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### 17 and 19 Balls, Colliding

```Date: 03/25/2003 at 16:26:10
From: Melissa
Subject: Math puzzle

There are 17 equally spaced spaced balls on the left side of a
horizontal line, each moving to the right at constant speed S. There
are also 19 equally spaced balls on the right side of the same line,
each moving to the left at constant speed S. The balls are all the
same size and mass and the spacing between the balls on the left is
the same as the spacing between the balls on the right. When two balls
collide they will reverse directions, but each will maintain the same
speed S.

How many collisions will occur in the described system?

It looks as if there can be only 17 collisions, since the number of
balls on the two sides is not equal ( i.e, 17 vs. 19).
```

```
Date: 03/26/2003 at 06:32:51
From: Doctor Jacques
Subject: Re: Math puzzle

Hi Melissa,

Look at what happens in a single collision. We start with two balls A
and B moving against each other:

A -->  <-- B

and we end up with the balls fleeing in opposite directions:

A <--  --> B

If the balls had been travelling on parallel tracks, we would have no
collisions at all, but the picture would be almost the same:

Before crossing:

A -->  <-- B

After crossing:

B <--  --> A

Note that the only difference is the name of the balls. We may imagine
that each ball has a sticker on it with its name, and that we exchange
stickers at each collision (this does not change the physics of the
collisions).

Now, the problem is the same as if we have two parallel tracks: one
with the balls going to the left, and one with the balls going to the
right:

A1 -> A2 ->.... A17 ->
<- B1.... <- B19

In this case, the end result is pretty obvious: all the A's will
continue to the right, and all the B's will continue to the left.

For each collision in the original problem, one A ball crosses one B
ball in the parallel track version. Now, note that, in the parallel
track version:

* A given A ball cannot cross the same B ball more than once, since
all balls continue in the same direction.

* We started with all A's on the left, and ended with all A's on the
right.

This means that each A ball has crossed each B ball exactly once, and
the number of crossings is the same as the number of collisions in
the original problem.

some more, or if you have any other questions.

- Doctor Jacques, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Logic
High School Puzzles

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