Age 14 to 16
Challenge Level

We have to prove the statement:

If a regular icosahedron has three red vertices then it has a vertex that has at least two red neighbours.

We can use an argument by contradiction. We suppose that NO vertex has more than one red neighbour and reach a contradiction thus showing that this statement must be false

Without loss of generality you can think about the top vertex being red and decide what that means for the other vertices around it.