What are some interesting mental math techniques that you know?

Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number minus $25$ and put that in the first two digits, and then add the square of $50$ minus the number: $$(\text{number}-25)\times100+(50-\text{number})^2$$ For example, to do $47^2$ we have $47-25=22$ for the first two digits and $(50-47)^2=9$ for the last two so we get $47^2=2209$.

Bonus points if you include justification! For this trick, $$100(n-25)+(50-n)^2=100n-2500+2500-100n+n^2=n^2$$