The four color theorem is a beloved result with a long and fascinating history.
The theorem says that four colors suffice to color any map so that no two bordering regions are the same color. The conjecture was made in 1852 by Francis Guthrie. After many, many failed proofs, the conjecture was finally put to rest in 1976 (using a computer) by Kenneth Appel and Wolfgang Haken. To this date there is no pencil-and-paper proof of this seemingly elementary theorem.
Today I was exploring the website of the Japanese publisher and puzzle-popularizer Nikoli. The website has a number of interesting Flash-based logic games (Kakuro, Nurikabe, Sudoku, etc.). They also have five four color theorem applets such as the one below.
The goal of each puzzle is to find a four-coloring of the given map. I wish they had more than five of these!
David Richeson: Division by Zero 
For pencil-and-paper proof(s) see my arXiv papers. For illustrations including one of the puzzle visit: http://www.flickr.com/photos/49058045@N00/
Thanks for the link. If your papers get published, please post (or send me) the publication information.
Nice applet
Hi,
I am also building a Java application (http://sourceforge.net/projects/maps-coloring) and a WordPress blog dedicated to the four color theorem, to share some ideas and to find a very easy “pencil and paper” proof (human checkable) of the four color theorem.
I think I’ve found an interesting fact about maps, which I would like to verify and share with people interested in the problem. I’ve found that, all regular maps can be topologically transformed into circular or rectangular maps, which are more manageable forms of maps.
Example of these maps with the proof of this transformation problem, can be found here (see T2 and “conversion of famous maps”): http://4coloring.wordpress.com
Bye, Mario Stefanutti
Here is a video that shows the nature of rectangular and circular maps.
All maps concerning the four color theorem (“regular maps” | “planar graphs without loops”) can be topologically transformed into rectangular or circular maps.