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
re: weighing in on the gravity solution (full solution?) by Tristan)
After BNE the ball is rising. When it reaches the SW edge it is still rising, but not as quickly because gravity has reduced its upward velocity somewhat. My idea was that there is a high enough velocity which would allow the ball to clear the NE edge after the 4th and 5th two bounces.
It may be easier with the 2d analogue
\ ______
XXXXXXXXXX 
X_\ X __
X \ X 
X  \ X 
X \ X 
X \ X 
X \X 
X VX \
XXXXXXXXXX
The ball enters at the upper left, and hits the bottom near the right edge, then immediately hits the right wall. Its path is actuallt a parabola, so instead of exiting the box it hits the left wall. Still rising, it exits the box over the right wall, then falls outside the box.

Posted by Jer
on 20051121 10:07:18 