345. The sides of an angle equal to $\alpha$ are the edges of a billiard table. What is the maximum number of reflections from the edges that a billiard ball (the size of the ball can be neglected) can make?
345. "Straightening" the path of the ball: instead of "reflecting" the ball off the side, we will reflect the billiard table itself mirror-like relative to this side. We will obtain a system of rays with a common vertex; any two adjacent rays form an angle $\alpha$. The maximum number of rays in the system that a straight line can intersect is the maximum number of reflections of the ball. This number is $\left[\frac{\pi}{\alpha}\right]+1$, if $\frac{\pi}{\alpha}$ is not an integer

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Fig. 61 $\left([x]\right.$ - the integer part of the number $x$ ); if $\frac{\pi}{\alpha}$ is an integer, then it is equal to the maximum number of reflections.