4 Color Map Theorem

The concepts behind the proof of the 4-Color Map Theorem can be discussed without actually doing any of the math. The concepts are:
  1. Every map can be reduced to a finite number of maps.
  2. An exhaustive examination of every one of these finite number of maps, with the aid of a computer, shows they all need only 4 colors.
Maps of countries are first converted to planar graphs as follows:
  1. For each country we select a capital (an arbitrary point inside that country) and join the capitals of every pair of neighboring countries.

Now we have a planar graph we can deal with.

A good discussion of the current status of the 4-Color Map Theorem can be found at:

Here is the article in "Notices of the AMS" describing the present-day situation:

Here is a very elementary presentation of the history of the problem (with photographs) and illustations to key ideas of the proof:

cover Four Colors Suffice: How the Map Problem Was Solved by Robin Wilson (2002)
An excellent and easy to read book.

Here is an interesting attempt at an alternate proof using spirals (I haven't read it thoroughly):

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