Short Answer 
There are 64 small cubes in a big cube arranged in a 4x4x4 formation. Only 24 small cubes on the surface, not including the edges, have exactly one painted face.
Step 1: Understand the Structure of the Big Cube
The big cube consists of 64 small cubes, arranged in a 4x4x4 formation. Each of the six faces of the big cube has equally divided small cubes, which creates a layered structure. When painting the cube, only some small cubes are affected based on their position on the cube’s surface.
Step 2: Identify Cubes with One Painted Face
To determine which small cubes have exactly one face painted, we note that only those on the surface of the cube, but not along the edges, qualify. The following conditions highlight this:
- Each face of the cube consists of 16 small cubes arranged in 4 rows and 4 columns.
- Excluding the edges on each face, there are only 4 inner cubes per face that have just one painted side.
- With 6 faces on the big cube, you’ll find 24 cubes in total that meet the criteria (6 faces x 4 inner cubes per face).
Step 3: Conclusion and Final Count
After calculating the cubes with exactly one painted face, we conclude that there are 24 small cubes out of the total 64 small cubes that possess this characteristic. This result confirms that only the non-edge cubes on the surface are counted in this total. Therefore, the small cubes with one painted face total to 24.