  The problem of the Hanoi Tower is a classic exercise in recursive
programming: the solution has a simple recursive definition, and its complexity
and the matching lower bound are the solution of a simple recursive function
(the solution is so easy that most students memorize it and regurgitate it at
exams without truly understanding it). We describe how some very minor changes
in the rules of the Hanoi Tower yield various increases of complexity in the
solution, so that they require a deeper analysis than the classical Hanoi Tower
problem while still yielding exponential solutions. In particular, we analyze
the problem fo the Bouncing Tower, where just changing the insertion and
extraction position from the top to the middle of the tower results in a
surprising increase of complexity in the solution: such a tower of $n$ disks
can be optimally moved in $\sqrt{3}^n$ moves for $n$ even (i.e. less than a
Hanoi Tower of same height), via $5$ recursive functions (or, equivalently, one
recursion function with $5$ states).
