4 colour map

[myron21]

you can use 4 colours to seperate all the countries of the world and never have the need to put like colour against like. I hear that this is accepted but impossible to prove, I cant understand why its impossible to prove, am I missing something.

[Gnu]

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.

[Hammer]

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."