The concepts behind the proof of the 4-Color Map Theorem can be discussed without actually doing any of the math. The concepts are:

- Every map can be reduced to a finite number of maps.
- 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.

- 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:

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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