Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted?
Answer & explanation
Answer: (b) 10
Treating paintings that can be turned into one another as the same, count by the number of black faces: 0, 1, 5 or 6 black faces give one pattern each, while 2, 3 or 4 give two each. Total 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10.
- Count by the number of black faces; rotations of the cube give the same painting.
- 0 black: 1 way. 1 black: 1 way (any face can be turned to the top).
- 2 black: adjacent or opposite → 2 ways.
- 3 black: all three meeting at a corner, or in a band (two opposite faces plus one between) → 2 ways.
- 4 black = 2 white → 2 ways; 5 black → 1 way; 6 black → 1 way.
- Total = 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10.
Remember · For two-colour cube patterns, count by faces of one colour; k and 6 − k faces give the same count.
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). ·