Card_Trick_Problem_Set - ArsDigita University Month 2...

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ArsDigita University Month 2 – Discrete Mathematics and Probability Professor Shai Simonson The Combinatorial Card Trick – Challenge Problem We showed a method in class based on combinatorics, which allows a magician and an accomplice to perform the following trick. The magician asks someone to choose five cards randomly from a standard deck. The accomplice then looks at these cards, and shows four of them in a particular order to the magician, who then immediately identifies the last card. At first thought the trick seems impossible, because there are only 4! ways to order four cards, and 24 pieces of information is not enough to identify a unique value among the possible 48 cards remaining. However, a more careful analysis reveals that the accomplice has more choices than 24 up his sleeve. The accomplice may choose which four cards to show, as well as their order. There are C(5,4) ways of choosing four cards from five, hence the accomplice has 120 different pieces of information he can send. The hard part is that the magician cannot easily decode which of these 120 was sent. She can easily decode 24 pieces of information, but not so easily the extra factor of five. One method that allows the magician to decode more than the obvious 24 is to notice that among the five cards chose, there must be two of the same suit (due to the pigeonhole principle). The first card shown by the accomplice is one of these two cards, and the second is never shown. If we number the cards in a suit from 1 to 13, and do arithmetic modulo 13, then given any two cards, there is always one which is six or less below the other (again the pigeonhole principle – if they were both 7 or more less than the other, then there needs to be 14 cards). The accomplice chooses the card that is 6 or less below the other (modulo 13). For example, given the 3 and the Jack, we choose the Jack. The accomplice then chooses an ordering of the last three cards to encode a number from 1 to 6. The magician decodes this number, (value 1-6) and adds it to the first card, in order to recover the identity of the missing card.
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Card_Trick_Problem_Set - ArsDigita University Month 2...

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