A cube 6 units on each side is painted on all sides then cut into smaller cubes by making 2 equally spaced cuts through each of the three sets of parallel faces.  What is the total volume of the smaller cubes with exactly two faces painted?

 

 

A cube with sides of 6 units has a total volume of 216 cubic units.  The cuts result in a cut-up cube like the one shown.

 

 

There will be a total of 27 smaller cubes.  The shaded regions indicate those smaller cubes with exactly two faces painted.  There are 4 on each layer of the original cube for a total of 12 smaller cubes with two painted faces.  We can then set up an equation to find the volume of these.

 

 

The volume of the smaller cubes with two painted faces is 96 cubic units.