Why Do Most Maps Only Use 4 Colors?

Have you ever noticed that any map you look at now a days only uses four colors?  Let’s take this example of the United States.

map

Not counting Mexico and Canada, all the states are colored with either green, red, purple or yellow.  Let’s zoom out and look at this map of the world:

world

The only colors needed here are blue, purple, red and yellow.  By colors needed we mean the map is colored in such a way that no two touching territories have the same color.  For instance, the United States and Canada are different colors above.

So why are four colors sufficient?  Although this problem seems so simple, it was one of the hardest problems to prove mathematically.  A man who got close was Charles Lutwidge Dodgason or as you may know him, Lewis Carroll the man who wrote Alice’s Adventures in Wonderland.  As Dodgason explains, four colors will always be needed for a map in which three regions surround a fouth like below:

fourcolors

He claimed every map took on this form at one point or another so four colors were sufficient.

This question was first posed  in 1852 in a letter written by Francis Guthrie to his brother.  His brother showed the problem to his math professor and the obsession over solving the four color problem commenced.  It took this math professor, Dodgson, Alfred Kempe and over 100 years before anyone who have a solid proof.

In 1976 two math professors at University of Illinois, Kenneth Appel and Wolfgang Haken, finally thought they had figured it out.  This proof however relied on complex computer software and trial and error.  This proof was hundreds of pages long and is one of the hardest proofs to read and understand.  To this day some mathematicians still believe this proof could have been flawed.  Proof that not all things are as simple as they seem.

Can you find a better proof to the four color problem?

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