### 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.