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# Prime and Composite

**A great simple card trick involving two spectators is this one. When disclosed, it reminds them of something they should have learned in school… **

*by Franka Miriam Brueckler*

…the difference between prime and composite numbers, which can be a good starting point for a discussion about cryptography or simply interesting fact about primes. Anyway, you need a standard card deck for this, the one with 52 cards, but it can easily be adjusted for a 32-card deck.

Before you perform the trick, asign values to your cards: A is 1, 2 to 10 have the usual value, J is 11, Q is 12 and K is 13. Now, some of the values are prime (2, 3, 5, 7, 11 and 13), some are composite (4, 6, 8, 9, 10 and 12), and there are aces (ones), which are neither.

Note that there are equally many prime and composite valued cards in the deck. Collect all prime ones and shuffle them. Repeat with the composite ones. To get two equal halves of the deck you could remove the aces, but that would easily be spottet and would seem suspicious to the spectators, so it is better to assign two of the aces, say A &diamonds; and A ♠, to the prime half, and the remaining two to the composite group (remember which aces you have put into which half!). After each half is well mixed up, put one on top the other.

Now, you ask two spectators to come out. You may show them the cards, but do not let them shuffle. You say something like “You are two, so it is fair that each gets equally many cards. This is a 52 card deck, so each will get 26.” Then you count the top 26 to the first spectator, and give the other half to the other (note that you known which one received the prime half, and who received the composite half!). Now you ask both to pick one card from their halves, look at it and remember it, and then place the chosen card among the other’s cards (i.e., they exchange two cards). You may let them shuffle their half-packs, and then you ask that they place one half on top of the other.

By going through the full deck, pretending that you are concentrating hard and having some doubts, but in fact just checking for the prime among the composite numbers and vice versa, you can easily discover which cards where exchanged.

Note that the division into primes and composite was irrelevant. It was important to have a distinction of one half from the other that is easy for you to spot, but not for the spectators. For example, if you would just take diamonds and spades in one half, and hearts and clubs in the other, it would be easy to find the exchanged cards, but the principle would be obvious to most spectators. The same would be if you separate the even from the odd values. Our suggestion is one of those which are not so easy to spot, but you are free to find your own – just let it be a mathematically based one 😉

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