The outer surface of a 4 cm x 4 cm x 4 cm cube is painted completely in red. It is sliced parallel to the faces to yield sixty four 1 cm x 1 cm x 1 cm small cubes. How many small cubes do not have painted faces?

The large cube is 4 cm x 4 cm x 4 cm. When it is sliced into 1 cm x 1 cm x 1 cm small cubes, it means each side of the large cube is divided into 4 smaller segments. So, there are a total of 4 x 4 x 4 = 64 small cubes. The outer surface of the large cube is painted. This means any small cube that is on the surface of the large cube will have at least one painted face. We are looking for small cubes that do not have any painted faces. These are the cubes that are completely internal to the large cube, not touching any of its original outer surfaces. Imagine removing the outermost layer of small cubes from all sides of the large cube. If the original cube has 'n' cubes along each edge (here n=4), then removing the outer layer from both ends of each dimension reduces the effective length by 2. So, the dimensions of the inner unpainted cube will be (n-2) x (n-2) x (n-2). In this case, n = 4. The dimensions of the inner unpainted cube will be (4-2) x (4-2) x (4-2) = 2 x 2 x 2. The number of small cubes that do not have painted faces is 2 x 2 x 2 = 8. Analyzing the options: A) 8: This matches our calculation for the number of unpainted inner cubes. B) 16: This would be the number of cubes with one face painted on an edge (e.g., 12 edges * (4-2) = 24, or faces with 4 cubes each). C) 24: This is often related to cubes with one face painted (6 faces * (4-2)^2 = 6 * 4 = 24). D) 36: This is not directly derived from common cube painting formulas for a 4x4x4 cube. The final answer is A.