In order to shuffle a pile of n poker chips. They are first separated into two equal piles of size n/2. Next, chips are added by alternating from each side. The side that is chosen to have the bottom chip is randomly selected with equal probability. For instance, an initial pile with the arrangement:
1 2 3 4 5 6 7 8 9 10 (where 1 is at the top of stack and 10 is at the bottom)
following a shuffle could end up as:
6 1 7 2 8 3 9 4 10 5 (if the pile with 5 was chosen to be the bottom chip)
or
1 6 2 7 3 8 4 9 5 10 (if the pile with the 10 was chosen to be the bottom chip)
If we are to start with chip #1 at the top, what is the expected number of shuffles until the chip at the top of the stack is chip #1 again.