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MatheMUSEments
Fair Shares
By Ivars Peterson
Muse, May/June
1999, p. 28.
The birthday party is over, and one
chunk of thickly
frosted,
richly decorated cake is uneaten. Your mother insists that you
and your sister slice the cake into two equal pieces, so that she
doesn't have to listen to you fight over which is bigger. What
should you do?
The simplest strategy is to let one
person cut the
cake into
two pieces, then let the other person choose first. That's sure
to give a result that appears fair to both people.
This divide-and-choose method of
settling arguments
over the
division of cake, goods, or land goes back thousands of years, to
biblical times. Mathematicians became interested in the problem
of fair division about 60 years ago when they began to wonder
what to do when three people want to divide a
cake
fairly. Getting a fair result for three people turns out to be
surprisingly complicated. Here's one way you might do it.
Mathematicians call it the "moving knife"
strategy.
Suppose that a knife floats above a
rectangular cake.
Starting
at the cake's left end, it moves slowly toward the right. Three
people are all told to shout "Stop!" as soon as the
knife reaches a point that, in their opinion, is one third of the
cake. The first one to shout gets the first piece, and the
remaining two people divide the rest using the "I cut, you
choose" scheme.
The moving-knife strategy will also
work for more
than three
people. As the knife moves to the right, more and more people
drop out with what they think is a fair share until only two
people are left to divide the last morsel.
Mathematicians have figured out
other methods of
sharing
things, many of which don't involve a moving knife. Such methods
are now used to help handle disputes between people and have even
been used among nations to determine offshore mining rights. It's
all in the math of fair cake cutting!
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