Draft:Bpow44

Take a deck of 52 playing cards consecutively numbered from 1 to 52. A perfect shuffle occurs when you divide the deck in two, one half in each hand, and riffle them together in an alternating fashion; after the shuffle the cards in the deck will be ordered as follows:

1, 27, 2, 28, 3, 29, ... , 24, 50, 25, 51, 26, 52

Notice that the top card always stays on top and the bottom card always stay on the bottom. Imagine now that you repeatedly shuffle the cards, performing a perfect shuffle every time. How many times will you have to do this before all the cards in the deck are restored to their original position?

(You may want to start by convincing yourself that, indeed, the deck will return to its original position eventually.)

For the more adventurous: What happens if you put in the two joker cards and perform perfect shuffles with 54 cards; that is, how many times will you have to perform perfect shuffles with a deck of 54 cards before all the cards return to their original position?

For the true daredevil: What happens with a deck of n cards?

Problem by Alberto L Delgado, from the now extinct Bradley Problem of the Week.