Perfect shuffle of 52 cards

Question

Prove: How many perfect shuffles of a deck of 52 cards do you need to do until the deck returns to its original order?

Can anyone please help me prove this?

Attempt: I have tried putting the deck of cards as a 2 line permutation .

(1 2 3 4 .....52) then by shuffling the first time, card 2 will be in position 1 and card 27 will be in position 1. So, (2 4 6 8 ......1 3 ......51). However, I don't know how to prove this.

Thank you.

Answer 1

There are two ways to do a "perfect" riffle shuffle. Label the initial cards as $1,2,3,\ldots,52$. First you split the deck into equal piles: $1,2,3,\ldots,26$ and $27,28,29,\ldots,52$. Then you interleave them: but which goes first? One way is $1,27,2,28,\ldots,26,52$. The other way is $27,1,28,2,\ldots,52,26$.

Answer 2

This is called a faro shuffle. It restores the original order in either 8 or 52 iterations, depending on whether the cards are permuted in the order [1, 27, 2, 28, ... 26, 52] or [27, 1, 28, 2, ..., 52, 26].