This problem is a good way to explore properties of the Platonic solids and gives children opportunities to visualise 3D shapes.

You might like to get children making their own Platonic solids using sheets of paper before trying this activity. This article shows you how to go about it.

Alternatively (or in addition) it would be useful to have some Polydron available for children to build their own solids as they work.

You could introduce the problem by asking children to visualise the cube, asking for justifications of the minimum number of colours needed before they are able to physically make it to check.

How many faces/edges/vertices does this shape have?

How many other faces does each face touch?