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

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 $p>4$.