by Bill Graham

Suppose you have a large number of small 1 inch cubes that you stack to make larger cubes. You could make larger cubes which measured 2 inches on a side (made with eight small cubes) or 3 inches on a side, etc. Suppose you painted the exterior surfaces of these various larger cubes. Some of the small cubes would have paint on one face, some on two faces, some on three faces or possibly, no paint on them at all.

1. How many small cubes are needed to make this larger cube?
2. How many cubes have no paint at all?
3. How many cubes have paint on one face?
4. How many cubes have paint on two faces?
5. How many cubes have paint on three faces?

Answer the same questions for a cube which measures 3 in. by 3 in. by 3 in.

Answer the same questions for a 4 inch cube.

Write a rule or a formula for a cube which measures n by n by n.