A cube of 12cm sides is painted red on each side. It is cut into cubes of 3cm side each. How many of the smaller cubes do not have any side painted red?
A. 8
B. 12
C. 16
D. 0
The large cube is cut into 4x4x4 cubelets. The outside faces on both sides have paint, so only the 2x2x2 inside cubelets have no red paint.
To find the number of smaller cubes that do not have any side painted red, we need to determine the number of smaller cubes that have at least one side painted red, and then subtract that from the total number of smaller cubes.
First, let's calculate the total number of smaller cubes. Since the original cube has sides of length 12cm, it consists of 12/3 = 4 smaller cubes along each edge. Therefore, the total number of smaller cubes is 4 * 4 * 4 = 64.
Next, we need to determine the number of smaller cubes that have at least one side painted red. Since each face of the original cube is painted red, the smaller cubes located on the faces of the original cube will have at least one side painted red.
The smaller cubes on the inside of the original cube will not have any side painted red. To determine the number of smaller cubes on each face, we need to subtract 2 from each edge since we are removing the outer layer of smaller cubes.
Each face will have (4-2) * (4-2) = 4 * 4 = 16 smaller cubes that have at least one side painted red.
Since there are six faces on the original cube, the total number of smaller cubes with at least one side painted red is 16 * 6 = 96.
Finally, we subtract the number of smaller cubes with at least one side painted red from the total number of smaller cubes to find the number of smaller cubes that do not have any side painted red.
64 - 96 = -32
Since a negative number of cubes is not possible, it means that all of the smaller cubes will have at least one side painted red.
Therefore, the answer is D. 0.