A few months ago, @preshtalwalkar at Mind Your Decisions showed off how he’d advise someone to work out $43 \times 67$ using one of my favourite tricks, the difference of two squares.

In fact, that’s how I’d have approached the question at first, too: the two numbers are 12 either side of 55, which is easy to square, so $43 \times 67 = (55 - 12)(55 + 12) = 3,025 - 144 = 2,881.$ Lovely – except the subtraction needs a bit more thought than I was happy with.

What’s all this?

Oh, hello, sensei, long time no see. How’s the world of Ninja mathematics?

I could tell you. But then I’d have to kill you.

I quite understand. I presume you have a nicer way to work out this sum?

But of course. Multiplying by 67 is a breeze!

For you, perhaps?

Ah! You see, $67 = \frac{201}{3}$.

It is? I mean, yes, yes it is.

And that makes multiplying by 67 easy. $43 \times 201 = 8,643$.

Oh! It’s especially easy for two-digit numbers, you just double it and stick the original number on the end.

I have killed men for less.

I don’t doubt it.

And then divide the result by 3, which is 2,881.

No borrowing, no messing about. Lovely! Thank you, sensei!