# Secrets of the Mathematical Ninja: squares near 50.

Difficulty: **
Impressiveness: ****

(Many thanks to Swar for pointing me at this one - and challenging me to explain it well!)

It's surprisingly easy to square numbers near 50. Here's the recipe:

1. Find the difference between your number and 50. (If you're looking at 46, it'd be -4. If you're looking at 59, it'd be 9).

2. Add this to 25. This would give you 21 (for 46) or 34 (for 59). This is your 'hundreds' number - so you really have 2100 or 3400.

3. Square the number from step 1. For these examples, it's 16 or 81.

Easy peasy! You can go further with it, if you like: to work out [pmath]65^2[/pmath], you could do 25 + 15 = 40 for the hundreds (4000) and [pmath]15^2 = 225[/pmath] to get [pmath]65^2 = 4,225[/pmath] - exactly what you get from the squaring fives routine.

### Why does it work?

Good question. It all comes down to algebra again. Consider [pmath](50 + x) (50 + x)[/pmath]. That multiplies out to: [pmath]2500 + 100x + x^2[/pmath] - which is exactly what the recipe works out, one step at a time! ## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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