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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

Alison has been playing with numbers again. She started by choosing a triangular number, multiplied it by 8, and added 1. She noticed something interesting about her results...


Try a few examples. Can you make a conjecture?

Once you've made a conjecture of your own, click below to see what Alison noticed:

"If $T$ is a triangular number, $8T+1$ is a square number."

Can you prove the conjecture?

You might like to have a look at this Scrambled Proof and see if you can rearrange it into the original order.

Claire thought that she could use a picture to prove this conjecture. Can you use her picture to create another proof to show that the conjecture is true?

I wonder if there are any integers $k$ where $8k+1$ is a square number but $k$ is not a triangular number...

Can you prove that if $8k+1$ is a square number, $k$ must be a triangular number?

Here is a Scrambled Proof for this conjecture.

Can you use your theorem to devise a quick way to check whether the following numbers are triangular numbers?

  • 6214
  • 3655
  • 7626
  • 8656


The title of this problem, "Iff", is sometimes used by mathematicians as shorthand for "If and Only If", which can also be represented by the double implication arrow $\Longleftrightarrow$. To explore the difference between "If", "Only if" and "Iff", try the problem Iffy Logic.