  We give a complete solution of the following problem: Find, classify and
count the (classes of) periodic orbits on an equilateral triangle. We prove
that Fagnano's period 3 orbit is the only periodic orbit with odd period. A
periodic orbit is either prime or some d-fold iterate thereof. We count prime
and iterate periodic orbits of period $2n$ via a bijection with a certain
partition of $n$, then count only prime orbits using the prime factorization of
$n$.
