Self Similarities of the Tower of Hanoi Graphs and a proof of the Frame-Stewart Conjecture

TL;DR

This paper proves the strong Frame-Stewart conjecture for 4 pegs in the Tower of Hanoi problem by analyzing graph symmetries and self-similarity, and suggests the method can extend to more than 4 pegs.

Contribution

It introduces a novel approach using graph symmetries to prove the minimal move formula for 4 pegs, confirming the conjecture.

Findings

Proves the Frame-Stewart conjecture for 4 pegs.

Derives a recursive formula for minimal moves.

Method potentially applicable to more than 4 pegs.

Abstract

Considering the symmetries and self similarity properties of the corresponding labeled graphs, it is shown that the minimal number of moves in the Tower of Hanoi game with $p =4$ pegs and $n \geq p$ disks satisfies the recursive formula $ F(p,n) = \min_{1\leq i \leq n-1} \{ 2F(p,i) + F(p-1,n-i) \} $ which proves the strong Frame-Stewart conjecture for the case $p=4$. The method can be generalized to $p>4$.