A solid cube is painted yellow, blue and black such that opposite faces are of same colour. The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are big. None of the faces of the bigger cubes is painted blue. How many cubes have only one face painted?

The correct answer is 8.

Let us break down the geometry and the cuts of the cube step-by-step:

Step 1: Determine the size of the original cube and the smaller cubes
The original solid cube is cut into 36 cubes of two different sizes: 32 small cubes and 4 big cubes.
Let the side length of a small cube be 1 unit, and the side length of a big cube be 2 units.
The total volume of these 36 cubes is:
$$\text{Total Volume}=(32\times 1^3)+(4\times 2^3)=32+32=64\text{cubic units}$$
Since the volume of the original solid cube is 64 cubic units, its side length must be:
$$\sqrt[3]{64}=4\text{units}$$

Step 2: Understand the arrangement of the cuts
To get 32 small cubes (size 1) and 4 big cubes (size 2) from a 4 × 4 × 4 cube, the layers must be arranged as follows along one direction (say, the height or Z-axis):

• Bottom layer (height 0 to 1): 16 small cubes of size 1 × 1 × 1.
• Top layer (height 3 to 4): 16 small cubes of size 1 × 1 × 1.
• Middle layer (height 1 to 3): 4 big cubes of size 2 × 2 × 2.

This layout accounts for all 36 cubes (16 + 16 = 32 small cubes, and 4 big cubes) and fits perfectly within the 4 × 4 × 4 dimensions.

Step 3: Determine the colors of the faces
Opposite faces of the original cube are painted the same color (Yellow, Blue, Black). We are given that "None of the faces of the bigger cubes is painted blue".
Since the 4 big cubes are situated in the middle layer (height 1 to 3), they do not touch the top face (height 4) or the bottom face (height 0). They do, however, touch all four lateral side faces.
Therefore, for the big cubes to have no blue paint, the Blue paint must be on the top and bottom faces of the original cube. Consequently, the lateral side faces are painted Yellow and Black.

Step 4: Count the cubes with only one face painted
Cubes with only one face painted must lie on the outer faces of the original cube but cannot touch any edges or corners:

• On the Top Face (Blue): This face consists entirely of small cubes. The inner 2 × 2 grid of small cubes (which do not touch the outer edges of the 4 × 4 face) will have only their top face painted. This gives 4 cubes.
• On the Bottom Face (Blue): Similarly, the inner 2 × 2 grid of small cubes on the bottom face will have only their bottom face painted. This gives 4 cubes.
• On the Lateral Faces (Yellow and Black): Any small or big cube on these sides either touches the top/bottom blue faces, or lies on the corners/edges, meaning they will all have at least two painted faces.

Adding these together, the total number of cubes with only one face painted is: