EMAT6690

Card Shuffling

by

Doug Westmoreland, Brad Simmons, and Kim Burrell

Card shuffling has intrigued gamblers and well as magicians for years. The perfect shuffle has roots back to a very old card game called faro.  It is stilled referred today to magicians as the faro shuffle. The faro shuffle is one in which the cards are cut into two equal stacks and then the cards are dropped one at a time and alternately from from each stack. The deck must, if it contains an even number of cards, be divided exactly in half before the shuffle begins. The deck must be cut as nearly in half as possible if it contains an odd number of cards.  With odd decks the smaller half (one card fewer) shuffles into the larger one so that the top and bottom cards of the larger half become the top and bottom cards of the deck after the faro shuffle is completed. With even decks you have a choice of dropping the first the bottom card of either half. If  the first card to fall is from the half that was formerly the bottom of the deck, the cards previously at the top and the bottom of the deck will remain at the top and bottom.  Magicians call this faro shuffle the "out-faro" because the top and bottom cards remain on the outside.  If the first card to fall is from what was formerly the top half of the deck, the former top and bottom cards go into the deck to positions second from the top and bottom.  Magicians call this the "in-faro."

For odd decks, a faro is an out-faro if it is cut below the center card.  This places the top card on the larger half, with the result that it remains the deck's top card after the shuffle.  The faro is an in-faro if it is cut above the center card.  This places the top card on the smaller half, with the result that it becomes the second card after the shuffle.  Both in-faro and out-faro shuffles of odd decks are called straddle shuffles.  This term was named by Edward Marlo, a Chicago card expert who has written several books on the faro shuffle.

A deck of n cards, given a repeated series of faro shuffles of the same type, will return to its original order after a finite number of shuffles.  If n is odd, the deck returns to its original state after x shuffles, where x is the exponent of 2 in the formula 2^x = 1 (modulo n).  "1 (mod n) means that the number has a remainder of 1 when divided by n.  For example, if a joker is added to a full deck, making it 53 cards, the formula becomes 2^x = 1 (mod 53). We must find an integral value of x such that 2^x has a remainder of 1 when divided by 53.  If we go up the ladder of the powers of 2 (2,4,8,16,32....), we do not reach a number that is 1 (mod 53) until we come to 2^52.  This tells us that 52 in-faros are required to restore the order of a 53 card deck.

If the deck is even, the situation is a bit more complicated.  The number of out-faro shuffles needed to restore the original order is 2^x (mod (n-1)).  The number of in-faro shuffles that does the trick is 2^x (mod (n+1)).  This sometimes makes a big difference.  For the normal pack of 52 cards, 52 in-faro shuffles restore order.  However, 2^8 = 1 (mod 51), so that is only eight out-faro shuffles needed to restore the deck.

The chart below gives the number of faro shuffles of both types required to restore the order of a deck of any size from two cards to 52 cards.  Note that for an odd deck the number is always the same for either type of shuffle, and equal to the number of out-faro shuffles required for a deck of one more card.  For an even deck the number of out-faro shuffles is the same as the number of in-faro shuffles for a deck of two cards fewer.  This reflects the fact that the top and bottom cards are never disturbed during out-faro shuffles and so you are in effect in-faro shuffling the rest of the deck.

.....No. of Cards......Out-Faros.........In-Faros.....

 2 1 2 3 2 2 4 2 4 5 4 4 6 4 3 7 3 3 8 3 6 9 6 6 10 6 10 11 10 10 12 10 12 13 12 12 14 12 4 15 4 4 16 4 8 17 8 8 18 8 18 19 18 18 20 118 6 21 6 6 22 6 11 23 11 11 24 11 20 25 20 20 26 20 18 27 18 18 28 12 28 29 28 28 30 28 5 31 5 5 32 5 10 33 10 10 34 10 12 35 12 12 36 12 36 37 36 36 38 36 12 39 12 12 40 12 20 41 20 20 42 20 14 43 14 14 44 14 12 45 12 12 46 12 23 47 23 23 48 23 21 49 21 21 50 21 8 51 8 8 52 8 52

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