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For more on marking an answer as the "Best Answer", please visit our FAQ.The 4 colour theorem was proved in 1976 by Appel and Haken. They reduced the problem to about 1500 basic patterns, and relied on computers to check each of these. This approach was not well received by mathematicians at the time, as it was not possible for other mathematicians to verify the proof.
There are details in http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.html
There are some theorems in mathematics that are impossible to prove or disprove. http://www.miskatonic.org/godel.html gives an outline of Godel's Incompleteness Theorem.
Mathematicians don't like proofs by example. Just because nobody has found a map that you can't colour using just 4 colours doesn't necessarily mean that one doesn't exist.
Reminds me of a 'joke' one of our lecturers gave (just google it):
An astronomer, a physicist and a mathematician (it is said) were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field.
"How interesting," observed the astronomer, "all scottish sheep are black!"
To which the physicist responded, "No, no! Some Scottish sheep are black!"
The mathematician gazed heavenward in supplication, and then intoned, "In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black."