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# Little riddle watch

1. I don't know if this is exactly the right board to put this in, but I saw this really great riddle the other day which I'm sure you'd enjoy

"A ball is a rest on a square billiard table. The ball is hit and starts moving. There is no air resistance and the collisions are smooth, so the ball carries on moving forever. Assuming the laws of physics are obyed (angle of incidence = angle of reflection when the ball hits a wall), what intial conditions are necessary (when the ball is hit) for its movement to be cyclic - ie, for the ball to describe the same cycle again and again rather than move erratically all over the table"

Here are a few clues and the answer in spoiler tags, but without any working - I found this in a book ("How to Solve it, Modern Heuristics"), and the answer they give there is extremely clever (and really very easy too once you know it!!) See if anyone here gets it...

Clue 1:
Spoiler:
Show
It's to do with the angle of the ball's intial motion once it's been hit.

Clue 2:
Spoiler:
Show
Think SOHCAHTOA

Spoiler:
Show
The movement will be cyclic as long as the Tan of the angle of the ball's initial motion is either rational (ie: can be expressed as p/q where p and q are integers) or infinite
2. Where about on the table does it start? (or are we meant to show that it doesn't matter)
Which direction is 0 degrees?
3. could you not just hit it exactly straight on, and it'll keep going back and forth forever?
4. See the diagram attached for what I meant by "the angle"

Also, yes, you're right - if you just hit it straight up it'll bounce back and forth forever (which corresponds to the tan(90) = ∞ in the answer). However, there are clearly some other possibilities for it to bouce around forever (for example, if you hit it into the midpoint of a side and it bounces around in an oblong shape forever) - the question is to find the GENERAL condition for the motion to be cyclic...

And yes, the answer doesn't depend on the starting position of the ball, it could be anywhere...
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