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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.

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Kite in a Square

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


Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2
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?

If you're finding it hard to prove the conjecture, you might like to print out these proof sorter cards, and then cut them out and rearrange them to form a proof. 

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?

Again, if you're finding it hard to prove, here is another set of proof sorter cards for you to print out and rearrange.

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.