  We consider systems of "pinned balls," i.e., balls that have fixed positions
and pseudo-velocities. Pseudo-velocities change according to the same rules as
those for velocities of totally elastic collisions between moving balls. The
times of possible pseudo-collisions for different pairs of pinned balls are
chosen in an exogenous way. We give an explicit upper bound for the maximum
number of pseudo-collisions for a system of $n$ pinned balls in a
$d$-dimensional space. The proof is based on analysis of foldings, i.e.,
mappings that formalize the idea of folding a piece of paper along a crease. We
prove an upper bound for the size of an orbit of a point subjected to foldings.
