Associated Topics || Dr. Math Home || Search Dr. Math

### 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

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.

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/
```
Associated Topics:
High School Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search