  For the two-dimensional Q-state Potts model at criticality, we consider
Fortuin-Kasteleyn and spin clusters and study the average number N_Gamma of
clusters that intersect a given contour Gamma. To leading order, N_Gamma is
proportional to the length of the curve. Additionally, however, there occur
logarithmic contributions related to the corners of Gamma. These are found to
be universal and their size can be calculated employing techniques from
conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the
thermal phase transition we find agreement with these predictions from
large-scale numerical simulations. For the spin clusters, on the other hand,
the cluster numbers are not found to be consistent with the values obtained by
analytic continuation, as conventionally assumed.
