# Flipping cards

**Discrete Mathematics**Level pending

You're given a deck of \(99\) cards.

To perform one perfect shuffle, you divide the deck into three parts: the top \(33\) cards, middle \(33\) cards, and the bottom \(33\) cards. Call these \({ P }_{ 1 }\), \({ P }_{ 2 }\), and \({ P }_{ 3 }\) respectively.

To finish the perfect shuffle, you choose a card from the top of \({ P }_{ 3 }\) and make it the first card in a new pile. Then, take a card from the top of \({ P }_{ 2 }\) and place it on top of the new pile, and so on. (This goes in a cycle, so once you get to \({ P }_{ 1 }\) you go back to \({ P }_{ 3 }\))

How many perfect shuffles does it take to get back to the starting arrangement of cards?

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