MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 1 [made Community Wiki]

It is currently unknown if all triangles have a periodic billiard path. (See, for example, http://en.wikipedia.org/wiki/Outer_billiard#Existence_of_Periodic_Orbits)