Lets play Carrom!

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.

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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.