3x3 doesn't have any virtual pieces and we called the 3x3x3 which included all the real and imaginary piece the Complex 3x3x3. I'm wonder what I should call a puzzle which includes all the real piece and just the virtual pieces. The 3x3x3 with just the real pieces is the Multi 3x3x3. The reason I ask is the Skewb animation I made is just the 2 real pieces and the 2 virtual pieces. As I recall it has many more imaginary pieces. Correct me if I'm wrong.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Mon Aug 02, 2010 8:40 pm 
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Allagem wrote:
Carl, I'm alittle confused by what you're asking here.... I'm sure I could help if I just understood your question.

Andreas has already answered the first question I asked. That being what the two "new" pieces I found were. One turned out to be part of the Dino Skewb Edge and the other WAS a virtual piece given a real physical volume.
Allagem wrote:
First off, remembering that I count orders differently than most people, can you make sure this is correct: You are looking for a puzzle with 3 parallel cutting planes: one that is exactly through the center (deepcut), and then one offset on each side. So that's what has been called an elite skewb correct?

Close... The Order=3 Corner-Turn MultiCube would look exactly like the elite skewb on the outside BUT those cuts ALSO define volumes INSIDE the puzzle which don't have to be solved on an elite skewb. On The Order=3 Corner-Turn MultiCube those pieces ARE solvable. I can and need to make an animation much like the order=2 one above to show everyone all these pieces. I can see them in my head and Andreas has already named them all but I can do more to help show them to everyone. I just need some free time, somthing I don't have much of these days.
Allagem wrote:
Second, you are looking for the multicube. My understanding of your definition of multicube is as follows: using dodecahedral cored puzzle as an example, as you make the cuts deeper and deeper through the puzzle (which is equivalent to keeping the cut planes stationary and "filling in" the space farther and farther away from the center, making the puzzle larger and larger) you run into different types of pieces that behave differently. This can be seen going from megaminx to pyraminx crystal to starminx to "master pentutlimate" as Scott I believe called it (a name I actually don't like for 2 reasons. Oh well...) Now then as per your discussion here a multicube includes all of these pieces simultaneously.

YES!!! As Johny Carson would say "You are correct Sir!"
Allagem wrote:
IMO the most direct way to make this is with a virtual program that allows an exploded view of the puzzle, including all internal pieces.

Its certainly A way... its NOT the only way. Say I want to play with a Multi-4x4x4. Am I confined to playing with a virtual copy of it via a program? No... we have the Crazy-4x4x4's to help us there. And in principle circle cubes in general can give us all the higher order Multi NxNxN puzzles. See here:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=14868
Again we run into issues rather fast as we may not be able to easily make these any time soon but Gelatinbrain COULD (and I hope they do) make these into playable programs soon. It would be easier then making Gelatinbrain capable of showing a 2x2x2 sitting beside a 4x4x4 for example. Even if we end up limited to playing with virtual copies I believe there is merit in making the interior pieces visible on the surface to simplify the appearance of the puzzle and it also clearly shows the link between the pieces. Say you simulated a Multi-4x4x4 but showing a 2x2x2 next to a 4x4x4. Its NOT as clear how the two interact. One wants to solve one and then the other. The Crazy-4x4x4 clearly shows how its all just one puzzle.
Allagem wrote:
I think this definition also implies all orientations are visible right?

No. That word is "Super". Here this is cut from an email I wrote to TomZ on July 26.

Quote:
Basically I view "Multi" as similiar to "Super" in the sense of Super Solving a 4x4x4 means each pieces has only one solve position and orientation. However you are only looking at the surface pieces. If you have an means of making the interior 2x2x2 visible so you can solve it too it becomes a Multi-4x4x4, i.e the crazy 4x4x4. And if you then give each piece only one solved position and orientaion then it becomes a Super-Multi-4x4x4. Three of the 4 versions of the Crazy 4x4x4 are Super-Multi-4x4x4s. The one with the smallest circle is just a Multi-4x4x4.

Allagem wrote:
Now then, it sounds like you want a puzzle that is equivalent to a (3 parallel cuts per axis)-order vertex turning cube (which will forever and always be equivalent to a (3 parallel cuts per axis)-order face turning octahedron assuming all orientations are visible on both). However you want a puzzle that has all necessary information visible from the surface only... Correct?

Yes, that is what I was after. Looks like I got that PLUS one virtual piece too.
Allagem wrote:
And in an attempt to pursue this, certain layers on this puzzle are forced to spin together?

