### Squaring two digit numbers in your head

`24 + 35`I'll see the

`50`first and then the

`9`. This even applies when there's a carry and I'll do something like

`36 + 56`as

`80 + 12`. 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 calculation

`124 + 353`you 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:

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

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

`27`. Of course, the brute force way to do that is to calculate

^{2}`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

`27`. Since you probably know that

^{2}= 30 x 24 + 3^{2}`3`this means you have to calculate

^{2}= 9`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,

`r`, which will be 1, 4, 9, 16 or 25.

^{2}This works because

`( X + r ) x ( X - r ) + r`.

^{2}= X^{2}- rX + rX - r^{2}+ r^{2}= X^{2}Example:

`67`is

^{2}`70 x 64 + 3`which 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.

^{2}The trick is especially impressive/easy with numbers near 100 because the multiplication becomes dead easy. For example,

`96`is

^{2}`100 x 92 + 4`which you (or at least I) can almost instantly see is

^{2}`9216`.

Labels: pseudo-randomness