Question

A cube whose two adjacent faces are coloured is cut into 64 identical small cubes. How many of these small cubes are not coloured at all?

• A. 24
• B. 36
• C. 48
• D. 60

Hint:

Visual Method: If you remove the two colored "outer skins" (the top slice and the front slice), you are left with a block of $4 \times 3 \times 3$. Calculation: $4 \times 3 \times 3 = 36$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find uncolored cubes, we subtract all cubes that have at least one colored face from the total number of cubes.

Step 2: Detailed Explanation:
1. Total small cubes = 64. 2. Edge length $n = \sqrt[3]{64} = 4$. 3. Imagine the $4 \times 4 \times 4$ grid. 4. Two adjacent faces are colored. Let's say the Top face and the Front face. 5. Total cubes in the Top layer = $4 \times 4 = 16$. 6. Total cubes in the Front layer = $4 \times 4 = 16$. 7. However, the cubes on the edge where the Top and Front faces meet are counted twice. That edge has $n = 4$ cubes. 8. Total colored cubes = $16 \text{ (Top)} + 16 \text{ (Front)} - 4 \text{ (Common Edge)} = 28$. 9. Uncolored cubes = Total cubes - Colored cubes \[ 64 - 28 = 36 \]

Step 3: Final Answer:
There are 36 small cubes that are not colored at all.