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### 15 Ball Players

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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,

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

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/
```
Associated Topics:
High School Discrete Mathematics

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