Date: 8/26/96 at 22:36:24 From: Anonymous Subject: Perfect Shuffles of Cards Take a standard deck of cards and do a "perfect shuffle." By "perfect shuffle," I mean you take the top half of the deck in your right hand and place the bottom card of the right stack down first, then the bottom card from the left deck, then the new bottom card from the right half, then the left, etc.... How many shuffles, until the cards are back in the order you started?
Date: 10/8/96 at 10:1:30 From: Doctor Ceeks Subject: Re: Perfect Shuffles of Cards Hi, Label the positions of the deck 1 through 52. After one of your perfect shuffles, the original top card will be in the second position, the original second card will be in the fourth position, the original fourth card will be in the eighth position, etc... We can write this process down as follows: ( 1 2 4 8 16 32 11 22 44 35 17 34 15 30 7 14 28 3 6 12 24 48 43 33 13 26 52 51 49 45 37 21 42 31 9 13 36 19 38 23 46 39 25 50 47 41 29 5 10 20 40 27) where the list means that a card originally in position n will be found in position m where m is the next number in the list. If n = 27, the last number in the list, then you cycle back to the beginning. In this way, you can see that it will take 52 of your perfect shuffles to restore the deck. However, there is another way to do a perfect shuffle which leaves the original top card in the same position. This shuffle is called the "out" shuffle, and yours is known as the "in" shuffle. The out shuffle returns the deck to its original order in 8 shuffles. You can read more about these perfect shuffles in the paper: The mathematics of perfect shuffles, by Diaconis, Graham, and Kantor, in Advances in Applied Mathematics, volume 4, 1983. -Doctor Ceeks, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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