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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 
would it take to return to the original order?

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

Please help me. Thank you very much for your time.

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 
think rather than look for a formula. So how can we think about this?

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 

    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 
Associated Topics:
High School Permutations and Combinations

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