Hidden Faces in a Set of Cubes

Date: 10/04/2000 at 06:05:23
From: Justin Roden
Subject: Hidden faces of cubes

Hello, 

I am doing coursework on the hidden faces of cubes. We have to come up 
with a formula that will tell you many invisible faces there are in an 
arrangement of cubes if you know how many visible faces there are.

The cubes can be arranged in any number or order, there could be 1653 
small cubes connected to form one big cube, but we have to be able to 
say, "Okay, there are x visible faces, so that means there are y 
hidden faces." Could you give me a hint as to what this formula may 
be? Just some help? NOT the formula itself, if there is one.

Thanks,
Justin

Date: 10/04/2000 at 07:56:07
From: Doctor Anthony
Subject: Re: Hidden faces of cubes

I will give one example from which you should be able to derive a 
general method.

A large cube is formed by gluing together 27 identical small cubes. 
What are the dimensions of the large cube? How many of the small cubes 
will have glue on all 6 faces, 5 faces, 4 faces, 3 faces, 2 faces, or 
1 face?

The size of the big cube is of course 3*3*3 = 27 small cubes. With 27 
cubes and 6 faces on each cube there are 6*27 = 162 faces altogether.

It will be easier to find the number of cubes with 0, 1, 2, 3, 4, 5 
faces exposed to the outside, and by subtraction find the number with 
6, 5, 4, 3, 2, 1 faces with glue on them.

The total number of outside faces is 9 x 6 = 54. We have:

    1 cube  in the middle with no outside face. 
    6 cubes in the center of each side with 1 outside face. 
   12 cubes in the center of each edge with 2 outside faces.  
    8 cubes at the corners with 3 outside faces.

Now we consider the number of OUTSIDE faces.

     Outside Faces   No. of Cubes   No. of Faces
     -------------  -------------   -------------
          0               1               0
          1               6               6
          2              12              24
          3               8              24 
          4               0               0
          5               0               0
                                     -----------
                                         54

Now we can make up the table for faces with glue on them, using the 
results from table above.

     Glue on Faces   No. of cubes   No. of faces
     -------------   ------------   ------------
          6               1               6
          5               6              30
          4              12              48
          3               8              24
          2               0               0
          1               0               0
                                     -----------
                                        108 

And 54 + 108 = 162, so we know that all the faces are accounted for. 

Try other sizes for the large cube and see if you can derive a general 
formula.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/