  Hironaka's concept of characteristic polyhedron of a singularity has been one
of the most powerful and fruitful ideas of the last decades in singularity
theory. In fact, since then combinatorics have become a major tool in many
important results. However, this seminal concept is still not enough to cope
with some effective problems: for instance, giving a bound on the maximum
number of blowing--ups to be performed on a surface before its multiplicity
decreases. This short note shows why such a bounding is not possible, at least
with the original definitions.
