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 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/
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