Question

How many smaller cubes have none of their faces green?

• A. 24
• B. 28
• C. 32
• D. 16
• E. None of these

Answer 1

The correct option is C 32

The larger cube is cut into 64 smaller cubes. This implies that the edge of a smaller cube is one-fourth that of the larger cube. We know that the two adjacent faces are painted red and the two faces opposite to these faces are painted black. This implies that the top and the bottom faces are painted green. Thus, there are two edges where the faces painted red and black meet, and each edge has four cubes. Thus, there are totally 8 cubes which have red on one face and black on the other.
All smaller cubes along the edges are painted on two of their faces and the smaller cubes at the corner are painted on three faces. Thus, there are only two smaller cubes on each edge that are painted on two of their adjacent faces. Since the cube has 12 edges, there are totally $(12\times 2)=24$ smaller cubes which are painted on only two of their adjacent faces.
There are $4\times 4=16$ smaller cubes on one face, which are painted black, and $4\times 3=12$ smaller cubes on the adjacent face, which are also painted black. Thus, there are totally 28 smaller cubes which have at least one of their faces black.
The top and the bottom faces of original cube are painted green. There are $4\times 4=16$ smaller cubes on each face. So, there are 32 smaller cubes which have at least one of their faces green. The remaining 64 - 32 = 32 smaller cubes have none of their faces green.