Square Large Numbers in Your Head Quickly

This just blew my mind:

[W]hat’s 27 [squared]? Of course, the brute force way to do that is to calculate 27 x 27 which is a bit of a pain because it involves doing something like 27 x 20 + 27 x 7 = 540 + 189 = 729. But there’s a much faster way.

Observe that 272 = 30 x 24 + 32. Since you probably know that 32 = 9 this means you have to calculate 30 x 24 + 9 which is relatively easy because the multiplication involves a multiple of ten which means it’s really 3 x 24 and then add a zero.

So the rule is that if you want to square number X you first round it to the nearest multiple of 10, called that X + r, and then calculate X – r (i.e. round the same amount in the opposite direction). You calculate (X + r) x (X – r) and add back the square of the amount you rounded by, r2, which will be 1, 4, 9, 16 or 25.

I convinced myself with a simpler example: 12 squared. Round down to ten and an equal amount up to 14; square the amount you rounded (2) and add it to the product: 10*14 + 2^2 = 144. Cool…