**EMAT
6700**

**by
Brad Simmons**

**Four Color Problem**

Use geometer's sketchpad to investigate the four color problem.

Mapmakers have long believed that only four colors are needed to distinguish among any number of countries on a plane map. This conjecture eventually attracted the attention of mathematicians and became known as "the four color problem." Finally, in the 1980's with the help of computers a proof was accomplished confirming the four color conjecture. The problems below attempt to show some to the complexities associated with the four color problem.

**Assume that each closed region is a different country on a
map. Countries that meet only in one point may have the same color
provided they do not share a common border (borders are line segments).
Countries that have a common border must have a different color.**

**What is the minimum number of colors necessary for each
map?**

**
1.**

Please click here for a geometer's sketchpad sketch of the solution for the figure above.

**
2.**

Please click here for a geometer's sketchpad sketch of the figure above.

**
3.**

Please click here for a geometer's sketchpad sketch of the figure above.

**
4.**

Please click here for a geometer's sketchpad sketch of the figure above.

**
5.**

Please click here for a geometer's sketchpad sketch of the figure above.

**Draw some plane maps other than the problems shown
above. Show how each can be colored using four colors. Can the map
be colored with less than four colors?**