Problem:
Let and be two faces of a cube with . A beam of light emanates from vertex and reflects off face at point , which is units from and units from . The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point until it next reaches a vertex of the cube is given by , where and are integers and is not divisible by the square of any prime. Find .
Solution:
Place a coordinate system on the cube so that , , and . Point is the first point where the light hits a face of the cube. Let be the second point at which the light hits a face, and consider the reflection of the cube in face . Then is the image of the point at which ray next intersects a face of the reflected cube. Continue this process so that the cube is obtained by reflecting the cube in the face containing for . Therefore, each intersection of ray and a plane with equation , or , where is a positive integer, corresponds to a point where the light beam hits a face of the cube. Thus the path will first return to a vertex of the cube when ray reaches a point whose coordinates are all multiples of . The points on ray have coordinates of the form , where is nonnegative, and they will all be multiples of if and only if is a multiple of . This first happens when , which yields the point . The requested distance is the same as the distance from this point to , namely, , so .
The problems on this page are the property of the MAA's American Mathematics Competitions