A neat trick for multiplying two numbers sometimes is the identity:
(n+1)(n-1) = n^2 - 1
So, for example, if you need to multiply 15 times 17, you just square 16, and subtract one. 18 times 22 is 396 for the same reason. If the two numbers have an even difference, and you know the square of the number halfway between, it's a quick way to do it and can be impressive.
The calculation trick that I'm pretty good at is calculating the day of the week for any arbitraty date. Say somebody asks what day of the week May 31, 1961 (my birth date) was on.
Start with the year - 1962 was a zero-year (I have a list of these memorized), so you subtract one for 1961, and there were no intervening leap years. So far, -1.
Now you add a number corresponding to the month, which I also have memorized. May is zero, so we still have -1.
Now add the day to it and you get 30, and modulo seven leaves a result of two. Monday is zero, Sunday is six, so I was born on a Wednesday.
I can do this in about five seconds, but if you start out asking someone the year and get that data first (or make an educated guess for the year), you can be calculating the hardest part while you then ask for the month, then the day. It's kind of a magician's trick in that way, but using that distraction technique I can often nail the correct weekday in a second or two. Then I tell people I'm an idiot savant, which they usually half-agree with.