On Friday, March 7, 2014 12:04:09 AM UTC-6, swtch...@gmail.com wrote: > That's a very good point, thank you. > > The theorem does not guarantee a pattern or an algorithm. > > Only that there are 4 colors that exist to fill the map. > > Then what are the approaches or methods that are used to > > actually prove the theorem via supercomputers and mathematical > > equations? How do they do it?
On the assumption that you will actually listen and on the hope that you will stop your incessant blathering on this:
The proof of the 4-color map theorem was done by contradiction, as follows:
1) Assume that there is at least one (finite) map that cannot be 4-colored. Then there must exist a *smallest* (finite) map that cannot be 4-colored. We will show that given such a map, it is possible to find a strictly smaller map that can also not be 4-colored, giving a contradiction.
2) Prove that there is a finite family of "unavoidable configurations" that any (finite) map must have; these are a family of maps (in fact, graphs since the map can be represented by a planar graph) such that any given map will contain this map "inside it."
3) Construct this finite family. Then show that each element of the family is "reducible": given a map M (graph) that contains an element A of the family, we can replace A with a strictly smaller submap with the property that the resulting smaller map M' can be 4-colored if and only if the original map M can be 4-colored.
4) Conclude that no "smallest counterexample" can exist, thereby proving the theorem.
Now, please stop your nonsensical endless blathering.