I suppose that a pack of cards is the most important prop for a populariser of mathematics. Or a mathematics teacher. Or a mathematics enthusiast. Or ... Playing cards can be used to teach numbers (obviously), introduce variables (or what else is a joker), learn basic logic skills (through various card games) to explain some basic probability principles (I guess there is no need to explain this?!), ... In this article we shall describe a magical trick with cards founded on elementary algebra.
There are several versions of this trick, but the basic one, as described in Martin Gardner's "Mathematics, Magic and Mistery" under the name "A Baffling Prediction", works best in public - at least to this article author's experience. Let me, the author, be your performer, and you, the reader, the spectator willing to participate in the trick!
I give you a deck of cards - a standard one, with 52 cards - to shuffle. I ask you to deal any 12 cards you choose face-up, and I will hold the rest of the pack while you arrange them.
Twelve cards are dealt face open on the table.
"Please, select four of them and leave them on the table, and return the other eight to me", I say. You do as I say. "Now, let us agree on the card values. If you want to suggest some, please do! No? O.K., say we use the standard values: cards with numerals printed on them have the corresponding value, aces have value 1, and jacks, queens and kings have value 10. Is this O. K. with you? Yes? Fine." Now I return the pack of cards to you.
Four cards are left on the table.
"Well?", you ask. "Well," I say, "now deal from the pack onto each of the face-up cards as many cards from the pack as is the difference between 10 and the card value." "Face up?", you ask. "I don't care, just take care to count correctly", is my answer. So, you deal 7 cards on the three of hearts, one card on the nine of hearts, 6 cards on the four of diamonds and no cards on the king of clubs. And you still have some cards left in your hand.
Onto each card as many cards are dealt as is the value of the difference between 10 and the card value.
"Now, I will concentrate!", is my line and you see me closing my eyes, putting my thinking-magician's-hat on. "Yes, I have it!" "You have what?", you ask. "I think I am able to see the twenty sixth card in the pack in your hand! Please, find it!" "Holding the pack face-up or face-down?" "Down, if you please." And you do as you have been told to do ... And I say: "I see... something red ... and a number with two digits ... and a man with a hat ... he looks somehow sleepy ... It's Robert Mitchum, i. e. the 10 of diamonds, isn't it?" And you are surprised, or at least you act as if you were :-), because I am right!
The performer knows the 26th card if the cards have been chosen as shown in the previous pictures.
Now, dear reader, you ask yourself, what has happened? First of all, remember the time you have returned the pack of cards to me? Yes? The 10 of diamonds with the picture of actor Robert Mitchum was at the bottom, and I had plenty of time to see that while you were choosing the four cards. But this it the only unfair thing I've done - I'm a mathematician, and I don't cheat (I let maths work for me!).
Does the previous information help you to find how I've known which number to name if I want to find the 10 of diamonds? No? Then read on! We had 52 cards in the beginning, thus you have returned to me 52 - 12 = 40 cards in the first step. So, the 10 of diamonds was the 40th card from the top in that pack. You have returned 8 cards to me, I have put them below the pack, so the Robert Mitchum is still at position 40. Everything clear?
You still don't see? But, your open cards were the 3 of hearts, 9 of hearts, 4 of diamonds, king of clubs, 26 in total value. Does this help?
No? Then I have to teach you some algebra. You have four cards open, with values a, b, c and d. According to my instructions, you place onto them 10 - a + 10 - b + 10 - c + 10 - d cards, i. e. 40 - (a+b+c+d). That many cards are still in the pack, and the 10 of diamonds (or whatever card I've seen when you returned the pack to me) is at the bottom. Independently on how we agreed on the values a, b, c, d, this card is now on the position a+b+c+d from the top, so all I had to do is to add the agreed-upon values of the four open cards to tell you where the queen of hearts is!