  In this note we solve the ``birthday problem'' for loops on random regular
graphs. Namely, for fixed $d\ge 3$, we prove that on a random $d$-regular graph
with $n$ vertices, as $n$ approaches infinity, with high probability:
  (i) almost all primitive non-backtracking loops of length $k \prec \sqrt{n}$
are simple, i.e. do not self-intersect,
  (ii) almost all primitive non-backtracking loops of length $k \succ \sqrt{n}$
self-intersect.
