The unilluminable room problem

Part of this problem, posed by one Ernst Straus in 1952, is to design a polygonal room lined with mirrors having at least two points such that any ray starting at one point can reflect around the room forever without hitting the other point, and vice versa. One such room, shown above, was demonstrated by George Tokarsky in 1995. It has 26 walls. An improved 24-wall version was produced by D. Castro in 1997.

There are more details, and a cool downloadable demonstrator app, over at Wolfram MathWorld.

More:
Check out George Hart’s Math Monday columns