MathDB

Six quadratic mirrors are put together to form a cube

ABCDEFGH

with a mirrored interior. At each of the eight vertices, there is a tiny hole through which a laser beam can enter and leave the cube. A laser beam enters the cube at vertex

A

in a direction not parallel to any of the cube's sides. If the beam hits a side, it is reflected; if it hits an edge, the light is absorbed, and if it hits a vertex, it leaves the cube. For each positive integer

n

, determine the set of vertices where the laser beam can leave the cube after exactly

n

reflections.

Problem Statement