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?
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