Yes... just like the inner circles of a higher order circle cube spin with the layers below them.
Allagem wrote:
This is where I get confused. If I am understanding this right so far, can you better indicate which layers spin together and which are separate? It sounds to me like you've started with a complicated idea and are now trying to shove it into basic terms you can understand when you should be doing it the other way around :wink:

Its Andreas's post I had to translate just so I could understand it. And yes its a complicated topic and as stated I owe everyone better pictures and some animations but from the pics here I think you can figure it out. See Andreas post above.
Allagem wrote:
As soon as I can figure out what you're getting at here I'll be happy to help :)

Peace,
Matt Galla

Thanks... it will be nice to have you join the mix.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Tue Aug 03, 2010 10:14 am 
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Quote:
Even just looking at order=2 puzzles I still have a question. Are the virtual pieces a subset of the imaginary ones? Or are there pieces which are virtual but NOT imaginary.
No. The "definition by twistability" for every piece can be transformed unambigously into Matt's system. The set of imaginary pieces is the superset of virtual pieces. Thats true at least for puzzle with 2 cuts per axis.
Quote:
I'm wonder what I should call a puzzle which includes all the real piece and just the virtual pieces.
Keep it self-expressing and call it "all-virtual-pieces XXn" where XXn could be CH3 or HV3 or whatever.
Quote:
The 3x3x3 with just the real pieces is the Multi 3x3x3. The reason I ask is the Skewb animation I made is just the 2 real pieces and the 2 virtual pieces. As I recall it has many more imaginary pieces.
I have no idea how to apply Matts system to a deepcut which means I have to look at CH2 instead.
01[] real - O
02[ULF] real - C1
03[ULF ULB] real - E3
04[ULF URB] virtual
05[ULF DRB] imaginary
06[ULF ULB URF] real - X9
07[URF ULB DLF] virtual
08[ULF URB DLF] imaginary
09[ULF ULB URF URB] real - F3
10[ULF ULB URF DLF] real - C4
11[ULF URB DLB DRF] virtual
12[ULF ULB URF DLB] imaginary
13[ULF ULB URF DRB] imaginary
14[ULF ULB DRF DRB] imaginary
15[ULB URF DLB DRF DRB] imaginary - inversion of piece 08
16[ULF URB DLB DRF DRB] imaginary - inversion of piece 07
17[URB DLF DLB DRF DRB] imaginary - inversion of piece 06
18[ULB URF URB DLF DLB DRF] imaginary - inversion of piece 05
19[ULB URF DLF DLB DRF DRB] imaginary - inversion of piece 04
20[URF URB DLF DLB DRF DRB] imaginary - inversion of piece 03
21[ULB URF URB DLF DLB DRF DRB] imaginary - inversion of piece 02
22[ULF ULB URF URB DLF DLB DRF DRB] imaginary - inversion of piece 01

Edit: Corrected the indices. Thank you Carl.
Edit2: Some more mistakes corrected.


Last edited by Andreas Nortmann on Wed Oct 06, 2010 12:03 pm, edited 2 times in total.

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Wed Aug 04, 2010 10:20 am 
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Andreas Nortmann wrote:
Keep it self-expressing and call it "all-virtual-pieces XXn" where XXn could be CH3 or HV3 or whatever.

That's what it IS. But its more like a definition then a name.

Just like Real+Imaginary=Complex (which also happens to equal Real+Imaginary+Virtual) we should have a name.

How about Real+Virtual=Augmented (see here http://en.wikipedia.org/wiki/Augmented_reality). It seems to fit.

So the puzzle this thread is about could be considered partially augmented as it has one virtual piece.

And we really should try to get the Complex 3x3x3 up on Gelatinbrain.
Andreas Nortmann wrote:
22[ULF ULB URF URB DLF DLB DRF DRB] imaginary - inversion of piece 00

I think your count is off by 1 as your list started with piece 01.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Thu Aug 05, 2010 2:12 am 
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Sorry guys, I keep meaning to add to this thread but ive been so busy lately. I'm studying for a p.d.e. final right now, but i wanted to let you guys know i am still here :lol:

-sorry no pretty pictures this time...- :(

It sounds like the puzzle wwwmwww came up with has been pretty well analyzed at this point. Andreas, you asked how to extend my system to find a complete list of complex pieces (both imaginary and real) [btw you guys are using 3 piece classifications... real, virtual, and imaginary... which ones are you calling virtual? the internal ones?]

This is simple in theory. First i need to state my ordering system: i start from the center of the puzzle and move out and count the number of layers to the outside of the puzzle. Thus 3x3x3 is order=1 in my book and 5x5x5 is order = 2. For deepcut puzzles this is clearly a problem, however i have learned that a deepcut puzzle has so much in common with the related, slightly less deepcut puzzle that i just group them together. thus a 2x2x2 is a deepcut order=1. And a 4x4x4 is a deepcut order=2 puzzle (or at least the inner layer has a deepcut plane defining it). Commonalities between a non deep cut puzzle and the related deepcut puzzle are everywhere- structure, potential imaginary pieces, algorithm-stuff, parity issues for a deepcut puzzle nearly always arise from the less deepcut puzzle within. Plus when using a core that does not have opposite parallel sides (excellent recent example: meteor madness) the deepcut ambiguity (i say there is no such thing as a deepcut meteor madness, at least not with the same consequences as usual deepcut puzzles, I know some won't agree with me there :wink: ) does not add hiccups in the pattern going from order 1 to order 2 to order 3, etc.

