# Perfect Shuffles

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### Perfect shuffles

By shuffling a deck perfectly, it is well-known that the deck can be returned to its original configuration. We will investigate this phenomenon here. Our main question will be, How many shuffles are require to return a deck of 52 cards to its original configuration. The approach I take here is that of one of my former students in a class project.

## An Easier Problem First

A basic problem-solving strategy is to solve a hard problem by first trying an easier problem. Let's suppose we had only eight cards, numbered 1-8. First let's write out the result of a perfect shuffle. This means that the deck is split exactly in half, and then the cards from each half of the deck are shuffled together so that they alternate. So assuming our deck begins with the cards in ascending order, here is the result of a few shuffles:

    Start:          1 2 3 4   5 6 7 8
First Shuffle:  5 1 6 2   7 3 8 4
Second Shuffle: 7 5 3 1   8 6 4 2
Third shuffle:  8 7 6 5   4 3 2 1


They are now in reversed order! Another three shuffles would return the cards to their original order. We now have some patterns to explore. How can we bring mathematics into the picture?

• Each of the numbers 1-4 is now under the number twice as big. So x goes to 2x works for these 4 numbers.
• If I multiply 5 by 2 I get 10. This is too big - somehow I have to make the numbers stay between 1 and 8.
• Maybe this should be done on a clock (see Clock Arithmetic.) But what size clock?
• Returning to the number 5, 2 times 5 is 10 = 1 on what size clock?
• We need a clock with 9 numbers (work mod 9). So how about the formula that x goes to 2x (mod 9)?
• Trying this for 6: $2(6) = 12\equiv 3$ (mod 9). Check that 6 is under the 3.
• Checking the formula for the other two numbers we see that this formula works to give us the place of each number.

Show how iterating the formula leads to the identity after 6 shuffles.

Now use 2x (mod 53) to find the number of shuffles needed for a full deck.

Find a link to a program to perform shuffles???