When you cut the 4-inch cube into the 1-inch cube, there will be a 4X4 grid on all sides.

So there will be 8 cubes who will be having 3 sides with paint on them.
24 cubes with two painted side
24 cubes with one painted side
8 cubes with no painted side.

Where n > 1, if you cut a painted n-inch cube into 1-inch cubes, you will have:

n^0:  8 * n^0 cubes with 3 sides painted
n^1:  12 * (n – 2)^1 cubes with 2 sides painted
n^2:  6 * (n – 2)^2 cubes with 1 side painted
n^3:  (n – 2)^3 cubes with 0 sides painted

So plugging in 4 for n gives:

8 * 4^0 = 8 * 1 = 8 cubes with 3 sides painted
12 * (4-2)^1 = 12 * 2^1 = 12 * 2 = 24 cubes with two sides painted
6 * (4-2)^2 = 6 * 2^2 = 6 * 4 = 24 cubes with one side painted
(4-2)^3 = 2^3 = 8 cubes with no side painted

The cubes with 3 side paint will be 8 – 1 on every corner with 8 corners in a cube.
The cubes with 2 side painted will be 24. – 2 on each edge with 12 edges in a cube.
The cubes with 1 side painted will be 24- 4 on each face with 6 faces in a cube.
With no paint- 8 cubes-  64 minus (8+ 24+ 24)