NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS Nov. 1990

Coloring Maps

Mapmakers follow two rules in coloring maps:

Each country or state should be colored with a single color.
Different colors should be used for countries or states that share a common border.

Figure 1 is a map of Utah, Colorado, Arizona, and New Mexico. We could use a different color for each state, but following the mapmakers' rules, we can use fewer colors. Color the map with the fewest possible colors. If the states touch at only a single point, they can be of the same color. 1. How many colors are required? _______________
	Figure 2 is a map of another portion of the western United States. How many colors do we need for this map if we follow the mapmakers' rules? Are two colors enough? ______________ If not, can the map be colored with three colors? _______________ What is the minimum number of colors required to color the map that combines figures 1 and 2? ______________

Test your conjecture by coloring figure 3.

For each of the regional maps in figure 4, determine the fewest colors necessary to color the states following the mapmakers' rules.

Which maps require only three colors? ______________

Which maps require four colors? ______________

Do any require five colors? ______________

Is it possible for a map to require five colors? ______________

Consider figure 5, in which country A is separated into two pieces. Are five colors necessary? ______________

Are any states in the United States broken into more than one piece? _________ Which ones? ______________

Is it possible to create a map that would require six colors? ______________

Is it possible to create a map requiring five colors in which none of the countries are broken? ______________

Figure 6 shows a map of the continental United States. Four colors are enough to color this map. Color it using the mapmakers' rules.

Can you...

color the map below at a minimum cost if the four chosen colors cost $1, $2, $3, and $4 per 1000 square miles, respectively? The number in each state represents thousands of square miles.

Did you know that...

if a map is drawn on a flat plane or a globe and if all the countries or states are single, unbroken regions, then four colors always suffice to color the map (four-color theorem)?

the four-color theorem was verified in 1976 at the University of Illinois with the help of computers?

if we draw our map on a doughnut, the map may require as many as seven different colors?

Bibliography

"Thinking with Ink." In Problem-solving Strategies. (Computer program) Available from Minnesota Educational Computing Corporation, 3490 Lexington Avenue North, St. Paul, MN 55126. Challenges students to color maps with inks of varying costs.

Answers:

1. Two	4. Four	7. No	10. Yes; Michigan and Virginia
2. No	5. a, b, c	8. Maybe	11. Yes
3. Yes	6. None	9. Yes	12. Not if the map is in a plane

NCTM STUDENTS MATH NOTES is published as a supplement to the NEWS BULLETIN by the National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091. This five issues a year appear in September, November, January, March, and May. Pages may be reproduced for classroom use without permission.

Editor:	Johnny W. Lott, University of Montana, Missoula, MT 59812
Editorial Panel:	Carol Findell, Boston University, Boston, MA 02215 Judy Olson, Western Illinois University, Macomb, IL 61455 Daniel J. Teague, North Carolina School of Science and Mathematics, Durham, NC 27705
Editorial Coordinator:	Joan Armistead
Production Assistants:	Ann M. Butterfield, Sheila C. Gorg

The editors wish to thank Thomas Dick, Oregon State University, Corvallis, OR 97331, for writing this issue of NCTM Student Math Notes.

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