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?

It seems to me that the 'bottom' does not have to be the third bounce with the help of gravity. 

Imagine the box with the bottom on the ground and the opening at the top.  Throw the ball so that it hits very close to a corner but a little to one side and actually hits the bottom first.  Without gravity if it hit B-N-E (as per Charlie's notation) it would exit the box without hitting S or W.  But with gravity pulling the ball down a bit it could hit these last two sides before exiting.

This gives 16 more possibilities BNESW, etc.

By similar reasoning it could hit B second and give 16 more: NBESW, etc.

Now picture the box with its 'bottom' on the ceiling.  Throw the ball so that it hits near an edge at the lower lip and bounces N-E.  Without gravity it would have to hit B before S-W to be able to exit without hitting any more sides.  But with gravity helping it could hit those other sides first, then B and just miss N-E on its way out.

This gives 16 more: NESWB, etc.

Similar reasoning gives NESBW, etc.

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We're up to 80 out of the 5!=120 possible ordering of the 5 sides.  I'm pretty sure the remaining 40 are also possible.  To have the ball hit N-E but then W before S the box would have to be oriented with a corner facing down.  My head hurts trying to picture the combinations here, so I'll leave it to someone else to describe these.

Posted by Jer on 2005-11-18 09:53:51