A painted cube cut up — Aptitude Question
A painted cube cut up: A cube is painted on all faces and then cut into 27 smaller cubes of equal size. Answer and worked solution on CodeKairo.
- Topic: Mathematical Reasoning
- Section: Logical Reasoning
- Difficulty: Hard
- Time target: 120 seconds
Question
A cube is painted on all faces and then cut into 27 smaller cubes of equal size. How many small cubes have exactly two faces painted?
Options
- 6
- 8
- 12
- 18
Show the answer and the worked solution
Correct answer: C. 12
Worked solution
- 27 cubes form a 3 × 3 × 3 block.
- Cubes with exactly two painted faces sit along the edges, excluding the corners.
- A cube has 12 edges, and each contributes 3 − 2 = 1 such small cube.
- Total = 12.
Answer: 12.
The idea
For an n × n × n painted cube: 8 corners have three faces painted, 12(n − 2) edge cubes have two, 6(n − 2)² face cubes have one, and (n − 2)³ have none.
Shortcut. In a 3 × 3 × 3 cut each of the cube's 12 edges has exactly one middle cube, and those are the two-face cubes: 12.
Common trap. "8" — counting the corner cubes, which have three painted faces, not two.