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USING A CARD TRICK TO TEACH DISCRETE MATHEMATICS 1. Introduction

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The Card Trick. A professor and teaching assistant (TA) enter class and announce that they will begin. class today by performing a magic trick. ...

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USING A CARD TRICK TO TEACH DISCRETE MATHEMATICS 1. IntroductionPage 1USING A CARD TRICK TO TEACH DISCRETE MATHEMATICSSHAI SIMONSON AND TARA S. HOLMAbstract. We present a card trick that can be used to review or teach a variety of topicsin discrete mathematics. We address many subjects including permutations combinationsfunctions graphs depth first search the pigeonhole principle greedy algorithms andconcepts from number theory. Moreover the trick motivates the use of computers inmathematical research. The ultimate solution to the card trick makes use of Halls DistinctRepresentative Theorem.1. IntroductionAn interesting card trick is presented while telling the story of how our discrete mathematics class analyzed attacked and solved some of its mysteries. The trick is a model forengaging students in mathematical research using computers and programming as tools.The presentation is based loosely on what actually transpired in class. We discover theorems refute conjectures verify others and leave work for the future. The story has arecurring theme of making progress just when it seems that the options have been exhausted.1.1. The Trick. The original trick appeared in 1950 in Math Miracles L]. It was inventedby Fitch Cheney. It was largely ignored until 1986 when Art Benjamin showed this trickat the Hampshire College Summer Studies in Mathematics. Several mathematicians havesince analyzed it discovered strategies and performed it around the country K2] M1]M2].The trick makes use of a variety of combinatorial ideas including a neat application ofHalls Theorem (see for example CL] BM] or almost any introductory graph theorytext). It provides a good review for the many concepts discussed in a discrete mathematics course. In addition the mathematics behind the trick is a natural candidate forexperimentation using computers. The trick is also fun and entertaining on its own.1Page 22SHAI SIMONSON AND TARA S. HOLM1.2. The Class. The students involved in this report were from ...

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