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Each chapter begins with the description of a card trick and ends with its explanation, usually using mathematics developed in the chapter. The book itself is designed as a prop for a trick, but you don't need to use or understand any of its mathematics to do some good magic.
1. The Perfect Shuffle
The Origins of the Perfect Shuffle
The Faro Dealer's Shuffle
A Mathematical Model of the Perfect Shuffle
The Stay-Stack Principle
Trick: "The Seekers," Paul Swinford
2. The Order of Shuffles
The Order of Shuffles
The Product of Shuffles
Moving a Card in a Deck
Trick: "A Spelling Bee"
3. Shuffle Groups
Randomizing a Deck of Cards
Shuffles and Cuts in Even Decks
Shuffles and Cuts in Odd Decks
Out- and In-Shuffles in an Even Deck
Trick: "A Challenge Poker Deal"
4. Generalizing the Perfect Shuffle
Out-Shuffling Several Packets of Cards
Looking for a Neat Formula
Generalized Shuffle Groups
Generalizing the In-Shuffle
Trick: "The Triple Seekers"
5. Dynamic Computer Memories
The Shift-Register Memory
Data Accessing Algorithm for a Shift-Register Memory
The Perfect-Shuffle Memory
The Shift-Shuffle Memory
Details, Details, Details
The Perfect-Shuffle Memory for N=2"
Random Accessing Algorithm for a Perfect-Shuffle Memory
Sequential Accessing in a Perfect-Shuffle Memory of Size N=2"
Properties of Tours
Trick: "Unshuffled" by Paul Getner
Appendix 1. The Order of Shuffles
Appendix 2. How to Do the Faro Shuffle
The Double or Ordinary Faro Shuffle
The Triple Faro Shuffle
Appendix 3. Tours on Decks of Size 8, 16, 32, and 64
Appendix 4. A Lagniappe
The Book of Theorems
The Mathematician, the Psychic, and the Magician
A Constant Function
Bibliography: Selected Perfect Shuffle References
1 The Perfect Shuffle
I was watching a guy kill time by shuffling cards. He cut the deck exactly in half, butted the halves together, and pushed slightly. The cards seemed to jump together, perfectly woven, every-other-one. He saw me watching him and said, "Come over here, and I'll show you a trick."
He had me choose a card and return it to the deck. The magician then reversed two cards side-by-side in the deck, and he asked me to cut the cards a couple of times. He then gave the cards a perfect shuffle, spread the deck, and pointed to a face-down card sandwiched between the two face-up cards. He asked me to turn over the sandwiched card. I did, but I couldn't believe my eyes. It was my card!
How did he do that?
I'm not sure how the idea came up of perfectly interlacing the halves of a deck. Maybe it was an enthusiast of some card game where the even cards won for the house and the odd cards won for the players. With a perfect shuffle, the cards from one half could be made to win for the house or the player, as the shuffler desired. Getting an edge in a card game can be a fierce motivator. Then again, a card player killing time might have decided to see if he could do a perfect shuffle- just for the sheer challenge of doing it. Stranger things have been mastered for no better reason. The simple, descriptive name perfect shuffle characterizes this action nicely.
The Origins of the Perfect Shuffle
The perfect shuffle traces its roots to an old card game, faro, and is still referred to by American magicians as the faro shuffle. The origins of the game of faro are unclear, but by 1726, The whole Art and Mystery of Modern gaming has a chapter devoted to "The Description of a Pharo-Bank, with the Expences and Attendants." [6,i] The game evolved in 18th-century France, and its name supposedly came from one of the cards of the pack at the time bearing the picture of a pharaoh [82, 781]. According to John Scarne, faro was the most popular gambling-house game from shortly after the Louisiana Purchase in 1803 until craps succeeded it in the early 1900s [91, 2670. The legendary lawman Wyatt Earp is said to have wanted to be a faro banker- that is, to run a faro game. Out West the game was advertised with a sign showing a tiger, and playing against a faro bank was known as "bucking the tiger." Its popularity has faded to the points that it is now virtually unknown, except at a few casinos trying to achieve an "old west flavor" by offering "old time" games.
Faro is a simple game with little strategy, that allows players to lose slowly, perhaps even gracefully. Players bet on any of the thirteen values to win (suits don't count). In some versions players could bet on a card to lose by placing a small copper token on the wager, but we won't complicate our analysis by allowing this variation. (The copper tokens are thought to be the origin of the expression "copper a bet" or "cop a bet") Bets on multiple cards are placed by putting the wager on the layout between cards (e.g., K-Q, 5-6, or 3-J) or in the center of four cards (e.g., 5-6-8-9 or 3-4-10-J). After bets are placed, the deck is shuffled and put face-up in a "shoe" (a box that allows cards to be dealt one at a time). See figure 1.
At the start of a game, the first card in the shoe, seen by everyone, is called "soda" and doesn't count in play. Cards are then dealt in pairs; the first revealed is a loser and the second a winner. Soda is dealt aside to start a "winning" pile. The next card is a loser and is immediately death into a "losing" pile. The third card, now exposed in the shoe is a winner. This completes a "turn"- one loser and one winner, and bets are settled. Now bets are made, and another pair of cards is dealt. (The first card of the next pair dealt is the previous winner, and it is placed on the winning pile.)
Bets on the winning card are paid at 1 to 1. Bets on the losing cards are lost; bets on unplayed cars must remain (though sometimes house rules allow these bets to be moved). Bets on multiple cards are paid proportionately, e.g. a bet of 1 on K-Q is treated like one-half on K and one-half on Q, and if the winning and losing cards are the same, a "split," the house takes half the wagers on that card. Splits are what give the house some of its advantage, and we'll see later how dishonest dealers contrived to create splits with the perfect shuffle. "case-keeper" or counting frame (usually thirteen wires, each with four beads) is used to keep track of the cars as they are played. A fully equipped faro bank requires a betting layout, a deck, a dealing shoe, and a case-keeper.
Let's follow the start of a simple game of faro, shown in figure 2. A shuffled deck is in the shoe, with the 8 of clubs as soda. The case-keeper records the one card displayed. Three bets have been place: two chips on any of the sixes, one chip on the twos, and one chip on the jacks.