Given Data

1. A cube is painted on all six outer faces and divided into 7 x 7 x 7 equal cubes.

Question

How many small cubes have no paint?

Answer

125.

Solution

1. Only completely internal cubes have no paint.
2. There are 7 - 2 = 5 internal positions along each dimension.
3. 5 x 5 x 5 = 125.