Painted octahedron
What is the smallest number of colours needed to paint the faces of a regular octahedron so that no adjacent faces are the same colour?
Age
11 to 14
Challenge level
Problem
Image
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?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
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.