A solid cube is painted yellow, blue and black such that opposite faces are of same colour. The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are big. None of the faces of the bigger cubes is painted blue. How many cubes have only one face painted?
Answer & explanation
Answer: (c) 8
The only cut that works is a 4 × 4 × 4 cube with four 2 × 2 × 2 blocks. To keep the big blocks off both blue faces they must fill the middle two layers, leaving 16 small cubes in the top layer and 16 in the bottom. Only the 4 central cubes of each of these layers have a single painted face.
- 36 cubes of two sizes fit a 4 × 4 × 4 cube: 4 big cubes of 2 × 2 × 2 (4 × 8 = 32 units) and 32 unit cubes; 32 + 32 = 64.
- Let the two blue faces be the top and the bottom. A big cube is 2 units tall, so it avoids both only if it sits in the middle two layers.
- The 4 big cubes fill the middle 4 × 4 × 2 block; the top and bottom 4 × 4 × 1 layers hold 16 + 16 = 32 small cubes.
- Each big cube stands at a corner of the middle block and shows one yellow and one black face — two painted faces.
- In the top layer, 4 corner cubes have 3 painted faces, 8 edge cubes have 2, and the 4 central cubes have only the blue top painted. The bottom layer is the same.
- Cubes with exactly one face painted = 4 + 4 = 8.
Remember · Fix the cut first, place the pieces with the restriction, then count painted faces layer by layer.
Question and answer: UPSC's official GS Paper II (2019, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·