A rectangular prism has dimensions of 4 by 4 by 5. The cube is painted on all faces. Then the block is cut into eighty 1 by 1 by 1 cubes. How many of these cubes have no paint on any face?
the volume of the prism = 4 * 4 * 5 = 80 cubic units
Imagine a shell that surrounds the prism that has a thickness of 1 cubic unit. All of the cubes in this shell have paint on them. All of the cubes inside the shell have no paint on them.
the volume of the prism inside the shell = (4 - 2) * (4 - 2) * (5 - 2) = 2 * 2 * 3 = 12 cubic units
So...there are 12 cubes without any paint on them.