Anyways, because of this, if you were to ask me to determine all of the potential pieces for the puzzle shown here i would look at the order=2 (by my definition) octahedron-cored family. Now in theory this is easy. Imagine you have an octahedron with paper sides and you have TWO markers this time, one red, one blue. Here are the rules: you have to color an entire face. a single face may have both colors. not coloring any faces or coloring every face is perfectly acceptable. The total number of potential complex pieces is exactly the number of unique colorings of this octahedron you can get. In fact each of the different colorings specifically correlates to one of the pieces. This is analagous to Andreas's use of UFL, URD, etc to determine all the pieces.

I started to draw these all out but was shocked to see just how many there was. There's at least 100! probably even near 200. I realized the pattern I was following actualled missed a few combinations and hit some others twice, so i scrapped my first attempt and have yet to take another. soon as i get an hour or so free i will 8-)

Now i havent had time to look into this but i suspect theres a problem here with determining exactly which of these pieces are imaginary and which are real. My intuition tells me that you can vary the "depth" of the two layers of cuts relative to eachother to push some real pieces out of existence and maybe even bring some virtual ones into existence much like my example of imaginary pieces with the lopsided octahedron i showed in the long post of the imaginary pieces thread. i need to look into this more... again when i have time.

oh and before i forget, i think the easiest way to interpret these multi-colors is as follows:
let red represent a turn of all pieces beyond the first cutting plane counting from the core and let blue represent a turn of all pieces beyond the second cutting plane counting from the core. Then you'll notice that on a 5x5x5, the internal "mathematical" pieces that exist only in the inner layers will have only red defining patterns. An extreme corner will have only purple (red+blue :) ) defining patterns. Some of the other external pieces on a 5x5x5 will have defining patterns of both red and purple. So it would appear that blue defining patterns don't exist naturally. But I certainly think we can consider them :wink: they would move when you spin the outer layer, but not when you spin two layers at a time... which is kind of bizzare actually :D

What do you guys think of this? Again this method gives well over 100 potential complex pieces.... :roll: but i think thats correct

Peace,
Matt Galla

PS: definitely agree with getting the complex cube on gelatinbrain. I'm finishing up a twisty puzzle simulator (currently does support shape-shifting but no jumbling, bandaging, offset cores, rotatable "rotations" [required to simulate Oskar's Mixup Cube] or partial rotations [Golden Cube, Ghost Cube]) I plan to include a special complex cube, as well as complex octahedron, complex dodecahedron, etc if the pattern holds true (haven't checked yet) so if gelatinbrain doesn't catch on to this, you'll still get it eventually 8-) though probably not for awhile at the rate im going at :roll:

PPS you can blame my lack of proofreading this for any grammar errors or typos. You can blame my lack of proofreading on the fact that I have a p.d.e. final in the morning and have to study! :lol: did i just use the word blame to get similar statements in opposite directions...? I need sleep...


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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Thu Aug 05, 2010 7:49 am 
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Allagem wrote:
It sounds like the puzzle wwwmwww came up with has been pretty well analyzed at this point. Andreas, you asked how to extend my system to find a complete list of complex pieces (both imaginary and real) [btw you guys are using 3 piece classifications... real, virtual, and imaginary... which ones are you calling virtual? the internal ones.


We need some definitions here... Lets start with CH2. To me CH2 is defined by 4 pairs of equidistant planar cuts centered on the origin of 3-space. Each pair allows rotation about one of these 4 axes: <1,1,1>, <1,1,-1>, <1,-1,1>, <1,-1,-1>.

Any piece that has real volume in this 3-space after it is cut up by these cut planes is REAL. There are 6 in CH2 and you can see them in the animation above and in Andreas's list.

All the other pieces are imaginary. The virtual pieces are a subset of the imaginary pieces that share a propery with the real pieces that the other imaginary pieces don't have. Let's look at the 3x3x3 for example. Just like CH2 each axis has 3 layers that can turn. However only 2 can be thought of as independant. A slice turn is equal to two face turns, or a face turn is equal to the opposite face turn and a slice turn, etc. As such we don't limit ourselves by just sticking to two layers per axis of rotation that we will turn...

On a 3x3x3 if we pick...

(1) Just the 6 face layers then the real core is always stationary. It doesn't move or rotate.
(2) If we pick 3 face layers and 3 slice layers there is a corner that is stationary.
(3) If we pick 4 face layers and 2 slice layers there is an edge that is stationary.
(4) If we pick 5 face layers and 1 slice layer there is a face center that is stationary.

