Diagram 1: The enchanted forest problem
How to place the circular and perfectly reflecting trees such that the light of a match cannot escape to infinity?
That mathoverflow question only asks for a single contained ray, which got an interesting answer by the Bill Thurston (who passed away last August).
It is where Joseph O' Rourke found a place to helpfully provide image, references and, for me, our problem. It seems it is yet unsolved. Considering a closed room instead, one can ask wether there is a point from which the light misses some places. This is indeed possible, see e.g. MathWorld on the "Illumination Problem":
If you like geometry problems you should look up Josephs Q&A's now: