Question Bank • QB_GA_142 NAT • 2 Marks

Section 8: General Aptitude

Analytical & Spatial Aptitude

Subtopic: Spatial Reasoning & 3D Representations

Question Statement:

A solid cube of side $3\text{ cm}$ is painted red on all six outer faces and then cut into 27 unit cubes of side $1\text{ cm}$. How many of the small unit cubes have EXACTLY TWO painted faces?

Answer Key:

12

Step-by-Step Solution & Explanation

For an $n \times n \times n$ painted cube: - 3 faces painted: 8 corner cubes. - 2 faces painted: cubes along the edges excluding corners $= 12 \times (n - 2) = 12 \times (3 - 2) = 12$. - 1 face painted: $6 \times (n - 2)^2 = 6 \times 1 = 6$. - 0 faces painted: $(n - 2)^3 = 1^3 = 1$. Total $= 8 + 12 + 6 + 1 = 27$. Thus exactly 12 cubes have 2 painted faces.
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