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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.

Kite in a Square

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


Age 14 to 18
Challenge Level

Why do this problem?

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

Proving the statements in both directions (if and only if) will be quite challenging for some students. To help them to develop these skills, we have provided two "proof sorter" activities which can be used to scaffold the task and offer the support that students need in order to complete the problem. 

Possible approach

If students are unfamiliar with triangular numbers, Picturing Triangle Numbers is a good starting point.

"Choose a triangular number, multiply it by 8 and add 1. What did you get?"
Collect a variety of answers on the board, and invite students to comment on anything they notice:
"The answer is always odd."
"The answer is always a square number."

Once they have made a conjecture, there are different routes to a proof. Some students may prefer a visual approach, looking for ways to show the triangular numbers fitting together to make a square. Others may prefer to work algebraically. 

Once students have had a go at proving their conjecture, you may wish to offer them this Proof Sorter, which has the steps of a possible proof mixed up for them to put in the correct order. There is a second Proof Sorter which proves the result that if $8k+1$ is square, $k$ is triangular. This could lead to a discussion about the difference between "If" and "If and only if", which can be explored further in the problem Iffy Logic.

Here are links to interactive proof sorters of the first conjecture and of the converse.

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?

If $8k+1$ is square, does that necessarily mean $k$ is triangular?

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.

Possible extension

Can you prove it? and Sums, Squares and Substantiation has a selection of similar results for which students could try to construct proofs.