IMC 2013 Q7
7. The faces of a regular octahedron are to be painted so that no two faces which have an edge in common are painted in the same colour. What is the smallest number of colours required?
- A 2
- B 3
- C 4
- D 6
- E 8
Intermediate Mathematical Challenge 2013, Question 7. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 2
Only two colours are needed for the upper four faces of the octahedron. If, for example, blue and red are used then these four faces may be painted alternately red and blue. Consider now the lower four faces: every face adjacent to an upper blue face may be painted red and every face adjacent to an upper red face may be painted blue. So only two colours are required for the whole octahedron.
Solution © UK Mathematics Trust, from the IMC 2013 solutions. Longer versions with investigations: Solutions and investigations (PDF).