These stationary pieces can be thought of as holding points. You can hold the puzzle by that piece and it never has to move and you still aren't bandaging the puzzle in any way. All real pieces have this property which makes sense as they have real volume and you can hold on to them. The 3x3x3 doesn't have any virtual pieces but the simpliest example that does is the Skewb. Its defined by just 3 cut planes that all intersect at the origin. They cut 3-space into 6 face pieces and 8 corners. So you just have those 2 real pieces. However if you take an off the shelf Skewb apart you find a core inside. This is one of the two virtual pieces that exist for the Skewb. Note if you scrable a skewb such that this piece stays stationary then none of the corners or faces are stationary. All virtual pieces can be thought of as holding points that never move under the proper choice of independant layers. In that sense they are more real then the other imaginary pieces.

The other imaginary pieces do NOT have this property. Regardless of your choice of independant layers these pieces will always be in atleast one of them so they will move. They can't serve as holding points... if you stick with one specific choice of independantly turnable layers.

If I recall correctly Andreas found other property that he could use to further subdivide the virtual pieces but I don't recall the details or even if I ever fully understood it at the time. I need to go back and re-read some of the older threads.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Thu Aug 05, 2010 8:17 am 
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Allagem wrote:
Now i havent had time to look into this but i suspect theres a problem here with determining exactly which of these pieces are imaginary and which are real. My intuition tells me that you can vary the "depth" of the two layers of cuts relative to eachother to push some real pieces out of existence and maybe even bring some virtual ones into existence much like my example of imaginary pieces with the lopsided octahedron i showed in the long post of the imaginary pieces thread. i need to look into this more... again when i have time.

You are looking at what Andreas would call CH4. I'd define this puzzle as:

To me CH4 is defined by 8 pairs of equidistant planar cuts centered on the origin of 3-space. 4 of the pairs are distance A apart and the other 4 are distance B apart. Each pair allows rotation about one of these 4 axes: <1,1,1>, <1,1,-1>, <1,-1,1>, <1,-1,-1>. Such that you have one A-pair and one B-pair per axis.

Any piece that CAN have real volume in this 3-space regardless of the choice of A and B IS real. I'm sure not all of these pieces can exist at the same time with any one specific choice for A and B but I still condider them real pieces. Just because you push a real piece out of existence I don't consider it virtual. It takes a little more work then that to bring a virtual piece into existence.

As seen above it can be done with disjoint layers being considered one layer. To bring imaginary pieces into existance I believe it requires considering a layer as part of two seperate independant layers as we did on the complex 3x3x3.

Outside of the 3x3x3 I'm not even sure its possible to bring ALL the imaginary pieces into existance in one puzzle at the same time. I'll try to think about this a bit more.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Thu Aug 05, 2010 8:21 am 
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I just realized that in the case of CH4 you can subdivide the real pieces into two sets.

1. Real pieces that always exist regardless of your choice for A and B.
2. Real pieces that only exist under certain conditions for A and B.

And then I guess you could almost consider the virtual pieces as... real pieces that NEVER exist regardless of your choice for A and B.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Thu Aug 05, 2010 12:09 pm 
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At first: Best wishes for your finals.
Ok. I have understood Matts system. At least I think so.
Allagem wrote:
I started to draw these all out but was shocked to see just how many there was. There's at least 100! probably even near 200. I realized the pattern I was following actualled missed a few combinations and hit some others twice, so i scrapped my first attempt and have yet to take another. soon as i get an hour or so free i will.
I made an estimate how much pieces you might expect:
There are 4 ways to color a face.
You have 8 faces.
You use a system of axis with at most 48 symmetries.
This means you will have at least 4^8/48 = 1366 types of pieces. The number will be higher but this is a lower bound.
The lower bound for real+virtual in CH4 (which is equivalent) pieces is 5^4/48 = 14.
Maybe I can come up with a full analysis over the weekend.

The complex faceturning dodecahedron with 1 pair of cuts per axis has at least 2^12/120 = 35 types of pieces.
The complex faceturning dodecahedron with 2 pairs of cuts per axis has at least 4^12/120 = 139811 types of pieces.


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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Thu Aug 05, 2010 2:56 pm 
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Andreas Nortmann wrote:
At first: Best wishes for your finals.
Ok. I have understood Matts system. At least I think so.


Yes, good luck on the finals from me as well. I think I understand Matt's system too however I don't really like how it deals with deep cut puzzles. If the 2x2x2 has the exact same pieces as a 3x3x3 then that means the 3x3x3 core, face centers, and edges are imaginary pieces on a 2x2x2. Or in effect the complex 3x3x3 is also the complex 2x2x2. Is that really necessary? To me a 3x3x3 has 6 layers that can be turned independantly namely R,L,U,D,F, and B. A 2x2x2 only has 3 layers that can be turned independantly namely R,U, and F. Is there no way to take that into account with the imaginary pieces. Shouldn't the complex 2x2x2 just have 1 independant layer per axis too?

Carl

P.S. Doing some more thinking if I use Andreas's method above to list all the pieces in a complex 2x2x2 could I just do this?

01[] real
02[R] real
03[U] real
04[F] real
05[R U] real
06[R F] real
07[U F] real
08[R U F] real

Note: These are just the 8 corners of the 2x2x2. Opposite corners are the inverse of each other. So if correct then the complex 2x2x2 is just a normal 2x2x2 as it doesn't have any imaginary pieces. Do you guys agree or disagree?

