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Iff

Age 14 to 18
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?

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?

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.

 

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.
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