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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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Can you make sense of the three methods to work out what fraction of the total area is shaded?

Iff

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 $1$?

If you're finding it hard to prove the conjecture, you might like to print out these proof sorter cards, and then cut them out and rearrange them to form a proof. Alternatively, you can use this interactive proof sorter.

 

For the second conjecture, if you're finding it hard to prove, here is another set of proof sorter cards for you to print out and rearrange, and another interactive proof sorter.