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?

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 that if $8n+1$ is a square number, $n$

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.