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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 the area of the kite in the square?


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?

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

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

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.