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?

http://mathoverflow.net/questions/38307/trapped-rays-bouncing-between-two-convex-bodies/38314#38314

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":

http://mathworld.wolfram.com/IlluminationProblem.html

If you like geometry problems you should look up Josephs Q&A's now:

http://mathoverflow.net/users/6094/joseph-orourke