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How to be a mathemagician

The seven and two of spadesFind out about a few simple magic tricks that reveal some of the hidden maths that makes the world go round.

Make the magic do itself

If you’ve never been able to master forcing a card on someone or tucking one into their pocket without them noticing, you might want to try ‘self-working tricks’. These are tricks that ‘just work’ as long as you follow the instructions, because they depend on maths rather than sleight of hand.

There are lots of different tricks you can do like this, but learning the steps is only half of it - there’s also the puzzle of working out what makes them tick. Take a look at some of these card tricks and see if you can work out why they work.

Predict the future

This requires a bit more effort, some quick mental arithmetic, and a barcode. It lets you predict the last number of a barcode from the rest.

Here’s what you have to do:

  • Get a friend to find a barcode 12 numbers long on something they’ve got with them, like a water bottle, and read out the numbers one by one.
  • Add the numbers together as you hear them, but for the first, third, fifth and so on, multiply them by three first. This might take a bit of practice!
  • Once you’ve got all the numbers, the last number will be whatever you have to add to the total to round it up to the next 10. So if the total was 62, the last number will be 8.

Once you’ve got the idea down, there are a few ways to make it easier. For example, because only the last number of the total matters when you round up at the end, you only have to remember the last number as you work it out.

This works because the last digit of a barcode isn’t there to give you information like the others: it’s a ‘check digit’. When the barcode is scanned, the computer does the same trick you just learned and compares the result to the actual last digit. If the two don’t match up, it can tell that the barcode didn’t scan properly. Making the process more complicated than just adding up the numbers helps to spot more errors. For example, if you just added up the numbers, you wouldn’t be able to tell if you had got two of them in the wrong order.

Create a world first

This one isn’t quite a magic trick - in fact, you do it every time you shuffle a pack of cards. But it is an easy way to claim a world first.

When you pick a card at random, each card has a 1 in 52 chance of being picked. If you put that card to one side and take another, each of the cards that are left has a 1 in 51 chance. You can work out the odds of picking exactly those cards in that order by multiplying those numbers together: it’s 1 in 2652.

So what if you keep going? If you carry on all the way through the pack, you’ll eventually put the cards in a random order - just like you would if you shuffled them. This means that the odds of getting any single order of cards when you shuffle is 1 in 52 × 51 × 50 × … × 3 × 2 × 1. So what does that come out as?

It comes out as 1 in 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,
000,000,000,000.

If you can’t wrap your head around how big that number is, don’t worry - nobody could. It means that if you could shuffle a deck of cards in a second, and you had been doing it since the beginning of the universe, you wouldn’t even be halfway to seeing all the possible orders the cards could be in. You wouldn’t even be a billionth of a billionth of the way there.

More usefully, it means that every time you shuffle a pack of cards you can be almost certain that nobody in the world has ever seen a pack of cards in that order before.

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