Cards. There are many possibilities that arise with a deck of cards. S. Brent Morris

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1 Cripe 1 Aaron Cripe Professor Rich Discrete Math 25 April 2005 Cards There are many possibilities that arise with a deck of cards. S. Brent Morris emphasizes a few of those possibilities in his book Magic Tricks, Card Shuffling and Dynamic Computer Memories. One of Morris s main topics of the book was the concept of a perfect shuffle, and how it relates to mathematics. The idea of the perfect shuffle has been around for quite some time now, and Morris also wrote about the progression of it and how it has changed over that years. The first idea of the perfect shuffle came about in the roots of an old card game known as faro. Some American magicians even refer to the perfect shuffle as the faro shuffle. The game of faro was one of the first major gambling games, but now it is very hard to find a casino with the game of faro still being played. The concept of the faro shuffle is that you want to get the cards into winning and losing positions. Suppose we have a standard deck of 52 cards, in order to perform the perfect shuffle you will have to know where every card is at when passed out. To make this example easier lets use eight cards and we will call them a, b, c, d, e, f, g, and h. The winning position would be to have the cards a, b, c, and d, thus the losing position would be to have the cards e, f, g, and h. In order to perform the perfect shuffle, the dealer would want the cards to come out in this order: [a, e, b, f, c, g, d, h]. If the dealer dealt the cards like this he would come out with the cards he wanted. A different version which accomplishes the same thing is

2 Cripe 2 known as the milk shuffle. In the milk shuffle the dealer uses his thumb and fingers to pinch the deck and draw off or milk the top and bottom cards as the rest of the cards are pulled away. The dealer would repeat this step until the entire deck of cards is shuffled, he would end up dealing them out in this order: [d, e, c, f, b, g, a, h]. This deal would result in the same cards as the deal of a perfect shuffle. In 1847 J. H. Green gave the first American idea of the perfect shuffle that was in print, however, he idea was very unclear. A better description of the perfect shuffle was printed in Koschitz s Manuel of Useful Information in In this description he introduces a new term butting-in. If one half of a pack be taken in each hand and their ends pressed slightly together, the cards of one half could be readily be made to enter one over another in the other half, interlapping each other by the momentary springing of the pressure given, a mode of shuffling which, in gambling parlance, is called butting in (Koschitz 27-28). In his description Koschitz gives the reader an idea of how to perform the perfect shuffle. There are two main types of the perfect shuffle, out and in. An out or outshuffle is where the top card of the deck is left out. An in or in-shuffle is where the top card of the deck is shuffled in. For an example we will only use 10 cards, numbered 0 through 9, and we will also represent an out-shuffle with O and an in-shuffle with I. The shuffle of each would look like this: O [0,1,2,3,4,5,6,7,8,9] = [0,5,1,6,2,7,3,8,4,9] I [0,1,2,3,4,5,6,7,8,9] = [5,0,6,1,7,2,8,3,9,4] It is easy to see the similarities of these shuffles, however, after shuffling you can also see that they are exactly opposite. These two ideas have a very strong mathematical

3 Cripe 3 background. For a deck of N= 2n cards, and let p represent the position of the card, the after an out-shuffle the position is moved to O(p) = 2p (mod N-1), where 0 p < N-1. An example would be the number 3 which is the fourth position. N=10 and p = 4 then the fourth position would move to 8 (mod 9). For the same deck N = 2n where p is still position, an in-shuffle would be represented I (p) = 2p + 1 (mod N + 1). An example would with the number 5 in the sixth position, would move to 13 (mod 11). The math changes if the total number of cards in the deck is odd. The definition which is stated by Morris of an out-shuffle is the following: Definition. The out perfect shuffle or out faro shuffle or out-shuffle on a deck of N cards, 0,, N-1, in the permutation that moves the card in position p to position O(p) where O(p) = 2p {(mod N-1) for N even and 0 p < N-1} O(p) = 2p {(mod N) for N odd and all p} The definition of an in-shuffle stated by Morris is the following: Definition. The in perfect shuffle or in faro shuffle or in-shuffle on a deck of N cards, 0,., N-1, is the permutation that moves the card in position p to position I(p) where I(p) = 2p + 1 {(mod N-1) for N even } I(p) = 2p + 1 {(mod N) for N odd p} The concepts of the in-shuffle and the out-shuffle have many similarities, but when shuffled are the exact opposite. The two techniques are very common among magicians and can also be used when performing the perfect shuffle.

4 Cripe 4 Another idea of card shuffling that Morris writes about is The Order of Shuffles. The concept of The Order of Shuffles is asking what is the smallest number of shuffles to return any deck to its original order? The answer to this question is not just a simple one, there is not one answer that satisfies the question on hand. The amount of shuffles that is takes to restore a deck to its original start depends on the amount of cards that you have in the deck. A standard 52 card deck (no joker) can be restored to its original start with eight perfect shuffles. It takes eight perfect shuffles to restore any size deck between 17 and 52. If the deck consists of 53 cards it would take 52 perfect shuffles to restore that deck to its original start. There is not an easy pattern or formula that answers the question of order shuffles. There is a formula, however, which is O^k (p) = 2^k p mod (2n-1), where k is the smallest number of perfect shuffles needed to restore the deck to its original value This equation also has other branches, but those would be to challenging to understand in this paper. In this book Morris explains that however abstract something may seem, there most likely is a mathematical connection to it somewhere. This concept of the perfect shuffle dealing with mathematics is a very challenging one and very difficult for someone of my math background to understand. I find the idea of this concept very interesting, but at my level of mathematics very confusing and something that I would be interested in pursuing in the future. To fully understand the concept of the perfect shuffle and the order of shuffles, one would need extensive background in this field of study. Also in order to perform a perfect shuffle it would take a lot of time and hard work practicing these

5 Cripe 5 techniques over and over again. This is not something just any old person can do by picking up a deck of cards for the first time.