Impress your friends with this simple technique: First, note that (50 + x)2 = (25 + x)100 + x2. Given this identity, the technique is almost obvious: take the difference (x) between your number and 50, add 25, then tack on two digits representing x2. The description is not at all clear until you see an example or two:

522 = (25+2) followed by 22 = 2704

592 = (25+9) followed by 92 = 3481

It doesn't matter if the number you're squaring is a bit under 50, either. Just subtract the difference from 25 instead of adding (of course, you're really just adding a negative difference):

432 = (25-7) followed by 72 = 1849

For numbers less than 41 or greater than 59, the "difference squared" term is longer than two digits; but if you're good enough, the same technique can be used to speed up mental squaring of these numbers also. You just have to perform an addition with carry instead of tacking the squared difference onto the end. For example, every computer nerd knows that 32 squared is 1024, so to square 82, instead of (25+32) followed by 1024, which the above procedure would dictate, just do this addition:

Of course, attempting three digit numbers would just be silly.