What is the least number of moves needed to solve the k-peg Towers of Hanoi problem?

Abstract

We prove that the solutions to the [Formula: see text]-peg Tower of Hanoi problem given by Frame and Stewart are minimal. The proof relies on first identifying that for any [Formula: see text]-disk, [Formula: see text]-peg problem, there is at least one minimal sequence is symmetric. We show that if we order the number moved required for the disks in the minimal symmetric sequence in an increasing manner and obtain the sequence [Formula: see text], then [Formula: see text]. We also prove that the maximum number of disks that can be moved using [Formula: see text] steps is [Formula: see text]. We use these to lower bound the telescopic sum (2) that is a lower bound on the number of moves required for any minimal symmetric sequence. This gives us the required result.