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Problem of the Week 953

Reflect On This

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Consider an isosceles triangle ABC with apex angle A of one degree and base BC of length 2. Let D, E bisect AB, AC, respectively, and then erase all parts of the figure except for the segments DB and EC. Thus we have formed a truncated one-degree angle, with the distance at one opening equal to twice the distance at the other.

Imagine these two lines act as reflective mirrors, and a light ray (that lives in the plane) enters the larger opening. It will bounce around the angle a certain number of times before exiting at the smaller opening. What is the maximum number of reflections that are possible?

Source: This is the second of two problems that we have permission to use from "50 Mathematical Puzzles and Problems, Red Collection", published by Key Curriculum Press (ISBN 1-55953-500-8). We repeat that this is a collection of 50 superb problems at the level of our PoW program, and I encourage problem enthusiasts looking for good problems for themselves, or their students, to take a look at this book. The publisher's URL is http://www.keypress.com.
© Copyright 2002 Stan Wagon. Reproduced with permission.

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20 February 2002