You’ve got a 10 x 10 x 10 cube made up of 1 x 1 x 1 smaller cubes. The outside of the larger cube is completely painted red. On how many of the smaller cubes is there any red paint?

First, note that the larger cube is made up of 1000 smaller cubes.  The easiest way to think about this is how many cubes are NOT painted?  8 x 8 x 8 inner cubes are not painted which equals 512 cubes.  Therefore, 1000 – 512 = 488 cubes that have some paint.  Alternatively, we can calculate this by saying that two 10 x 10 sides are painted (200) plus two 10 x 8 sides (160) plus two 8 x 8 sides (128).  200 + 160 + 128 = 488.