Question

How many smaller cubes have none of their faces coloured?

• A. 0
• B. 1
• C. 2
• D. 4
• E. None of these

Answer 1

Answer: (C)

2

Two cubes are joined together and painted red. Thereafter, one cube is cut into 8 smaller cubes and the other into 27 smaller cubes. The smaller cubes that are at the corners are painted on three of their faces, and there are 8 such cubes.
Since the two cubes are joined face-to-face before painting, one face of each original cube. remains uncoloured. The cube which is cut into 8 equal smaller cubes has 4 smaller cubes painted on two of their faces. The cube which is cut. into 27 equal smaller cubes has $(n-2)\times 8$, i.e., $1\times 8=8$ smaller cubes painted on two of their faces. Further, the cube which is cut into 27 smaller cubes has 4 smaller corner cubes painted red on two of their faces. Thus, there are 4 + 8 + 4 = 16 such cubes which are painted red only on two of their faces.
In respect to the cube which is cut into 27 smaller cubes, there are 2 smaller cubes on the inner side which are not painted red. Thus, the total number of cubes that have at least one of their faces painted red is 27 + 8 - 2 = 33. Out of these 33 smaller cubes, 16 cubes are coloured on two faces and 8 cubes are coloured on three faces. So, the remaining 9 cubes are coloured only on one face.