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Colorful cubes


Best Answer Prime, 15 March 2013 - 02:31 PM

Actually, we should split the difference and go with 27.

Spoiler for missed symmetry

 

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11 replies to this topic

#1 bonanova

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Posted 13 March 2013 - 12:51 PM

I have 12 sticks and I have painted 8 of them red and 4 of them blue.

I then glued the sticks together to form the frame of a cube.

 

So now I'm wondering how many total ways could I have assembled the sticks into cubes that could be distinguished from each other.

Assume I can pick up the cube and move it around any way I like, but I cannot deform or re-assemble it


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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
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#2 BobbyGo

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Posted 13 March 2013 - 02:49 PM

Spoiler for guess


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#3 k-man

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Posted 13 March 2013 - 03:38 PM

Spoiler for I found


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#4 bonanova

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Posted 14 March 2013 - 04:43 PM

Is there a consensus?

Can you construct the possible configurations in groups, using words or a sketch?

 

My first try to solve this was to take 12 things 4 at a time, then remove the symmetrically equivalent solutions.

But I kept removing too many cases. Next try was to group the edges and distribute 4 blues among the groups.

I'm not convinced yet of my answer.


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#5 k-man

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Posted 14 March 2013 - 08:07 PM

Spoiler for my approach

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#6 bonanova

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Posted 15 March 2013 - 04:47 AM

Spoiler for my approach

 

Very nice approach. Nice categories and elimination of symmetric cases.

 

Spoiler for I have one question


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#7 Prime

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Posted 15 March 2013 - 09:42 AM

Good spacial vision exercise. Without looking at an actual model of a cube, I am not sure I got it right.

Next, let's try it in 4-D.

Spoiler for connections


Edited by Prime, 15 March 2013 - 09:50 AM.

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Past prime, actually.


#8 bonanova

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Posted 15 March 2013 - 11:20 AM


Spoiler for my approach

 
Very nice approach. Nice categories and elimination of symmetric cases.
 
Spoiler for I have one question
Never mind, I think I see two cases after all. I think that's it.
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#9 Prime

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Posted 15 March 2013 - 11:54 AM

Good spacial vision exercise. Without looking at an actual model of a cube, I am not sure I got it right.

Next, let's try it in 4-D.

Spoiler for connections

Correction: It is 10 arrangements for the variation 4.

Then the answer is the same as K-man has found.


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Past prime, actually.


#10 Prime

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Posted 15 March 2013 - 02:31 PM   Best Answer

Actually, we should split the difference and go with 27.

Spoiler for missed symmetry

 


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Past prime, actually.





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