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.

4. Add this on to your answer in step 2. ${46}^2=2116$; ${59}^2=3481$.

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

Why does it work?

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