Tracy Bledsoe
MATH 6161
The Four Color Map Problem
This learning activity packet (LAP) requires students to understand the idea of a common vertex and of common boundaries. This LAP lets students discover the Four Color Map Problem explored by mathematicians since 1852 with the proof completed in 1976. The Four Color Map Problem states that any map can be colored with a maximum of four colors including the idea that if two areas of the map share a common boundary, they must be colored differently. In exploring the Four Color Map Problem the students are introduced to a new branch of mathematics called topology. A connection to geography is made by including a map of the United States and a map of Georgia. This LAP is designed for students in 9th through 12th grade.
The Four Color Map Problem
Have you ever thought about how many colors it takes to color a map? Guess what? Mathematicians have been thinking about this question since 1852 and have only completed the proof since 1976. The mathematicians who have looked at this problem have been studying a branch of mathematics called topology. Let’s look at how to color a map with four or fewer colors. If two regions have a common boundary, they must be colored differently. If two regions only share a vertex (a single point), then they may be the same color. You need to have four separate colors or write the first letter of the color in each region (suggested colors are: b/blue, r/red, g/green, and y/yellow).
If you had a map with two regions, how many colors would it take to color the map? Use the space below to show a figure with two regions. Indicate the colors you would use.
Can you draw a map with three regions that would take a minimum of three colors? Use the space below to show a figure. Indicate the colors you would use.
Can you draw and color a figure which has three regions, but only takes two colors? Use the space below and indicate the colors you would use.
Can you draw and color a figure below which has four regions, but only takes two colors?
Can you draw and color a figure below which has four regions, but can be colored with a minimum of three colors?
Can you draw and color a figure below which has four regions, but can not be colored with fewer than four colors?
Look at the figures on the next page. What is the minimum number of colors needed? Remember you should be able to color the figures with four or fewer colors.
Figure (a) requires a minimum of _____ colors.
Figure (b) requires a minimum of _____ colors.
Figure (c) requires a minimum of _____ colors.
Teaching Secondary Mathematics, pp. 290-291
Let’s look at some actual maps. Color the map of the United States with four or fewer colors.
http://worldatlas.com/webimage/counrys/namerica/ustates/usa50out.htm
Let’s look at another map. Color the map of showing the counties of Georgia with four or fewer colors. http://worldatlas.com/webimage/counrys/namerica/ustates/usa50out.htm
You as a student will have successfully completed this LAP once you complete each section by following the rules stated in the Four Color Map Problem. I hope you enjoyed the challenge of using the Four Color Map Problem.
References:
Posamentier, A. S. and Stepelman, J. (2002). The four color map problem. Teaching secondary mathematics: Techniques and enrichment units (pp. 289-291). Columbus, Ohio: Merrill Prentice Hall.
Taylor, G. (2003). Four colors suffice: How the map problem was solved. The Booklist, 99(14), 1262.