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Impossible Problems


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?


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, 
and that's been done. You could learn more about geometry, for 
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," 
which you can read about in the MacTutor Math History archives:

  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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