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Exploring the Four-Color Theorem

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Part 1: Exploring the Four-Color Theorem by coloring a United States map

The four-color theorem states that four colors are sufficient to color any map drawn in the plane or on a sphere so that no two regions with a common boundary line are colored the same color. (Saaty, T.L., & Kainen, P.C. (1977). The four-color problem: Assaults and conquest. New York: McGraw-Hill, p.4)

  • Import the map of the United States within the maps folder built into MW.

To import the United States map on a Macintosh computer:
  • Choose the File menu and highlight Import and click on Picture
  • Open the media folder and then open the maps folder
  • Scroll horizontally until you locate Usmap.bmp, highlight this map and click on the open button
 
To import the United States map on a PC computer:
  • Choose the File menu and highlight Import and click on Picture
  • Pull down the drive menu bar and click on :C
  • Open Program Files and then open Lcsi
  • Open MicroWorlds 2.0, Open Media, Open maps, scroll to Usmap.bmp and double click.
  • In your own words describe the four-color theorem as it applies to this map.

  • Open the color palate and color the U.S. map to verify the four-color theorem. Where is a good place to begin coloring the map? Why?

To color:
  • In the command center, click on the button resembling a paint brush
  • Choose the paint bucket and a color of your liking by clicking on it.
  • Click in a region you wish to color.
  • To uncolor a region, simply color the region with its background color.

  • What kind of system did you follow while coloring?

  • Did you notice any patterns or insights related to the four-color theorem while you were coloring?

  • What difficulties did you run into? How did you remedy the problems?

  • Does this constitute a proof of the four-color theorem? If so, why? If not, explain how it can be extended into a proof.

Part 2: Importing and Coloring Maps

In MW you can import maps from other sources and use them to further explore the four-color theorem. For example, the Los Alamos homepage has a number of maps useful for this exploration.

  • Download and import the GIF, Boxville.

To Download the Boxville image using a Mac:
  • Highlight Boxville, by double clicking on the map.
  • Under File choose Save As.
  • Save Boxville, by using Boxville as your filename, as a Source file, instead of Text, on the Desktop.
  • In MW, choose Import Picture under the Edit menu.
 
To Download the Boxville image using a PC:
  • Click on the Boxville map.
  • Right click on the map.
  • Choose Save Image As…
  • Save the image on the Desktop once you choose a filename.
  • In MW, under File choose Import Picture.
  • Find your image on the desktop and click Open.

  • Resize this map by clicking on a corner and dragging. Once you click outside the image it’s size becomes fixed. You are now ready to paint your map.

  • Explore the Four-Color theorem with the Boxville map. How many colors did you use?

  • Attempt to color the map with one fewer color. Note: For the second coloring open a New Project under the File menu. Now, import a new Boxville map.

  • What is the minimum number of colors needed to color the map?

  • Conjecture what would happen if we modify the conditions so that regions cannot share boundaries nor share corners. Explore the theorem under these new conditions.

  • Discuss your strategy for coloring the map and your results.

  • Return to the Los Alamos site and import another map.

  • Now explore the Four-Color Theorem under both the original conditions and the modified conditions of not allowing shared boundaries and corners.

  • What conclusions can you draw about the Four-Color Theorem under both sets of conditions?

Extensions:

  • Explore the modified Four-Color Theorem with the U.S. map. What is the minimum number of colors for the United States map?

  • Search the World Wide Web for maps that may be copied and downloaded into MicroWorlds.
 


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Last modified on August 15, 2001.