# Secrets of the Mathematical Ninja: Squaring three-digit numbers (Part I)

OK, I’ll admit that this one is difficult: I’m still trying to get good at it myself. Inspired by this TED talk by Art Benjamin, I wondered: could I do that?

The answer is, probably. With a bit of practice. There’s no particular trick to squaring three-digit numbers, except maybe a bit of algebra, namely:

$(a + b)^2 = a^2 + b^2 + 2ab$.

(Don’t forget the +2ab, or I’ll have that ninja sword back off of you right now.)

The trick to squaring three-digit numbers is splitting them up into two numbers whose squares you know and doing the arithmetic in your head quickly. For an easy example, let’s take 81. You can split it up as:

$(80 + 1)^2 = 80^2 + 12 + 2 × 80 × 1$.

That’s obviously 6,400 + 1 + 160 = 6,561. The only two slightly hard bits are the multiplication at the end (not that 2 × 80 × 1 is particularly hard, just that when the numbers get bigger it can be) - and then keeping track of all the numbers you need to add together. Again, that’s not too hard with something like 81, but with three-digit numbers, it’s a pain.

Let’s take another example: $214^2$. Depending on your mood, you can split this up as 200 + 14 or 210 + 4 - I generally go for hundreds where possible, as I seem to screw up slightly less that way. So, 200 + 14 squared is 40,000 + 196 + 5,600, or 45,796. Written down, not so bad; in your head, possibly worse. So here’s how I do it:

- I work out the square of 2. With some mental strain, I get 4, and imagine it two places to the left.
- I work out four times 14, which is 56. That’s under 100, so I can just put it directly after the 4 to get 456. If it was over 100, I’d need to carry the hundreds and add them to the 4. I move the number two places to the left.
- I square the 14 to get 196 and do the same again: here, I carry the hundred on to the 456 to get 457 and write down the rest: 45796. Job done!

That’s enough, if you know your squares up to 99 and are happy enough multiplying biggish numbers by whatever you need. But if you’re good with subtraction, you can do better! More next week…

* Edited 2023-04-04 for formatting reasons.