The Reflection Puzzle
Solution

At first glance, it seems possible that the photon might rattle around in the mirror trap forever, bouncing faster and faster. A little thought shows, though, that each bounce is adjusting the direction of the photon by 1 degree increments.

If you draw a semicircular fan shape, of 180 1-degree sectors, you can straighten out the track of the photon, and then the solution will be obvious.

Here is an image of a fan, drawn for the case of a 10-degree reflector, which shows how you can straighten out the path of the photon, count the reflections and determine the distance traveled.

"Unreflecting" the path of the photon in this way allows you to determine that:

• the photon hits the mirrors 179 times;
• The x coordinate of the first hit is 2617/sin(1 degree), and the total distance traveled is 2*x, so the total time is 2*x/c where c is the speed of light.

Back to the reflection puzzle..

This puzzle appeared in Yan Fridman's Puzzles column in Contingencies magazine, November/December 2006.