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 Post subject: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 3:46 am 
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Playing around with GAP I stumbled upon a variant I implemented like this:
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This variant has a fourfold number of permutations compared with the unbandaged cube.
Explanation is as follows: If you consider the bridge fixed in space, you can rotated the 4 unbandaged faces. Furthermore you have all abilities of the unbandaged cube. This brigde is equivalent to making visible the orientation of one face.
This anomaly doesn't occur if orientations of all faces are considered. In that case this bandaging will result into a number of permutations one fourth of the original number.


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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 3:55 am 
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:shock: :lol:
Thats a nice one :P

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 4:34 am 
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It looks very easy ... to carry around. :lol:

Nice idea! :D

I wonder if you could link up several 3x3s in this way, either in a line or a loop, or maybe even a symmetric 3D network/lattice of 3x3s, for example:

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EDIT: This puzzle has now been made here.

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Last edited by KelvinS on Thu Oct 01, 2009 4:15 am, edited 1 time in total.

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 5:00 am 
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Correct me if I'm wrong but wouldn't the picture above, even tho cool looking only be a set of 8 connected "cube-in-a-cube"-cubes? :scrambled:

But to Andreas, nice idea and I love the mathematics you present that is relevant to this mod :)

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 5:16 am 
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eMp_dWa wrote:
Correct me if I'm wrong but wouldn't the picture above, even tho cool looking only be a set of 8 connected "cube-in-a-cube"-cubes? :scrambled:

But to Andreas, nice idea and I love the mathematics you present that is relevant to this mod :)

Wouldn't you get a similar effect due to the implicit orientation of faces with bandaging?

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Last edited by KelvinS on Thu Jul 30, 2009 2:37 am, edited 2 times in total.

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 5:37 am 
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Kelvin Stott wrote:
It looks very easy ... to carry around. :lol:

Nice idea! :D

I wonder if you could link up several 3x3s in this way, either in a line or a loop, or maybe even a symmetric 3D network/lattice of 3x3s, for example:

Attachment:
Lattice.jpg



That's very similar to a weird poor quality picture I found on the net and posted some time ago. Can't seem to find it now. (Kelvin Stott's pic, It doesn't appear when I quote for some reason).

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 5:41 am 
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Tony Fisher wrote:
That's very similar to a weird poor quality picture I found on the net and posted some time ago. Can't seem to find it now.

Probably, but I wans't aware of anything: the idea just came into my head as soon as I saw this post and so I thought I would share it.

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 11:17 am 
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are the surrounding pieces FF?
like a 1x1xY?

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Wed Jul 29, 2009 1:10 pm 
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This cube is great- the idea of bandaging opposing pieces to one another has been floating around my head for a while now. I imagine the cube looks like it's jumping rope as you solve it, haha.


Kelvin Stott wrote:
Attachment:
Lattice.jpg

At first glance, I called you ridiculous and moved on my way.

Coming back... I think it could work... and that scares me, haha. I don't see why it would'nt work.


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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Thu Jul 30, 2009 1:40 am 
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Kelvin Stott wrote:
Wouldn't you get a similar effect due to the implicit orientation of faces with bandaging?
You mean, by bonding an edge to a face? In this case the number of permutations would be multiplied by four for the orientation made visible but it would be divided by 24 for the edge which can't float free any longer.

ulmboy556 wrote:
are the surrounding pieces FF?
like a 1x1xY?

No. Those are just some cubies left over and glued together.
Do you want to build version 2 ?


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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Thu Jul 30, 2009 2:08 am 
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Andreas Nortmann wrote:
Kelvin Stott wrote:
Wouldn't you get a similar effect due to the implicit orientation of faces with bandaging?
You mean, by bonding an edge to a face?

No, I mean if you bandage the faces of different cubes with connectors, as shown in my diagram above. The connectors would define the orientation of 3 faces on each cube, so you would get 4^3 = 64 times more permutations per cube.

Also, if the connectors themselves were bandaged you would be forced to turn opposite faces of two connected cubes at the same time, so you would have to solve all 8 cubes together. This would be much more difficult than solving each cube independently (in the case where the connectors are not bandaged and are free to twist).

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Fri Jul 31, 2009 12:49 am 
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Kelvin Stott wrote:
I mean if you bandage the faces of different cubes with connectors, as shown in my diagram above. The connectors would define the orientation of 3 faces on each cube, so you would get 4^3 = 64 times more permutations per cube.
Yes. Thats true.
Kelvin Stott wrote:
Also, if the connectors themselves were bandaged you would be forced to turn opposite faces of two connected cubes at the same time, so you would have to solve all 8 cubes together. This would be much more difficult than solving each cube independently (in the case where the connectors are not bandaged and are free to twist).
That is an interesting idea but no new challenge. You could still solve every cube on its own. While solving one cube you would only twist one side per adjacent cube. An already solved adjacent cube wouldn't become mixed up again.


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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Fri Jul 31, 2009 3:10 am 
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Andreas Nortmann wrote:
That is an interesting idea but no new challenge. You could still solve every cube on its own. While solving one cube you would only twist one side per adjacent cube. An already solved adjacent cube wouldn't become mixed up again.

I know what you mean, but this is not quite right. Remember you would have to get the right orientation of two opposite faces in adjacent cubes AT THE SAME TIME. The probability of achieving this is 1/4, so for all 12 connectors this would be 1/4^12 = 1/16,777,216. You really would have to solve all eight cubes together because they are linked, as orientation information passes between the cubes through the bandaged connectors. It would be the ultimate parity problem! :D

The total number of permutations for this puzzle would be:

(4.325 x 10^19)^8 x 4^12 = 2.54 x 10^161

Or in long form:

254, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000.

I may be out by a couple of factors of two, but that's more than the total number of atoms in the universe (about 10^80) ... SQUARED!!

:shock:

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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Sat Aug 01, 2009 12:39 pm 
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Kelvin Stott wrote:
Remember you would have to get the right orientation of two opposite faces in adjacent cubes AT THE SAME TIME.
That is not that difficult. After you solved the first cube 3 connectors are oriented in the correct way. After solving an adjacent cube this connector will once again be correct in its orientation because otherwise the faces of this second cube wouldn't be solved.
Anybody a better explanation?


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 Post subject: Re: A bandaged cube with MORE permutations than the unbandaged
PostPosted: Sat Aug 01, 2009 1:04 pm 
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Aha, yes, I see now that you're right. As someone who doesn't do solving very often I didn't quite understand what you meant. :roll:

Still, I like the fact that it has "more permutations than the total number of atoms in the universe, squared". :D

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