Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Iff

"If $T$ is a triangular number, $8T+1$ is a square number."
## You may also like

### Proof Sorter - Quadratic Equation

### Proof Sorter - Geometric Sequence

### Proof Sorter - the Square Root of 2 Is Irrational

Or search by topic

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?**

*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.

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?

Try this interactivity to familiarise yourself with the proof that the square root of 2 is irrational. Sort the steps of the proof into the correct order.