Question

Two adjacent faces of a cube are coloured and cut into 64 identical small cubes. How many of these small cubes are not coloured at all?

• A. 42
• B. 48
• C. 24
• D. 36

Hint:

When dealing with a cube divided into smaller cubes, the cubes not touching the coloured faces form an inner cube with a side length reduced by 2.

Answer 1

The Correct Option is D

We are given that a cube is divided into 64 smaller identical cubes. The total number of small cubes is 64. The cube is cut into smaller cubes, and each side of the larger cube has 4 small cubes (since \( 4 \times 4 \times 4 = 64 \)). Step 1: Determine the total number of cubes along one edge of the larger cube: Since \( 64 = 4^3 \), the cube has 4 small cubes along each edge. Step 2: Count the total number of small cubes on the two coloured faces. There are 2 adjacent coloured faces of the cube. Each face contains \( 4 \times 4 = 16 \) cubes. Thus, the total number of cubes on the two coloured faces is \( 16 + 16 = 32 \). However, we must account for the small cubes where the two coloured faces overlap. Step 3: Account for the overlapping cubes. The two adjacent faces overlap along the edge where they meet. The edge contains 4 small cubes. So, the total number of cubes counted twice is 4. Step 4: Subtract the overlapping cubes from the total. The number of small cubes on the coloured faces is: \[ 32 - 4 = 28. \] Step 5: Find the number of uncoloured cubes. The total number of small cubes is 64, so the number of small cubes that are not coloured is: \[ 64 - 28 = 36. \] Thus, the number of small cubes that are not coloured is \( \boxed{36} \).