Concerning the bookshelf problem

TL;DR

This paper analyzes a folklore problem involving a process of reordering books on a shelf, proving bounds on the maximum number of moves needed for the process to terminate.

Contribution

It establishes the maximum number of moves required for the process to stop and proves that the process always terminates within a specific exponential bound.

Findings

The process can last up to 2^{N-1}-1 moves.

The process always stops within less than 2^N moves.

The process is guaranteed to terminate.

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.