Similarly to differentiate between a complex 4x4x4 and a complex 5x5x5 I just list all possible combinations of turns using 3 independant layers per axis for the 4x4x4 and 4 independant layers per axis for the 5x5x5.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Thu Aug 05, 2010 6:36 pm 
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Ok... knowing the set size was 3 for a 2x2x2, 6 for a 3x3x3, 9 for a 4x4x4, etc. I can calculate the number or real and imaginary pieces in each of the Complex NxNxN puzzles. Here is a table:

Attachment:
ComplexNxNxN.png
ComplexNxNxN.png [ 11.67 KiB | Viewed 4442 times ]


I'm NOT counting piece TYPES just total pieces. Andreas might be able to give the piece type counts from this but I'm not sure. I think this says the Complex 2x2x2 has 4 types but they are ALL corners so should we call that 1 type?

I think this generalizes the notion of imaginary pieces to deep cut puzzles. Its a good thing too. If the complex 4x4x4 was the same as a complex 5x5x5 I doubt we'd ever see a simulation of any complex NxNxN higher then N=3. As is N=4 just might be doable. Note it takes the 125 pieces of a 5x5x5 to simulate the 64 pieces of a complex 3x3x3. What size cube do we need to have to make a complex 4x4x4. The 8x8x8 with 512 pieces has just the right number of pieces but I doubt it will work as many pieces of the 5x5x5 are joined together to make a single piece in the Complex 3x3x3. I'd LOVE to be proven wrong though. My guess is you'll need a deep cut puzzle and you'll either need a 10x10x10 or a 12x12x12 just to model the Complex 4x4x4.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Thu Aug 05, 2010 7:18 pm 
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Wow! During the 5 minute drive home from work the solution hit me. It's remarkably simple and I'd show you a pic but I'm home and don't have internet yet so this is being posted from my iPhone. Here is a hint.... Best pic I can make with ASCII.

1 1 1 1 0 0 0 0
0 2 0 2 0 2 0 2
0 0 3 3 0 0 3 3

Draw an 8x8 grid over this. The turnable layers are numbered 1, 2, and 3. The 0's are just place holders.

Now I want a complex 4x4x4.

Carl

P.S. Hope this picture helps.
Attachment:
Complex4.png
Complex4.png [ 6.95 KiB | Viewed 4412 times ]

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Fri Aug 06, 2010 9:44 am 
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Ok... assuming what I've done above makes any sense I think I've come accross something very odd. Remember the Complex 3x3x3. Here is a picture Matt made.

Attachment:
Complex3x3x3SimulationMidTurn.jpg
Complex3x3x3SimulationMidTurn.jpg [ 73.54 KiB | Viewed 4411 times ]


This is built on a 5x5x5 with 64 pieces. But 64 pieces will fit into a 4x4x4. So... look at this?

Attachment:
Complex3.png
Complex3.png [ 6.81 KiB | Viewed 4411 times ]


The puzzle on the left is the Complex 3x3x3 that Matt shows mid-turn above. What is the puzzle on the right that is built out of a 4x4x4? Andreas, how would you classify its pieces?

I'm tempted to call it the Complex DeepCut 3x3x3. Mathematically the puzzles are very similar but from a solving stand point the puzzles are very different. The DeepCut 3x3x3 doesn't have any face centers for example.

What have I done? Did I make a wrong turn someplace and end up in la-la land talking about DeepCut 3x3x3's?

Help...
Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Sat Aug 07, 2010 11:38 am 
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Driving even further into la-la land I find some even stranger creatures.

Attachment:
Complex4b.PNG
Complex4b.PNG [ 14.13 KiB | Viewed 4375 times ]

A non-deep cut 4x4x4!?

It still has the correct 512 pieces as many pieces of the 9x9x9 stick together to form one piece. Note this still isn't the Complex 5x5x5.

And oddest of all... this puzzle has actually been built.

Attachment:
Complex2c.PNG
Complex2c.PNG [ 2.49 KiB | Viewed 4374 times ]

Its called the Fused Cube. Its order=1 with one independant layer of rotation per axis. But it has 9 edges and 3 face centers that don't appear in the twistability analysis. So they don't appear to be real, virtual, or imaginary. So what are they?

The simple answer is the fused cube is a subset of an order=2 puzzle, namely the 3x3x3. But shouldn't this puzzle be understandable in terms of an order=1 puzzle? If not, why not?

