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### Impossible Problems

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Date: 08/07/2001 at 17:22:22
From: David Khourshid
Subject: Impossible Problems

Is there a cash prize for finding solutions to impossible problems?
If so, what are they?
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Date: 08/08/2001 at 13:34:23
From: Doctor Rick
Subject: Re: Impossible Problems

Hi, David.

No one can earn a prize by finding a solution to an impossible
problem. If he finds a solution to a problem, then that problem is not
impossible.

Certain easily stated problems have been *proved* to be impossible.
Take a look in our Dr. Math FAQ:

"Impossible" Geometric Constructions
http://mathforum.org/dr.math/faq/faq.impossible.construct.html

There is no point in asking about prizes for solving these problems.
The proof that they are impossible is the only solution they can have,
instance, by trying, but you won't solve the problem, and you won't
get money out of it.

There are other problems that are not necessarily impossible, but (so
far) unsolved. Some may have solutions that no one has found yet, and
some may be truly impossible. Of these, some may one day be *proved*
impossible like the impossible geometrical constructions, but some
impossible problems can't even be proved to be impossible. A
mathematician named Kurt Goedel proved that there must be at least one
such problem:

Goedel's Incompleteness Theorem
http://mathforum.org/dr.math/problems/seldon2.8.8.97.html

Some significant previously unsolved problems have been solved in
recent years: the four-color map problem and Fermat's last "theorem,"

Fermat's last theorem
http://www-history.mcs.st-and.ac.uk/history/HistTopics/Fermat's_last_theorem.html

The four colour theorem
http://www-history.mcs.st-and.ac.uk/history/HistTopics/The_four_colour_theorem.html

A quick Web search on the words "unsolved problem" and prize, brought
up this site listing some major unsolved problems, and giving the
rules for winning a \$7 million prize:

Clay Mathematics Institute Millennium Prize Problems
http://www.claymath.org/prizeproblems/index.htm

These are not simple problems! You might deserve a prize at your age
just for understanding what the problem means, let alone solving it.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
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