Proximity

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

Kite in a Square

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

Iff

Why do this problem?

This problem gives an opportunity to try something out, make a conjecture, and then prove it. There are multiple approaches to a proof, both algebraic and visual, which can lead to fruitful discussion on different methods of proof.

Possible approach

If students are unfamiliar with triangular numbers, some preliminary work would be useful. This could include representing triangular numbers diagrammatically, and deriving the formula for the nth triangular number. The problem Picturing Triangle Numbers is a good starting point.

Explain that the task is about what we notice when we multiply a triangular number by 8 and add 1. Give students some time in small groups to try a variety of triangular numbers, and then discuss as a class or in their groups anything they notice about their answers.

Once they have made a conjecture, there are different routes to a proof. Some students may prefer a visual approach as suggested in the hint, and others may want to work algebraically. For students who find it hard to construct a proof, a proof sorting activity is available here which allows them to put in the correct order statements which prove the conjecture. Alternatively, the statements could be used as a card sort, available to print out here.

If different groups prove the result in different ways, they could present their proofs to other groups at the end of the session. This is a good opportunity for students to try to understand someone else's mathematical reasoning and be critical of the logical steps.

Key questions

What do you notice when you work out 8T+1 for a triangular number T?
Will this always happen?
Can you prove it will always happen?

Possible extension

Prove the statement the other way round, that is if a square number can be written in the form 8n+1 then n must be a triangular number.

Possible support

The problem Picturing Triangular Numbers is a good introduction to visual proof, and can be used as a foundation to a pictorial proof of this result.