I tried to formulate the following problem in a more mechanical way involving soccer balls, but the physics got too unrealistic. I know that what follows could be made more precise, but I hope the idea gets across, and I don't want to add exact conditions because I think it would make the problem more clunky.
Suppose we have a 2-dimensional maze with mirror walls. (No sneaky mazes, involving fractals or space-filling curves or infinitely thin corridors or any of that. You should be able to draw this one on a piece of paper.) Suppose that the maze has one entrance and one exit, by which I mean short line segments such that if we turned them into walls, the maze would be an enclosed room. As in any good maze, the exit is reachable from the entrance. (Of course, from a mathematical point of view, our maze is simply a very strangely shaped room, enclosed by some curve. When the problem occurred to me, I was think about mazes.) And we'll assume that any corners are slightly rounded, i.e. walls should be differentiable curves, to avoid complications about sharp corners.
My question: Suppose I'm at the entrance with a laser. Is there necessarily a point on the entrance line segment and an angle such that if I shine the laser at that angle, it will (after bouncing around) eventually reach the exit?
This is closely related to the "Illumination Problem" to which Roger Penrose apparently found an interesting solution. (http://mathworld.wolfram.com/IlluminationProblem.html) However, I think his solution depends on having sharp corners. Less importantly, my problem is more specifically about illuminating boundary points from boundary points.