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?

Well, I've trouble imagining but a single path for the uniform motion part ... unless one counts that path for each of the four front sides ... however, for the second part, although it seems it would need the same solution, does allowing gravity in necessitate considering ball speed and a specific elasticity factor? I mean, wouldn't this generate a family of parabolic segments over the five bounces? That would certainly open way to critical cornering ... i mean, if you did it right ... hmmm?

Posted by CeeAnne on 2005-11-17 12:44:59