Mathematical Card Tricks
The extraordinarily productive and influencial (not to mention eccentric but charming) late mathematician Paul Erdös apparantly never developed any interest in magic, not even the mathematical variety. Yet his own research with fellow Hungarian Gyorgy Szekeres in the 1930s suggests one prediction trick, which we now explore.
Effect: Two volunteers are sought from the audience, following which you turn your back on the proceedings until the very end of the trick. One person is given a deck of cards and invited to shuffle it and then give five cards to the second person. You ask this person to mix the cards thoroughly and hand them back to the first volunteer. The first person, in turn, is instructed to place the cards in a face up row on the table, without changing their order, and then silently indicate which three cards the second volunteer should turn face down. You turn around for the first time since the cards were chosen, survey the scene, and promptly identify all of the face down cards, one by one.
Method: Note the subtle change of language here: we said "volunteer" not "victim." Smell a rat? You're on the right track. Whatever about the second volunteer, the first one is no victim -- (s)he's a confederate of yours. This trick is rather dependent on team spirit, but unlike most magic stunts which use a stooge (i.e., an accomplice), there's no physical or verbal signalling here: the communication is entirely mathematical. Note that your back is turned during the crucial stages, the accomplice does not speak after getting the cards, and the final card handling (the flipping over) is done by the totally innocent second volunteer!
That still leaves a lot to explain. Before we go into the details, we note that if the fake volunteer idea strikes you too unreasonable to try to pull off, your accomplice may reveal up front that you are in cahoots. In that case, the trick may be presented as a ``thought transfer'' or ``mind reading'' stunt, but many questions remain. Even if the audience believes that the five cards used are known to both of you in advance, it's far from clear to them how you could determine the precise order of the three face down cards.
First we consider a simple version of the trick which is too transparent to actually perform, especially if you are upfront about your collusion, but which illustrates the principle. Suppose that the top five cards of the deck at the outset are the Ace, 2, 3, 4 and 5 of any one suit. The first volunteer -- your accomplice -- shuffles the deck in a way that preserves the positions of these five cards. They are then given to the real (second) volunteer, who mixes them and hands them back. Your accomplice how fans the cards from left to right, and checks that there is an increasing subsequence of length three (or more). For instance, if the cards are ordered 2, Ace, 4, 5, 3, we have 2, 4, 5 as the desired subsequence. Following your directions to preserve this order, the cards are placed one by one in a row on the table. Then you indicate that some of the cards must be turned face down. Your accomplice silently asks the other volunteer to flip over the 2, 4 and 5. When you turn around and survey the scene, you see the Ace and 3, so that you not only know which cards are face down, you know what order they are in.
What if there is no ascending subsequence of length three? A theorem of Erdös et al comes to the rescue: guaranteeing the existance of a descending subsequence of length three or more! For instance, if the cards are ordered Ace, 4, 5, 3, 2, then 4, 3, 2 (or 5, 3, 2) is a descending sequence of length three. This time, your accomplice places the cards one by one on the table, while surreptitiously reversing the card order -- there are numerous ways to do this with little risk of being noticed. Proceed as before: the second volunteer is silently asked to flip over the 2, 3 and 4, and you correctly -- and without hesitation -- identify the face down cards.
Of course, the audience may catch on to this if it is performed as above, so to throw people off the scent one should use another ordering of the cards employed, such as alphabetical (Ace, 5, 4, 3, 2) -- although that's not so helpful with these particular five values! Here's a better idea: use any five cards in an order you can easily remember. For instance, we could start with 3 Clubs, J Hearts, 4 Spades, Q Diamonds, and 5 Clubs at the top of the deck, the suits cycling in the usual CHaSeD order. Fan out those cards from left to right, and look at the lower right hand corners: with a little imagination, the upsidedown 3, J, 4, Q, 5 can be seen to resemble a distorted version of the word "Erdös". Now you and your accomplice use the same idea explained earlier. If the cards are in the order 4, 3, Q, J, 5, they are placed in a face up row in the table without change and the 3, J and 5 turned over.
As long as you remember the 3, J, 4, Q, 5 in order, all is well.
If you wish to be able to repeat the trick right away, you should load the deck with ten pre-arranged cards on top: the above five followed by five others whose values have some significance, such 6, Ace, 7, K, 10 - to represent the letters in Palkó, the name Erdös's beloved mother used when addressing her son - or 3, Ace, 4, Ace, 6 -- think of . You you may need to vary the suit cycling.
This trick can also be done with ten pre-chosen cards being given to the real volunteer to mix, in which case four (or more) may be flipped over and correctly identified by you. Here the alphabetical ordering of Ace, 2, 3, ..., 10 within one suit works well, or you can really be ambitious and use ten seemingly random cards which you memorize. Phone numbers such as the AMS's (800) 321 4267 make good mnenonics!
Mathematics & Source:
In each list of ten there is bound,|
To be four that do rise; is that sound?
In a paper with Erdös,
By Gyorgy Szekeres,
A counterexample is found.
The above card trick and limerick were inspired by a remark of Martin Gardner's in his Riddles of the Sphinx and Other Mathematical Puzzle Tales (MAA, 1987) to the effect that: in any ``a row of 10 soldiers, no two of the same height ... no matter what the order, there will always be at least four among the ten, not necessarily standing next to each other, who will be in ascending or in descending order.'' This, as Gardner points out, is a special case of a result of Erdös and Szekeres, which tells us that provided k is the smallest positive integer whose square is at least n, then in any list of n distinct numbers, there will always be at least k that will either be in ascending or in descending order. Above, we met the cases k = 3 and n = 5, and then k = 4 and n = 10. Clearly, the result is really about totally ordered sets, which means that identical card values pose no problem provided we have a total ordering in mind on the set of cards being considered.
For a proof of the Erdös-Szekeres theorem see page 124 of Martin Aigner & Gunter M. Ziegler's Proofs from The Book (Springer, 1998). Eric Weissman's World of Mathematics site lists the statement of a more general version of the theorem.
Bonus points: Consider, once more, the 3 Clubs, J Hearts, 4 Spades, Q Diammonds, 5 Clubs set-up recommended above. This arrangement has a pleasing symmetry to it, and consists of two interesting intertwined increasing sequences. One of these involves the smallest Pythagorean triple, the other an impish Jack (Prince) and a maternal Queen. Many of those who knew Erdös considered him to be a prince among men (as well as among mathematicians), and his legendary affection for his mother was rivalled only by his love for number theory, which has often been described as the Queen of Mathematics. The values 3, J (11), 4, Q(12) and 5 add up to 35, and it was in '35 (plus or minus , where is less than all primes p) that the theorem on which the trick is based was first published.
Magic or coincidence? You do the math.