Have computersbeen used to solve
mathematical proofs?
Yes, there have been many mathematical
proofs solved with the help of computers.
One example is the four color theorem,
which stated that it is possible to have a
geographic map colored with only four
colors so that no adjacent regions will
have the same color. Another way of look-
ing at the problem is: What is the small-
est number of colors needed to color any
flat map so that any two neighboring
regions always have different colors? This
idea was first presented in 1852, when
Francis Guthrie (1831–1899) colored a
map of English counties using only four
colors. The idea of only four colors took
on a mathematical bend and ended up
being a theorem to be proved. It took until 1976, with the help of modern computers,
before the four-color conjecture was finally proven to be true. But some mathemati-
cians are troubled by this computer proof, feeling that the theorem is so easy to
understand that it should have been proven by hand. Thus, anyone who can prove the
theorem without using a computer may win the Fields Medal, the math equivalent of
the Nobel Prize.
Another proof solved with computers is the double bubble.The double bubble
refers to a pair of bubbles that intersect; they are also separated by a membrane
bounded by the intersection of the two bubbles. This is similar to two bubbles stuck
together when a child blows bubbles using a water and soap mixture. Since the
ancient Greeks, mathematicians have worked on the problem of finding a mathemati-
cal proof of the efficiency of a single round bubble. The problem became even more
rigorous when considering enclosing two bubbles—or two separate volumes. The
problem was solved around 1995 by mathematicians Joel Hass, Michael Hutchings,
and Roger Schlafly. They used a computer to calculate the surface areas of the bubbles
and found that the double bubble has a smaller area than any other when the enclosed
volumes are the same. But this isn’t the last word: Scientists are currently working on
triple bubbles.
How are algorithmsconnected to computers?
Algorithms are essentially the way computers process information. In particular, a
computer program is actually an algorithm that tells the computer what particular
steps to perform—and in what order—so a specific task is carried out. This can include 367
MATH IN COMPUTING
In the problem of the double bubble, ancient Greek
mathematicians worked on a proof that would
demonstrate the efficient use of space created by two
joined bubbles.