Concerning the bookshelf problem

Abstract

We discuss the following folklore problem. On a bookshelf, there are $N$ tomes of the Encyclopedia in random order. Each hour, a librarian takes a tome which stands not on its place, and puts it in its place. Show that the process will stop. A natural additional question is how many moves are required for the process to stop. We show that the process can last $2^{N-1}-1$ moves, and that it will stop anyway in less than $2^N$ moves.