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Sun Aug 08, 2010 12:23 am 
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What about this complex 2x2 and what do you call it? It is a complex 2x2 right? It has all the pieces of a 3x3. Not sure if it has the same state space though.
Attachment:
complex-2x2.png
complex-2x2.png [ 1.3 KiB | Viewed 4352 times ]

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Sun Aug 08, 2010 9:42 am 
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wwwmwww wrote:
I think I understand Matt's system too however I don't really like how it deals with deep cut
puzzles. [snip]
The problem with your proposed extension is that pieces of identical type now become different signatures.
Two opposite corners of the 3x3x3 have e.g. [URF] and [DLB]
In your extension two opposite corners could have [URF] and []
You rediscover this problem in your table where you have 4 lines for 1 kind of piece.
wwwmwww wrote:
The puzzle on the left is the Complex 3x3x3 that Matt shows mid-turn above. What is the puzzle on the right that is built out of a 4x4x4? Andreas, how would you classify its pieces?
I wouldn't because my systems is usable just for puzzles where one piece isn't moved by more than one entity per axis. You CH3-puzzle which started the thread fits this definition but your complex 4x4x4 doesn't.
wwwmwww wrote:
A non-deep cut 4x4x4!?
Another reason to not extend Matts system for deepcut puzzles.
wwwmwww wrote:
Its called the Fused Cube. Its order=1 with one independant layer of rotation per axis. But it has 9 edges and 3 face centers that don't appear in the twistability analysis. So they don't appear to be real, virtual, or imaginary. So what are they?
The simple answer is the fused cube is a subset of an order=2 puzzle, namely the 3x3x3. But shouldn't this puzzle be understandable in terms of an order=1 puzzle? If not, why not?
I don't know if I have stated this before but I prefer to "make" the order (however it is defined) of a puzzle as low as possible. In most cases it is simple. The fused cube is a counterexample. The twistability analysis for FH1 doesn't work for it. Thats why I go to a bandaged FH2. Please note that this depends on the specific variant. A 4x4x4 could be bandaged into a 2x2x2 and then it is FH1 because it suffices. I have no problem with classifying the fused cube as a subgroup of FH2 because there are others like [L R F] which nobody would classify as FH1. And so its just seems logical to classify the FusedCube (aka [L F U]) as subgroup of FH2 too.
It is fascinating that you have done something similar. In your first post you have presenting something which could be called a bandaged CH9 although you use bridges. But with your kind of bandaging it CH3 suffices.

Meanwhile I have finished my analysis of CH4: There are 38 types of pieces. I don't know how much are real.


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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Sun Aug 08, 2010 3:54 pm 
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GuiltyBystander wrote:
What about this complex 2x2 and what do you call it? It is a complex 2x2 right? It has all the pieces of a 3x3. Not sure if it has the same state space though.

That puzzle is equivalent to the puzzle I predented above. You just chose the opposite corner as the holding point so instead of holding the big corner on a fuzed cube you are holding the small corner opposite it. It's still a fuzed cube. And now that I think about it... I don't think this puzzle is the complete picture of the Complex-2x2x2.

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Sun Aug 08, 2010 4:15 pm 
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Andreas Nortmann wrote:
The problem with your proposed extension is that pieces of identical type now become different signatures.
Two opposite corners of the 3x3x3 have e.g. [URF] and [DLB]
In your extension two opposite corners could have [URF] and []
You rediscover this problem in your table where you have 4 lines for 1 kind of piece.


Fair enough... But does that mean we can't have a Complex-2x2x2? What if the Complex version of all the NxNxN puzzles are built from MxMxM puzzles with M odd? I think the Complex-4x4x4 above built from a 9x9x9 might work.

I went back and looked at the Complex 2x2x2. And I think I see where I fall short. Consider the Complex-2x2x2 as being made from two 3x3x3's sitting side-by-side. A turn of the red layer on the left copy is tied to a turn of the red layer on the right copy.

Attachment:
Complex2A.png
Complex2A.png [ 35.05 KiB | Viewed 4311 times ]


Andreas Nortmann wrote:
I wouldn't because my systems is usable just for puzzles where one piece isn't moved by more than one entity per axis.


So lets look at the Complex-2x2x2 then as there is only one turnable entity per axis.

Andreas Nortmann wrote:
Another reason to not extend Matts system for deepcut puzzles.


Just because its not easy doesn't mean we shouldn't try. I'm not sure its impossible yet.

I wanted to see if I could merge these two 3x3x3's into one 5x5x5. Here is what I have so far:

Attachment:
Complex2B.png
Complex2B.png [ 22.74 KiB | Viewed 4311 times ]


I've matched up all the pieces of the inner 3x3x3 but I'm not sure what to do with the Face centers, the T-centers, and the Edges of the 5x5x5.

I think the T-Centers may be a third type of edge with the [RU] signature or they may be part of one of the already existing edges.

The Face-Centers and Edges I think have the [RUF] signature so they may be part of an already existing corner or two new types of corners.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Sun Aug 08, 2010 8:05 pm 
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wwwmwww wrote:
That puzzle is equivalent to the puzzle I predented above. You just chose the opposite corner as the holding point so instead of holding the big corner on a fuzed cube you are holding the small corner opposite it. It's still a fuzed cube. And now that I think about it... I don't think this puzzle is the complete picture of the Complex-2x2x2.

