Lets play Carrom!

Geometry Level 5

This problem reminded me of something I found and proven a long time ago.

I have a 2D box in which I shoot a ball and it starts bouncing around the box infinitely. The starting shoot makes an angle \(A\) with the vertical:

Given that \[A=\arctan { (a+0.1b) } \]Where \(a\) and \(b\) are non-negative integers and \(a+0.1b\le 10\)

Find the maximum number of distinct points where the ball hits the box.


Details and Assumptions

  • The size of the ball is negligible

  • The collisions the ball makes with the box are elastic, that is, angle of incidence is equal to angle of reflection.

  • This case has \(4\) distinct points where the ball hits the box.

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