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 Post subject: 3x3x3 subgroups by twistability
PostPosted: Mon Oct 05, 2009 12:27 pm 
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Hi,
I don't like long posts but here it seems unavoidable. Sorry for that.

after I presented the B12C111 it is time to reveal the common idea behind my latest "mega-bandagings":
The 3x3x3 is well known and can be defined by <U D L R F B>.
The Fused cube can be defined by <U L F> or some other symmetric definition.
The Siamese cube doesn't seem to fit in this scheme. But it does if we help ourselves with slices: <U R Us Rs> What happens here is that we don't assume the core as fixed any longer but the shared block and twist two sides and two slices independently.
The advantadge of doing so is, that we have "unbandaged" the Siamese cube: Every mentioned side and slice is twistable every time. Jaaps solution makes use of this, too.

If we now try to generalize this scheme we get 35 different subgroups of the 3x3x3 and the last year I worked to implement all of these subgroups in at least one mod. The last one was the B12C111.
The remaining post lists these 35 variants:

Lets start with the easy ones:
    <U L F Ls Fs> Bandaged (C112)
    <L F Ls Fs> Bandaged (C113)
    <U L F Fs> Bandaged (C122)
    <U F Fs> Bandaged (C123)
    <U L F> Bandaged (C222)
    <U L> Bandaged (C223)
These can be considered as generalized Siamese cubes and that is a good way to implement them:
Attachment:
MonoBandagedCorners.jpg
MonoBandagedCorners.jpg [ 62.65 KiB | Viewed 1302 times ]


Some other classic bandagings:
    <U D L F Fs> Bandaged (E112)
    <U D F Fs> Bandaged (E113)
    <U D L F> Bandaged (E122)
    <U D L> Bandaged (E123)
Implementation is pretty straightforward:
Attachment:
MonoBandagedEdges.jpg
MonoBandagedEdges.jpg [ 61.61 KiB | Viewed 1303 times ]


There are two direct bandagings left
    <U D L R F> Bandaged (F112)
    <U D L R> Bandaged (F113)
These are implementable fairly easily but it is hard to visualize them.
Attachment:
MonoBandagedFaces.jpg
MonoBandagedFaces.jpg [ 15.33 KiB | Viewed 1988 times ]


Now some less obvious ones:
    <U L Us Ls Fs> Bridge (Corner 111) => BC111
    <U L Ls Fs> Bridge (Corner 112) => BC112
    <L Ls Fs> Bridge (Corner 113) => BC113
    <U Us Ls Fs> 4 x Bridge (Corner 111) => B4C111
These are all implemented in the QuadrupletsCubes:
The B4C111 might be hard to see but if you think about what can twist and what not it should become clear. The one on the right is technically obsolete but did already exist when I had the idea for the other ones.
Attachment:
QuadrupletsCubes.jpg
QuadrupletsCubes.jpg [ 37.54 KiB | Viewed 1302 times ]


Seven other variants are implemented with the ColumnCubes
    <U Ls Fs> 4 x Bridge (Corner 112) => B4C112
    <Ls Fs> 4 x Bridge (Corner 113) => B4C113
together with B4C111 in ColumnCubes C+C+C+C
    <U D L Fs> Bridge (Edge 112) => BE112
    <U D Fs> Bridge (Edge 113) => BE113
    <U D L Ls Fs> Bridge (Edge 111) => BE111
all at once in ColumnCubes E+E
    <U L Fs> Bridge (Corner 122) => BC122
    <U Fs> Bridge (Corner 123) => BC123
together with BC112 in ColumnCubes CE+CE
Attachment:
ColumnCubes.jpg
ColumnCubes.jpg [ 75.71 KiB | Viewed 1302 times ]


There are other possibilities for ColmnCubes (with throughgoing bandaged blocks) but all we would get were trivial variants (see below) or variants we have already seen. We have 3 exoctic ones which can't:
    1. <U D L R Fs> Bridge (Face 111) => BF111
no comments
Attachment:
BF111.jpg
BF111.jpg [ 64.29 KiB | Viewed 1302 times ]

    2. <Us Ls Fs> 12 x Bridge (Corner 111) => B12C111
Here implemented two times.
Attachment:
B12C111.jpg
B12C111.jpg [ 64.54 KiB | Viewed 1302 times ]

    3. <U D Ls Fs> 4 x Bridge (Edge 111) => B4E111
This one is obsolete because a B4C111 implies a B4E111. which means this variant is implemented in ColumnCubes C+C+C+C and in QuadrupletsCubes-2-2+2+2 though in disguise.

Interestingly there are 4 groups which all represent the 3x3x3
    <U D L R F B> Original Cube (Fixed Centre)
    <U D L R F Fs> Original Cube (Fixed Face)
    <U D L F Ls Fs> Original Cube (Fixed Edge)
    <U L F Us Ls Fs> Original Cube (Fixed Corner)
no picture here.

To complete the list of 35 groups these groups are missing
    <U D> 1x1x3(Fixed inner cubie)
    <F Fs> 1x1x3(Fixed outer cubie)
    <Fs> Bridge (C133)
    <U> 1x1x2 or Siamese233 or whatever
    <> 1x1x1
Since they are all trivial, I haven't implemented them.

I hope you enjoyed.
And I am glad to dedicate my time to another project.


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 Post subject: Re: 3x3x3 subgroups by twistability
PostPosted: Mon Oct 05, 2009 5:37 pm 
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Nice to see these all together and I like your analysis. I'm curious... can you solve all of these? I wonder which would be the most challenging to a typical 3x3x3 solver.

Carl

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 Post subject: Re: 3x3x3 subgroups by twistability
PostPosted: Tue Oct 06, 2009 11:01 am 
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Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
I think the hardest is the E113, although I haven't tested BF111 and the column cubes yet.


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 Post subject: Re: 3x3x3 subgroups by twistability
PostPosted: Tue Oct 06, 2009 11:10 am 
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Location: Greece, Australia, Thailand, India, Singapore.
This is an extremely interesting result. Well done! :)

I wonder if a mathematical (group) scheme could found, such that
it can be built around those 35 cases, without using the speed-cubing terms.

:D



Pantazis

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 Post subject: Re: 3x3x3 subgroups by twistability
PostPosted: Wed Oct 07, 2009 11:21 am 
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Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Seems like this topic was to abstract. :cry:
kastellorizo wrote:
I wonder if a mathematical (group) scheme could found, such that
it can be built around those 35 cases, without using the speed-cubing terms.
I think you mean this:
The 1x1x3 is isomorph to C4^2, the cyclic group of order 4 squared.
Since the 3x3x3 is a finite group they can all be written like tried here.
But I have to confess that I haven't understand everything of "David Joyner, Adventures in Group Theory, Johns Hopkins University Press, 2002" so that goes beyond my horizon.


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