Carl
I don't think it is the same. Here's a quick example to show they are different using the sexy move (R' U R U).
If you do (R' U R U) with just one layer like in your example, you will be holding onto a 2x2x3 block. It takes (R' U R U)x6 to get back to your starting point.
If you do (R' U R U) with two layers like in my example, you will be holding onto a 1x1x3 block. It takes (R' U R U)x18 to get back to your starting point.

Or are they the same and the virtual pieces on the 1 layer version are real on the 2 layer version?

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Mon Aug 09, 2010 2:31 am 
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wwwmwww wrote:
Andreas Nortmann wrote:
I wouldn't because my systems is usable just for puzzles where one piece isn't moved by more than one entity per axis.
So lets look at the Complex-2x2x2 then as there is only one turnable entity per axis.

I still can't classify that puzzle with my twistability analysis because there is no way to let the corner LDB stay fixed.
I should write up all the conditions for my system. And check them for redundancies. Sometime...


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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Mon Aug 09, 2010 7:04 pm 
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GuiltyBystander wrote:
I don't think it is the same. Here's a quick example to show they are different using the sexy move (R' U R U).
If you do (R' U R U) with just one layer like in your example, you will be holding onto a 2x2x3 block. It takes (R' U R U)x6 to get back to your starting point.
If you do (R' U R U) with two layers like in my example, you will be holding onto a 1x1x3 block. It takes (R' U R U)x18 to get back to your starting point.

Or are they the same and the virtual pieces on the 1 layer version are real on the 2 layer version?


Well I'm re-thinking what I mean about the Complex 2x2x2 so lets call this the Fused Cube. The one layer verson and the two layer version are equivalent. But R'URU on one doesn't map to R'URU on the other.

Here is R'URU on the one-layer version:
http://www.randelshofer.ch/cube/rubik/?R'URU

It maps to TL'TFTUTL on the two layer version:
http://www.randelshofer.ch/cube/rubik/?TL'TFTUTL

Same puzzle... just different notation. Both are the Fused Cube the choice is simply which corner do you hold on to.

Note these two examples use the [] corner and the [R U F] corner. You could also choose one of the other 6 corners and it would still be the Fused Cube.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Mon Aug 09, 2010 7:42 pm 
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Andreas Nortmann wrote:
I still can't classify that puzzle with my twistability analysis because there is no way to let the corner LDB stay fixed.
I should write up all the conditions for my system. And check them for redundancies. Sometime...

Ok... let me backpeddle a bit. Instead of using the allowed rotations to analysis each pieces type by twistability lets use it to simply name the pieces in the complex 2x2x2. We are back to this:

01[] corner - real
02[R] corner - real
03[U] corner - real
04[F] corner - real
05[R U] corner - real
06[R F] corner - real
07[U F] corner - real
08[R U F] corner - real

So there can only be 8 pieces in the Complex 2x2x2 and that means your plan jane 2x2x2 is the Complex 2x2x2.

Just as these are the 64 pieces that are present in the Complex 3x3x3.

01 [] core - real
02 [R] face - real
03 [L] face - real
04 [U] face - real
05 [D] face - real
06 [F] face - real
07 [B] face - real
08 [RL] imaginary
09 [RU] edge - real
10 [RD] edge - real
11 [RF] edge - real
12 [RB] edge - real
13 [LU] edge - real
14 [LD] edge - real
15 [LF] edge - real
16 [LB] edge - real
17 [UD] imaginary
18 [UF] edge - real
19 [UB] edge - real
20 [DF] edge - real
21 [DB] edge - real
22 [FB] imaginary
23 [RLU] imaginary
24 [RLD] imaginary
25 [RLF] imaginary
26 [RLB] imaginary
27 [RUD] imaginary
28 [RUF] corner - real
29 [RUB] corner - real
30 [RDF] corner - real
31 [RDB] corner - real
32 [RFB] imaginary
33 [LUD] imaginary
34 [LUF] corner - real
35 [LUB] corner - real
36 [LDF] corner - real
37 [LDB] corner - real
38 [LFB] imaginary
39 [UDF] imaginary
40 [UDB] imaginary
41 [UFB] imaginary
42 [DFB] imaginary
43 [RLUD] imaginary
44 [RLUF] imaginary
45 [RLUB] imaginary
46 [RLDF] imaginary
47 [RLDB] imaginary
48 [RLFB] imaginary
49 [RUDF] imaginary
50 [RUDB] imaginary
51 [RUFB] imaginary
52 [RDFB] imaginary
53 [LUDF] imaginary
54 [LUDB] imaginary
55 [LUFB] imaginary
56 [LDFB] imaginary
57 [UDFB] imaginary
58 [RLUDF] imaginary
59 [RLUDB] imaginary
60 [RLUFB] imaginary
61 [RLDFB] imaginary
62 [RUDFB] imaginary
63 [LUDFB] imaginary
64 [RLUDFB] imaginary

The pieces that show up in the Fused Cube really are order=2 pieces and it IS best to think of them as a subset of the 3x3x3. At least that is what I'm thinking at the moment. At this point that makes the 8x8x8 I show above the Complex 4x4x4, NOT the 9x9x9 as it has order=4 parts. The Complex 4x4x4 should just have all the order=3 parts. You could think of the 9x9x9 as a higher order Fused Cube.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Tue Aug 10, 2010 9:01 am 
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Andreas Nortmann wrote:
The problem with your proposed extension is that pieces of identical type now become different signatures.
Two opposite corners of the 3x3x3 have e.g. [URF] and [DLB]
In your extension two opposite corners could have [URF] and []
You rediscover this problem in your table where you have 4 lines for 1 kind of piece.


