24 + 35

50

9

36 + 56

80 + 12

124 + 353

124 + 353 = 400 + 24 + 53 = 400 + 70 + 4 + 3 = 477

272

27 x 27

27 x 20 + 27 x 7 = 540 + 189 = 729

272 = 30 x 24 + 32

32 = 9

30 x 24 + 9

3 x 24

X

X + r

X - r

(X + r) x (X - r)

r2

( X + r ) x ( X - r ) + r2 = X2 - rX + rX - r2 + r2 = X2

672

70 x 64 + 32

962

100 x 92 + 42

9216

All my life I've done mental arithmetic the 'wrong way': I've calculated from left-to-right, instead of right-to-left. So when I do something likeI'll see thefirst and then the. This even applies when there's a carry and I'll do something likeas. I do the same thing for multiplication as well.Turns out I'm not so weird after all (well, apart from the finding doing mental arithmetic fun bit). I've been reading Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Mental Math Tricks and the author, Michael Shermer, is just like me: he works from left-to-right.He, like me, has found this to be a good system because it lets you discard digits early and not hold some enormous calculation in your head. For example, in the calculationyou can immediately say "four hundred" before doing the rest of the sum. This seems to free up headspace (at least for me) and let's me do the rest of the calculation. I'd do it like this:The books is filled with tricks for doing all sorts of mental calculations, including a nice section about estimation (I've always been an estimator) and working out things like tips and sales taxes. But the most fun part to me was a trick to let you do two-digit squares in your head really, really fast.Quick, what's. Of course, the brute force way to do that is to calculatewhich is a bit of a pain because it involves doing something like. But there's a much faster way.Observe that. Since you probably know thatthis means you have to calculatewhich is relatively easy because the multiplication involves a multiple of ten which means it's reallyand then add a zero.So the rule is that if you want to square numberyou first round it to the nearest multiple of 10, called that, and then calculate(i.e. round the same amount in the opposite direction). You calculateand add back the square of the amount you rounded by,, which will be 1, 4, 9, 16 or 25.This works becauseExample:iswhich is fairly easy to do in your head. And naturally the same trick works no matter how many digits you have, it's just that the multiplication gets harder.The trick is especially impressive/easy with numbers near 100 because the multiplication becomes dead easy. For example,iswhich you (or at least I) can almost instantly see is

Labels: pseudo-randomness