A Mathematical Card Trick

Card tricks fall into several categories, and my favourite are those which look amazing but actually have some pretty simple maths behind them. Here’s one of the simplest!

1.     Take an ordinary pack of cards and shuffle it. Deal out 26 cards, face up, and remember the 7th card.

2.    When you’ve dealt out the 26 cards, pick up the pile and turn them face down on the table, to one side. (If you’re brave, shuffle them first, but make sure that you leave the top 7 in order. This takes a bit of practice.

3.     The next bit you can either do yourself, or give to someone else to do (just to show you’re not doing some sleight of hand). Using the cards left in your hand, deal out one card face up. Then deal more cards face down on top of it to make the total 10. In other words, if the card was a 6, deal 4 more. If the card was a 10, you don’t need to deal any more. For this trick, Jack, Queen and King count as 10.

4.     Now repeat step 3 twice more. This is what it might look like at this stage.

5.     Add up the total of the three face up cards. In my photo, this would be 6 + 4 + Q = 6 + 4 + 10 = 20.

6.     Put the remaining cards on top of the 26 you dealt earlier, and tell your audience that the 20th card (or whatever the total was) will be the …. (name the 7th card from step 1). Carefully deal them out – and your prediction comes true!

Why does it work? See if you can work it out before looking at my explanation below.


It works because of the link between the total of the three cards, the number of extra cards you dealt, and the number left in your hand. Suppose the first number dealt was a: you then deal 10 – a cards on top of it. If the other numbers were b and c, then the total cards you have dealt out will be (10 – a) + (10 – b) + (10 – c) = 30 + a – b – c. But you also dealt out the three face up cards, so the number of cards which you have dealt is 33 – a – b – c. You had 26 in your hand, so the number left in your hand is 26 – (33 – a – b – c) = a + b + c – 7. (Can you see where this is going?) 

So if you now deal out a + b + c cards from your hand, you will need another 7 to make up the total, and these are the 7 from the other half pack.

In my example, we must have 33 – 20 = 13 cards in hand. So to deal out 20, we need another 7 from the original half pack.


It always looks spectacular! You can vary the trick in all sorts of ways. For example, start by looking through the pack and seeing what the eighth card is, then take out the card with the same number and colour and put it in an envelope. (For example, if the seventh card is the 8 of spades, take out the 8 of clubs.) Shuffle the pack, taking care to leave the top 8 in order. Now do the trick as normal, and reveal the two cards at the end. (Why the eighth card and not the seventh?)

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