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Challenge 116: Card Shuffle

Can you answer these questions about shuffling cards?

A pack of cards is shuffled by putting the second card on top of the first, the third one underneath these, then the fourth on top of the pile, the fifth underneath, and so on until all the cards are used up.

Show that when a pack of 9 cards is repeatedly shuffled in this way, after 4 shuffles every card is back in its original position.

How many shuffles are needed for a pack of 7 cards to return the cards to their original order? Or for a pack of 5 cards?

Do all packs with odd numbers of cards eventually return to their original order? How do you know? What about packs with even numbers of cards?