  We modify the rules of the classical Tower of Hanoi puzzle in a quite natural
way to get the Fibonacci sequence involved in the optimal algorithm of
resolution, and show some nice properties of such a variant. In particular, we
deduce from this Tower of Hanoi-Fibonacci a Gray-like code on the set of binary
words without the factor 11, which has some properties intersting for itself
and from which an iterative algorithm for the Tower of Hanoi-Fibonacci is
obtained. Such an algorithm involves the Fibonacci substitution. Eventually, we
briefly extend the study to some natural generalizations.
