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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:
 

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