  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 $p>4$.
