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.