Real World Math: Exploratory Lessons - Four Color Map

This is a fairly simple puzzle for students to solve but it took over a century for it to be proven with computers. 

The Four Color Problem is often related to a mapmaker’s design dilemma as to how many colors to use on a map.  Three colors are sometimes sufficient to color a map but not all.  Any map, however, can be colored with at least four colors.  This problem is typically categorized with the mathematical study called Graph Theory; that is, the study of models used to illustrate the relationships between objects in a collection.

This activity is presented in a SketchUp file.  Ironically, SketchUp enables students to color regions much more easily than with Google Earth.  Little training is needed to use SketchUp with these exercises but students do need to have the SketchUp application on their computers.  A link to the free Trimble SketchUp download can be found here.
As with most puzzles, the students will be using logic to color the maps.  This is the type of activity where students will typically ask “Why are we learning this?” or “What does this have to do with math?”  Developing deductive reasoning skills is an important thought process for students to learn not only for math, but for science and other disciplines as well.  These maps can’t be colored without it.  The ultimate goal, however, should be moving from deductive to inductive reasoning; that is, how can these situations be generalized?  Why are four colors necessary?  Don’t get caught up on terminology.  Often you’ll hear nuggets of wisdom from the students as they are working.

I’ve also included a tessellated cube as a download.  Four colors are needed to decorate the cube so that no color shares the same border with itself.  This problem is more difficult though, because each pattern wraps around three of the cube’s faces.

One possible extension for this activity is to have students color blank paper maps of the United States or regions where they live using only four colors.  Historically, maps of England and France were used.  I typically print two copies on each side of the paper because once a student has made an error the only solution is to start over.  Check for available materials online or with your Social Studies teacher.  Currently, this problem is also related to the coverage provided by cell phone towers.  Let me know if you can think of more extensions.  And finally, you can find some links to the history and proof of the theorem below.

You may find these links helpful:

Objectives

  • Color mapped regions with four colors so that no two regions with a common edge have the same color
  • Develop deductive and inductive reasoning

4 Color Cube Problem
Colored Cube example


Problem-Solving
K-6, Middle School, Educator

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