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# A big brain, big number trick

A Turing machine manipulates long sequences of 1s and 0s to make calculations. Let’s do it bigger. Get a friend to set their phone to calculator mode, and then multiply together any ten single digits. This will create a really, really big number. The friend should keep this big number a secret so you have no idea what it is. But even though you don’t know the big number, your big brain will be able to spot something missing. Get your friend to read out nine digits of their number, in random order, and to hold one of those digits back. That’s the number you have to figure out. Oh, and to make it more difficult, that number shouldn’t be zero; it’s too easy to predict nothing. Your friend reads out their numbers, and after a dramatic pause you correctly reveal the secret digit they have held back. Your brain’s like a Turing machine, only clearly bigger and better, as it was able to crack the hidden digit code. Or is it all a trick? (Hint: yes it is.)

So how can you spot the missing number from the list your friend gives you? Well, part of it is to do with the fact that you ask them to enter and multiply together ten single digits. Chances are that at some point they will enter a 9, or maybe a 3 and another 3. Which means the chances are high that the final big number will be a multiple of nine, and the trick depends on that. Any number that is a multiple of nine has the amazing property that the sum of the digits is also a multiple of nine. All you need to do is add up the digits your friend tells you one by one. When you get over nine, throw that nine away (it’s called casting out the nines) and keep the number left. For example, let’s say that as you’re adding the digits, you get to 11. Forget the nine and remember the 2, then keep adding. The number in your head at the end, after all the nines have been cast out, is the digit your friend has held back. Reveal it and enjoy the astonishment.

## What if it didn’t work? Do it again and do it better

We bet that since we said this trick depends on entering a 9 or two 3s, you’ve been wondering what happens if your friend doesn’t do what you want. Don’t worry, we’ve got you covered. As long as you make the revealing of the final hidden digit look like a strain, you can always say that mind reading isn’t easy, and that you’ll try again with 15 digits, to make it more of a challenge. Of course with 15 digits they will be even more likely to type a 9 or two 3s, and your final reveal will be even more spectacular.

## A bit of history

“Abjectio novenaria”, which is the Latin way of saying “casting out nines", was known as early as the third century by the Roman bishop Hippolytos. The technique was also used by Hindu mathematicians.