Painting Cubes

Date: 01/06/2003 at 11:43:04
From: Nia Lynn 
Subject: Making Choices

Katie, a student at the world's first orbiting school, noticed that 
her unit cubes floated in zero gravity.  If she was careful, she could 
even build floating geometric shapes with them.

One day Katie used the unit cubes to build a larger floating cube that
was three units long on each edge. She then painted all the exposed
faces of the cube red. Unfortunately, just after she finished, she
sneezed, scattering the unit cubes. No cubes remained attached.

When Katie gathers the cubes, how many cubes will she find that have
exactly 6 faces painted red? 5? 4? 3? 2? 1? 0 faces?  

I know that each unit cube does not have the same number of faces
painted red because some are on the corners and they should only have
3 faces painted (the top and the two sides) and there are 6 faces on 
the cube.

Date: 01/06/2003 at 12:35:11
From: Doctor Ian
Subject: Re: Making Choices

Hi Nia, 

To find the number of unit cubes, imagine slicing the larger cube into
three layers:


  

How many cubes are in each layer?  How many layers are there?  The
total number of cubes has to be the product of those two numbers. 

Now, what about the number of painted faces on the unit cubes after
the larger cube is broken up? 

Note that for a unit cube to have all 6 of its faces painted, it would 
have to have all 6 of those faces exposed while part of the larger 
cube. Is there _any_ position in the larger cube where that would be 
true?  

  

One way to approach this problem is to figure out how many different 
_kinds_ of locations there are. There are 8 corner locations (only 7 
are visible in this view), 12 edge locations (only 9 are visible in 
this view), and 6 center locations  (only 3 are visible in this view).

 


We can use a table to keep track of this information:

   Location   unit cubes
   --------   ----------
   Corner         8
   Edge          12
   Center         6

The main reason for grouping locations like this is that each location
is equivalent to every other location of the same kind. (If you don't
see why, imagine that while you're looking away, someone picks up the
larger cube and rotates it so that a different corner is facing you. 
Would you be able to tell?) And that means that once we figure
something out regarding _any_ unit cube, we've figured it out for
_every_ unit cube in the same kind of location. 

For example, look at one of the unit cubes located at a corner of the
larger cube. As you've noticed, it will have three of its faces
painted. But since any corner is just like any other corner, _every_
unit cube located in a corner will have three of its faces painted:

   Location   unit cubes   faces painted
   --------   ----------   -------------
   Corner         8             3
   Edge          12             ?
   Center         6             ?

I'll leave the rest of the table for you to fill in.

Finally, note that so far, we've accounted for 26 unit cubes.  But
from analyzing the layers, we know we have to have 27 unit cubes. So
where is the final one? And how many of its faces would be painted? 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
 
For a related, interesting problem, see How Many Cube Faces Were Painted?