“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.

### 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.