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### Perfect Shuffle

```Date: 09/09/2002 at 21:19:18
From: Gina
Subject: Introduction to Calculus

If you were to shuffle a deck of cards perfectly, how many repetitions

I attempted to use some type of formula, but to no avail. I also need
to figure out why this is the answer.

```

```
Date: 09/09/2002 at 23:22:26
From: Doctor Peterson
Subject: Re: Introduction to Calculus

Hi, Gina.

I don't know what formula you could use; most math works better if you

First, what does a "perfect shuffle" do? Have you defined it? My guess
is that it would mean you divide the 52 cards exactly in half and
interleave the cards like this: (I'm numbering the cards as they start
out.)

left right   new
hand  hand  order
----  ----  -----
1            1
27     27
2            2
28     28
3            3
29     29
4            4
30     30
5            5
31     31
...
25          25
51     51
26          26
52     52

The first card will always stay the same this way; card 2 moves to
position 3. Since card 3 moves to position 5, the second shuffle will
take card 2 to position 5. Make a list of where it will be after each
shuffle; don't forget it will move to the right half of the deck
eventually, and move a little differently. You might want to make a
formula telling you where card N goes in a shuffle, which will depend
on which half it is in. Once you find out when it will be back in
position 2, you can consider whether all the other cards will be back
in place too.

This is an interesting challenge, so have fun and don't forget to
think about what you discover, and perhaps look for shortcuts to the
answer. This kind of exploration is a lot of what real math is all
about, and there are many intriguing things to see along the way.

Incidentally, I checked our archives to see if we have discussed this
before, and found that I happened to define a shuffle the right way:
if I had defined it a little differently, it would take much longer
to get back to the original position! So you'll want to check how you
are supposed to define it. There's a good chance that my shuffle
would be not considered perfect, since the top and bottom cards
always stay in place.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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