Suppose a billiard table has a length twice its width and that a rolling ball loses no energy to bounces or friction but simply caroms around the table forever. Call the angle between the launch direction and the long side of the table α.
At what angle(s) should the ball be hit so that it will arrive back at the same point on the table and traveling in the same direction, so that its motion is cyclic, following the same path repeatedly.

Source: Colin White’s Projectile Dynamics in Sport (2010)

My remarks:
1. Assume ball's arbitrary initial position.
2. Ignore the trivial answers (0 & 90)