15 Ball Players

Date: 12/07/2001 at 13:29:01
From: Gerrit van Buren
Subject: 15 ball players and no player receives no more than 5 balls

Dear Dr. Math,

Thank you for previous excellent answers. I have this problem about 
ball players:

15 softball players, each with one ball, are standing at different 
distances from each other. Each pair (two players) has a different 
distance between them. Each player throws his ball to the player who 
is the closest by in distance.

a. Prove that no player receives more than 5 balls.
b. Can this problem be generalized for n players, and what then is the 
   minimal number of balls a player can receive?

Thank you for helping,
Kindest regards,
Gerrit van Buren

Date: 12/08/2001 at 05:27:47
From: Doctor Mitteldorf
Subject: Re: 15 ball players and no player receives no more than 5 
balls

Dear Gerrit,

Here's a way to see what this problem is about: Draw a regular 
hexagon. Divide it into 6 equilateral triangles. Put 6 ball players 
at the corners of the hexagon, and put number 7 in the center. Now all 
6 players are equidistant from the guy at the center, and what is 
more, the distance between each outer player and his closest neighbor 
is the same as the distance to the player at the center. 

Now we have to make all the distances different. Let's see if we can
make all 6 closer to the central guy than to the nearest hexagon guy.  
If we adjust the positions of the 6 outer players in such a way as to 
make them each a little closer to the guy in the middle, they're going 
to end up closer to one another as well. No way around it. To get the 
guys on the outside ALL closer to the central guy than to each other, 
there have to be fewer than 6 of them. 5 at most. This is the basis of 
your proof.

You can construct your proof like this: Let's assume player A gets 
6 balls. Draw a circle around player A at the radius of the 6th 
closest guy. Draw spokes out to all 6. The closest of these spokes 
must be 60 degrees or less apart. But whichever two guys are on these 
two spokes must then be closer to each other than they are to the 
central guy. (Prove this.)

When you've sorted out this proof, it should be clear that it doesn't
depend on the number of ballplayers being 15 - the proof is the same 
for any number.

It would be interesting, however, to extend the theorem to three-
dimensional space. Imagine ballplayers floating out in space - the 
maximum number of balls is now a lot greater than 5. But there's still 
a maximum number - what is it?

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/