  The billiard table is modeled as an $n$-dimensional box $[0,a_1]\times
[0,a_2]\times \ldots \times [0,a_n] \subset \mathbb{R}^n$, with each side
having real-valued lengths $a_i$ that are pairwise commensurable. A ball is
launched from the origin in direction $d=(1,1,\ldots,1)$. The ball is reflected
if it hits the boundary of the billiard table. It comes to a halt when reaching
a corner. We show that the number of intersections of the billiard curve at any
given point on the table is either $0$ or a power of $2$. To prove this, we use
algebraic and number theoretic tools to establish a bijection between the
number of intersections of the billiard curve and the number of satisfying
assignments of a specific constraint satisfaction problem.
