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MatheMUSEments
Tricky Tables
By Ivars Peterson
Muse, July/August 2000, p. 26.
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 here:
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?
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