On pinned billiard balls and foldings

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 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, in terms of $n$, $d$ and the locations of ball centers. As a first step, we study foldings, i.e., mappings that formalize the idea of folding a piece of paper along a crease.