There's a spaceperson with a very bouncy ball and a rigid box in the form of a cube with one face missing. One day she throws the ball into the box and notices the ball bounces off each face exactly once before exiting through the missing face.

(The ball travels in a perfectly straight line, being unaffected by air resistance, spin or any other forces other than the reactions with the box. Also the ball bounces symmetrically such that the incoming angle is identical to the outgoing angle and again is unaffected by spin. Also, the box cannot be moved while the ball is in motion.)

How many different combinations are there of the order in which the ball can bounce off all five faces?

On returning to Earth our spaceperson notices that new combinations are possible.

(All conditions are the same except the ball is now affected by gravity.)
How many different combinations are there of the order in which the ball can bounce off all five faces now?

(In reply to weighing in on the gravity solution (full solution?) by Jer)

I may have misunderstood, but you seem to say that the ball may bounce out of the box after hitting BNE, and then come back down  and hit the SW faces.  If this is allowed, I agree that 120 possible orderings are likely.

However, strictly interpreted, the ball bounces out after hitting each face exactly once.  At the end of this BNESW sequence of bounces, the ball is still in the box, and about to bounce on the bottom before exitting again (at which point it may or may not re-enter).  Lastly, it should hit each face before exiting, and should not exit anywhere in the middle of its sequence.

Posted by Tristan on 2005-11-18 23:11:54