(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. [pmath]46^2 = 2116[/pmath]; [pmath]59^2 = 3481[/pmath].
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.
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!