## 7. (Smells like) Team spirit 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 ## The Erdös five card trick 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 What if there is no ascending subsequence of length three? A theorem of Erdös 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!
The above card trick and limerick were inspired by a remark of Martin Gardner's in his For a proof of the Erdös-Szekeres theorem see page 124 of Martin Aigner & Gunter M. Ziegler's Magic or coincidence? You do the math. |
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