“You would not be certain that $17 \times 24$ is not 568.” - Daniel Kahneman, Thinking Fast And Slow

Thanks to Alice for pointing out that yes, she bloody well would.

Most people under 50 in the UK would reach for a calculator, or possibly a pen and paper to work out $17 \times 24$. It’s not the world’s trickiest sum, but at the same time, you probably never chanted your 17 times tables, no matter when you went to school.

The answer, as it happens, is 408 - but that’s not the point of this article; instead, it’s about how you know it’s not 568.

1. Estimation

The first, most obvious thing: 17 is a bit less than 20. 24 is a similar amount more than 20. That means $17 \times 24$ is somewhere in the region of $20 \times 20 = 400$. It’s definitely not off by 150 or more.

2. Factorisation rules

I recently saw a ‘counter-example’ to Fermat’s Last Theorem, that claimed $3987^{12} + 4365^{12} = 4472^{12}$. Both work out to $6.397~665~635 \times 10^{43}$ - however, they’re clearly not equal. Why not? The digits in the first term add up to 27, which mean it’s a multiple of 9. The digits in the second add up to 18, which mean it’s also a multiple of 9. In fact, they’re both multiples of $9^{12}$ - and adding them together gives you something that’s also a multiple of 9. The right hand side, however, has digits that add up to 17 - which means it’s not even a multiple of 3, let alone 9.

You can do a similar trick with Kahneman’s example: 24 is a multiple of 3, so $17 \times 24$ is as well. However, the digits of 568 add up to 19, which isn’t a multiple of 3. ((Note: this trick only works for multiples of 3 and 9.))

3. The Last Digit Test

This example actually passes the Last Digit test, but it’s a good one to use if you’re trying to narrow down answers: if you multiply the last digits together, you get $7 \times 4 = 28$ - which means the last digit of the big sum ought to be 8 (and, in this case, it is). If it hadn’t worked out, you’d know for sure it was wrong - although, as you can see, the last digit test being right doesn’t guarantee that the big sum is right ((If you like, it’s a necessary but not sufficient condition.)) .

4. Seventeens

Like I say, you don’t chant your 17-times table, so - unless you’re the Mathematical Ninja - you probably don’t know that $6 \times 17 = 102$. ((If you play darts, you might know that treble-17 is 51, though.)) That means you can work out $17 \times 24$ in a flash, because 24 is $6 \times 4$.

$17 \times 6 \times 4 = 102 \times 4 = 408$.

Kahneman is probably right, 99% of the time, that his readers won’t immediately spot that $17 \times 24$ isn’t 568. It could just be that his readers aren’t as smart as mine.