Ok. I've solved this problem. Think about the 3x3x3, why do two opposite corners of the 3x3x3 have the same signature e.g. [URF] and [DLB]?

It is due to your choice of a holding poing, i.e. which cubie you hold that doesn't move or twist. You picked the core cubie such that your turnable layers are RLUDFB. But that isn't the only choice you could have made. You could have held the 3x3x3 from a corner for example using 3 face turns and 3 slice turns and then you'd have the exact same signature problem you have with the 2x2x2.

Your choice of holding the core works out well as there is only one piece of that type. The corner doesn't work well as there are 8 of them.

The problem with the 2x2x2 is there isn't a core, or any one piece that is of its own type. So you MUST pick a corner. As such to analyze the twistability of the 2x2x2 you must pick a piece to serve as the core and then analyze the puzzle in such a way that all pieces of the same type are also considered to be a holding point.

So the twistability analysis of the 2x2x2 looks like this:

Attachment:
Twistability.png
Twistability.png [ 11.75 KiB | Viewed 4210 times ]


This will allow you to analyze the Complex 4x4x4 and see how many piece types its has and the count of each. I haven't done that yet but it should work. An easier exercise would be to look at the 3x3x3 and look at the 12 cases where an edge was picked as the holding point or the 6 cases where a face is picked as the holding point. In these cases you SHOULD end up with the same Complex-3x3x3.

Carl

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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Wed Aug 11, 2010 11:58 am 
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Okay Carl,
you could walk the way you proposed here to find your complex 2x2x2 but I have to mention these points:
1. It is not my system. Matt invented it.
2. "twistability analysis" is the name of MY system. You still want to modify Matts system to represent deepcut puzzles differently. At least that is what I assume.
3. In Matts system there is no such thing as a "holding point", more something like a "reference point". E.G. the inverted core of the 3x3x3 can't be held still in Matts system if I understood it.
4. To proove that the reference point could be placed arbitrarily just test it on the 3x3x3: Pick a piece different from the core as reference point and see if you get 64 pieces of 10 different types.

Andreas


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 Post subject: Re: The Order=3 Corner-Turn MultiCube (Plus One)
PostPosted: Sun Aug 22, 2010 3:39 pm 
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Ok... I'm back.

Andreas Nortmann wrote:
Okay Carl,
you could walk the way you proposed here to find your complex 2x2x2 but I have to mention these points:
1. It is not my system. Matt invented it.


I'm not trying to re-invent the wheel. You have a very nice system and I'm just trying to generalize it so it can handel Matt's imaginary pieces.

Andreas Nortmann wrote:
2. "twistability analysis" is the name of MY system. You still want to modify Matts system to represent deepcut puzzles differently. At least that is what I assume.


I'll start a new thread in the Complex NxNxN to present this in detail and try to get this tread back on subject. But in short Matt defines order such that the 2x2x2 and the 3x3x3 are the same. To me these puzzles are obviously different puzzle. Within the framework of your "twistability analysis" I simply believe there is a way to generalize things and have a well defined Complex 2x2x2, Complex 3x3x3, Complex 4x4x4, etc. and leave order the way we've defined it. The Complex NxNxN for N=odd would still be the same as those proposed by Matt but this system can deal with N=even now.

Andreas Nortmann wrote:
3. In Matts system there is no such thing as a "holding point", more something like a "reference point". E.G. the inverted core of the 3x3x3 can't be held still in Matts system if I understood it.


Agreed... none of the imaginary pieces can serve as holding points if one set of independant twists is used. Its what makes the imaginary pieces different from the virtual pieces in general. Not that any of the NxNxN puzzles have any virtual pieces.

Andreas Nortmann wrote:
4. To proove that the reference point could be placed arbitrarily just test it on the 3x3x3: Pick a piece different from the core as reference point and see if you get 64 pieces of 10 different types.


Done... and yes you do. Though I'm not sure I understand your distinction between "reference point" and "holding point". The choice that is actually made is a pick of 6 independant twists. For example [RLUDFB] leaves the core stationary so its a holding point. In each of the x, y, and z directions you have 3 layers that can turn (talking about a 3x3x3), two faces and a slice layer. You are free to pick any two you want in each of the 3 directions and that choice determines your "holding point". An imaginary piece will never be a holding point.

New thread will start shortly,
Carl

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