Question 51
An encyclopedia consists of $2000$ numbered volumes. The volumes are stacked in order with number $1$ on top and $2000$ on the bottom. One may perform two operations with the stack:
(i) For $n$ even, one may take the top $n$ volumes and put them in the bottom of the stack without changing the order.
(ii) For $n$ odd, one may take the top $n$ volumes, turn the order around and put them on top of the stack again.
How many different permutations of the volumes can be obtained by using these two operations repeatedly?
(i) For $n$ even, one may take the top $n$ volumes and put them in the bottom of the stack without changing the order.
(ii) For $n$ odd, one may take the top $n$ volumes, turn the order around and put them on top of the stack again.
How many different permutations of the volumes can be obtained by using these two operations repeatedly?