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Painting Cube Faces Green

Date: 08/19/2003 at 18:43:32
From: Mike
Subject: Combinations

Mrs. Jones had some white paint and some green paint, and a bunch of 
wooden cubes. Her class decided to paint the cubes by making each 
face either solid white or green. Juan painted his cube with all six 
faces white. Julie painted her cube solid green. Herman painted 4 
faces white and 2 faces green. How many cubes could be painted in 
this fashion so that each cube is different from the other? Two 
cubes are alike if one can be turned so that it exactly matches, 
color for color on each side, the other cube.  


Date: 08/20/2003 at 14:27:13
From: Doctor Ian
Subject: Re: Combinations

Hi Mike,

Interesting question.  I don't think I've seen this before.  

The first thing that occurs to me is to break it into cases, where
each case involves some number of green faces.  The number of green
faces can be 0, 1, 2, 3, 4, 5, or 6, so those are the cases we have to
consider. 

The extreme cases are the easiest. There is only one way to paint a
cube so that no faces are green, or so all the faces are green:

  0: 1
  1: ?
  2: ?
  3: ?
  4: ?
  5: ?
  6: 1

What about painting one face green? If we have two cubes with one face 
painted green, we know that we can always rotate one to look like the 
other. Do you see why? So there is only one way to paint one face 
green:

  0: 1
  1: 1
  2: ?
  3: ?
  4: ?
  5: ?
  6: 1

For the same reason, there is only one way to paint 5 faces green,
because that's the same as having one way to paint one face white.

  0: 1
  1: 1
  2: ?
  3: ?
  4: ?
  5: 1
  6: 1

This, by the way, is an illustration of why mathematicians are so fond
of symmetry. Anything that's true for the green faces has to be true
if we switch white to green and vice versa. That's handy to know,
because once we know how many ways we can have two green faces, we
automatically know how many ways we can have four green faces. 

Anyway, we're left with the trickier cases: two faces green, and
three faces green.  

Can we figure out a way to do this without just trying every
possibility?  We might be able to, if we can find a way to _describe_
how a cube is painted in a way that is independent of how it's 
rotated.  

For example, suppose we have two faces painted green. They can be
touching, or they can be across from each other. This is true no
matter how the cubes are oriented. (Convince yourself that this is
true.) So there are just two ways to paint two faces green, and two
ways to paint four faces green:

  0: 1
  1: 1
  2: 2
  3: ?
  4: 2
  5: 1
  6: 1

I'll leave the last case for you, but you can figure it out using the
same technique that I used for the pairs.  Let me know if any of this
wasn't clear, or if you have other questions. 

I hope this helps!  Thanks for the question.  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Permutations and Combinations

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