When the big cube is cut into smaller cubes, the cubes on the interior will have some of their faces touching the faces of other cubes. Therefore, only the cubes on the surface will have faces painted.
The big cube has 6 faces. When it is cut into 27 smaller cubes, each face of the big cube is divided into 9 smaller faces (3 x 3). Thus, there are 6 x 9 = 54 smaller faces in total.
Since each smaller cube has 6 faces, there are 27 x 6 = 162 smaller cube faces in total.
Now, we need to count the number of smaller cubes that have no face painted. These are the cubes that are completely on the inside of the larger cube and have no face on the surface.
Since the small cubes are all of the same size, the smaller cubes that have no face painted are those that are completely surrounded by other small cubes. There are 3 x 3 x 3 = 27 small cubes that are completely surrounded by other small cubes.
Therefore, there are 162 - 27 = 135 small cubes that have at least one face painted.