Suppose you have a pool table that measures $165\times297$. If you shoot a ball from the lower left corner at a $45^\circ$ angle, assume it will continue moving until it lands in a corner pocket and that each time it hits the wall, it will bounce off at a $45^\circ$ angle. Also, assume there are only corner pockets.
How many bounces will it take before it goes into a pocket?
Please show the work and patterns you found, and also please include a general rule.
I have tried multiple times to find where the ball bounces too but I keep getting $4$.
My equation is $\frac{l}{GCF} + \frac {h}{GCF} – 2$ but I can't seem to figure out the answer to the problem.
Best Answer
When the ball rebounds on an edge, reflect the entire board on that edge. Then the path of the ball is the straight line extension to the reflected board. Thus when the ball lands in a corner of the board, the number of horizontal reflections $m$ and the number of vertical reflections $n$ must satisfy $297 m = 165 n$. Thus $m=5, n=9$. The ball lands in a corner after $14$ rebounds.