Example

How many shuffles are required to return a deck of 16 cards to their original positions if the deck is "shuffled" in the following way:

  1. Place the deck face down on the table
  2. Split the deck into two piles, the top half and the bottom half. Place the top half to the right side of the table and the bottom half on the left side of the table.
  3. Merge the two piles together, by pulling the top card from either the right or left pile (as dictated by the order: R,L,L,R,R,L,L,L,R,R,R,L,L,R,R,L) and placing it in the center of the table.

Suppose we number the cards 1 through 16, where the 1 is on the bottom of the deck. Shuffling the cards once in the described way changes the order as shown below: $$\begin{array}{cccccccccccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ 16 & 8 & 7 & 15 & 14 & 6 & 5 & 4 & 13 & 12 & 11 & 3 & 2 & 10 & 9 & 1\end{array}$$ Now we can find the cycles: $$\begin{align} &(1,16),\\ &(2, 8, 4, 15, 9, 13),\\ &(3, 7, 5, 14, 10, 12),\\ &(6) \textrm{ and } (11)\end{align}$$ Note, the cycle lengths are 2, 6, 6, 1, and 1 respectively. So the deck will return to its original order in 6 shuffles (the least common multiple of the cycle lengths seen: 2, 6, and 1).