To calculate the number of small cubes that remain unpainted, we need to determine the number of small cubes on the surface of the big cube.
The big cube has dimensions of 12 mm on each side. Since it is made up of smaller cubes with dimensions of 3 mm, we can divide each side of the big cube by the side length of the small cube to find the number of small cubes along each edge.
12 mm / 3 mm = 4 small cubes
Since there are 4 small cubes along each edge of the big cube, the total number of small cubes on the surface can be found by multiplying the number of small cubes on one face by 6 (since there are 6 faces on a cube).
Number of small cubes on one face = 4 small cubes * 4 small cubes = 16 small cubes
Total number of small cubes on the surface = 16 small cubes * 6 faces = 96 small cubes
Since the big cube is painted on all its sides, the number of small cubes that remain unpainted is the total number of small cubes minus the number of small cubes on the surface.
Number of unpainted small cubes = Total number of small cubes - Number of small cubes on the surface
Number of unpainted small cubes = 64 small cubes - 96 small cubes
Number of unpainted small cubes = -32 small cubes
Since we cannot have a negative number of small cubes, it seems that there is an error in the given information or answer options. Please double-check the question and answer options.