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Shuffling Cards

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


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, 

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
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