Four Color Map of the World

Every week at our Tuesday-morning meeting, one member of the Center staff brings a creative, thought-provoking activity with which we kick off our meeting. One of our more recent meetings featured a fun twist on an old math problem—the Four-Color Mapping Problem. Here’s the backstory, from YoungZones.org:

Francis Guthrie, a 21-year-old mathematics student at University College in London, was mapping the counties of England in 1852 when he noticed that he only needed four colors for the map. He asked his younger brother, Frederick Guthrie, if this was true for any map. Frederick took the problem to his professor, Augustus de Morgan.

In 1878 Arthur Cayley presented this problem to the London Mathematical Society. He and various others tried to prove it over the next century. In 1976 Wolfgang Haker and Kenneth Appel of the University of Illinois proved the theorem using a computer. It took them four years to write the computer program for the Cray computer, which took 1200 hours to check 1476 configurations.

Some mathematicians are troubled by the proof by computer. They feel that a theorem so easy to understand should be able to be proved by hand. Anyone who can prove the theorem without the computer may win the Fields Medal, the math equivalent of the Nobel Peace Prize.

 

From left: Karen and Sarah, Emily and Matthew work in pair

The rules of the game are as follows:

• Each partner has 2 colors.
• Each player must lay the tiles out so that no two of the same color touch. Our problem had one additional twist—because we had square tiles, we added the stipulation that the corners couldn’t touch either.
• Taking turns, place one tile at a time on the table.
• Continue placing tiles to create a rectangle. The width and length can never vary by more than one tile. (So you can have 3 rows and 4 columns, but not 3 rows and 5 columns.

The team that uses all of their tiles first, following the above rules wins.

Oh, and two more important rules – no touching your partner’s tiles, and no talking for the first minute!

Alex (left) and Frank place tiles on the table

The Keep Away! A Four-Color Mapping Challenge is a great activity for every classroom, setting, and subject. It’s easily adaptable, because you can use pretty much any material with different colors—cut-up paper, M&Ms, colored pencils/crayons, markers and a white board, etc. It can be used as an ice breaker, a brainstorming activity, a team building activity, or a problem-solving assignment.

Here is a virtual coloring map: www.subtangent.com/maths/ig-mapcolouring.php

More information on the Four-Color Mapping Problem:

Wikipedia: Four Color Theorem

Wolfram Alpha: Four Color Theorem