## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## April 3, 2007

### Tricky Tables

The shape of a billiard table has a lot to do with the types of shots you can make in a game of billiards. Which of these odd tables do you think would be your best bet for hitting another ball?

With the help of a little geometry, an expert billiards player can figure out exactly where a ball will go. Unless it's a trick shot, the ball will travel in a straight line until it hits a cushion. When it bounces, it obeys a basic law of physics: the angle between the incoming ball's path and the cushion is the same as the angle between the outgoing ball's path and the cushion.

But if one bounce is predictable, many bounces may not be—especially on a tricky table.

Suppose you have a circular table. The mathematician Charles L. Dodgson, who as Lewis Carroll wrote Alice's Adventures in Wonderland, once published a set of rules for a two-player game of circular billiards. In his game, you had to hit other balls as well as bumpers to rack up points quickly.

Suppose you had a ball in the center of the table. To figure out what a ball will do on a table, mathematicians imagine what would happen if there were no friction and the ball could travel forever. It turns out that a ball can follow a path on a circular table that never passes anywhere near the center of the table. So a ball sitting there would never get hit. Dodgson's game isn't as easy as it sounds.

What about a rectangular table? Try setting a large circular pan, hoop, or other round object in the middle of a rectangular poil table or air-hockey table. Put a weight on the object to keep it in place, then see how it affects the movements of a ball or hockey puck during a game.

You'll find that balls or pucks that start off in nearly the same direction soon are on wildly different paths. So even though each bounce is predictable, after many bounces it is hard to say where a ball or puck will end up!

What about the ellipitical table? If the rectangular table with a circular obstacle is unpredictable, the elliptical table is totally predictable. Suppose balls are placed in the spots shown below:

If a player shoots one of the balls in any direction, it will hit the edge, bounce off and collide with the other ball. The player doesn't even have to aim. One ball will always hit the other!

So the best bet for hitting another ball is the elliptical table. Did you guess right?

Muse, July/August 2000, p. 26.