We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.
Can you make sense of the three methods to work out the area of the kite in the square?