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Mathematical billiards

Published online by Cambridge University Press:  01 August 2016

Harris S. Shultz  and
Ray C. Shiflett
Harris S. Shultz
Affiliation:
Department of Mathematics, California State University, Fullerton, California 92634 U.S.A.
Ray C. Shiflett
Affiliation:
School of Science, California State Polytechnic University, Pomona, California 91768, U.S.A.
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Extract

A ball is shot at an angle of 45 degrees from the corner of a rectangular billiard table of size m × n (where m and n are integers). It always bounces off from a side at 45 degrees until it lands in a corner pocket. Will it always land in a corner or will it keep bouncing for ever? And how many times will it bounce? And which corner pocket will it land in?

Try it for yourself. For example in the 7 × 4 table shown the ball will bounce 9 times and then fall into the top left-hand corner pocket.

This is a well-known problem but we now present a simpler proof than we've seen elsewhere of the fact that the ball always ends up in a corner pocket, and we derive a formula for the number of bounces and we determine the finishing pocket.

Type
Research Article
Information
The Mathematical Gazette , Volume 72 , Issue 460 , June 1988 , pp. 95 - 97
Copyright
Copyright © The Mathematical Association 1988

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