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Proximity

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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Kite in a Square

Can you make sense of the three methods to work out the area of the kite in the square?

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Age 14 to 18 Challenge Level:
If $T$ is the $n^{th}$ triangular number, how could you express $T$ in terms of $n$?

What happens if you multiply that expression by 8 and add 1?

There is an interactive version of each proof sorter here: 
Interactive Proof Sorter
Interactive Proof Sorter (Converse)