This post is very much not about the problem that stole my weekend:

Show that the sum of the digits of $3^{1000}$ is even, without calculating the sum directly.

That is supposedly from a 2004 Tunisian 10th-grade textbook; my working hypothesis is that it’s a much harder question than they meant to ask (and the greatest minds of r/mathematics have still to come up with a decent solution.)

In tackling it, I had cause to multiply a few things by nine, and either dreamt up or remembered a method that felt worth sharing.

It’s to do with the nine’s complement — number bonds to nine, if you’ve had experience with modern maths teaching, and it’s best to start with an example: suppose I want to multiply 531,441 by nine, because who wouldn’t?

I start by writing down one less than the first digit, 4.

Then I add the nine’s complement of the first digit (5 -> 4) to the actual second digit (3) — 4 + 3 = 7.

Follow the same pattern: add the nine’s complement of the second digit (3->6) to the actual third — 6+1 = 7.

We hit a bit of a snag with the next: we have (1->8) and (4), making 12. That’s fine, it just means we need to handle a carry. I usually write down the 2 and an arrow to remind me to carry. ((Sorry, exchange.))

Keep going: (4->5) and (4) make 9. (4->5) and (1) make 6. And the last digit is just (1->8) and a (1) that comes from nowhere, making 9.

I’ve written down: 4 7 7<-2 9 6 9

This becomes: 4 7 8 2 9 6 9, and $531,441 \times 9 = 4\,782\,969$.

The only tricky bits are remembering to drop down the first digit, bump up the last, and handle the exchanges. It’s significantly less mental load than doing it ‘properly’, even if it is fundamentally the same thing.

Do you have a favourite method of multiplying by 9? I’d love to hear about it!