Question

In a solid cube which is made up of 27 small cubes, two opposite sides are painted red, two painted yellow and the other two with white. How many cubes have two colours?

• A. 8
• B. 12
• C. 16
• D. 24

Hint:

The total number of "2-face painted" cubes is always $12(n-2)$. Even if the colors change, as long as each pair of adjacent faces has different colors, this formula will give you the count of multi-colored cubes.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In a painted cube, different types of small cubes are formed based on their position: - 3 faces painted: Corner cubes. - 2 faces painted: Edge cubes (excluding corners). - 1 face painted: Face center cubes. - 0 faces painted: Inner core cubes.

Step 2: Key Formula or Approach:
Number of cubes with exactly 2 faces painted is given by: \[ 12(n - 2) \] where $n$ is the number of small cubes along one edge.

Step 3: Detailed Explanation:
1. Total small cubes = 27. 2. Edge length $n = \sqrt[3]{27} = 3$. 3. Two-colored cubes are found on the edges of the large cube. A cube has 12 edges. 4. Using the formula: \[ 12(3 - 2) = 12(1) = 12 \] 5. Since opposite faces have the same color and adjacent faces have different colors, every edge cube (excluding corners) will necessarily have two different colors.

Step 4: Final Answer:
There are 12 cubes that have two colors.