Monday Morning Math: Four Colors Suffice

The Four Color Theorem is a theorem that says that every map on a plane can be colored, with different colors in adjacent areas, using just four colors: purple, gold, pink, and teal. Or whatever your favorite colors are. The known mathematical history of this theorem began exactly 171 years ago, and by “exactly” I mean exactly. Specifically, in 1852, a law student named Francis Guthrie was coloring a map of the counties of England (for…some reason? Maybe for some law assignment, maybe just for fun) and he noticed that he only needed 4 colors. Then he thought that might be true for all maps, and tried to prove it, but wasn’t happy with his proof. So he asked his brother Frederick, who offered to ask his math teacher. And Frederick did just that, on October 23, 1852. How do we know the date? Well, Frederick’s math teacher was August de Morgan, a famous mathematician, and de Morgan wrote a letter that day to William Hamilton, another famous mathematician, which began:

My dear Hamilton
A student of mine asked me to day to give him a reason for a fact which I did not know was a fact—and do not yet. He says that if a figure be any how divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured—four colours may be wanted, but not more…

De Morgan’s letter to Hamilton, public domain

That was the start of the Four Color Conjecture making its way through the mathematical community! De Morgan wrote to more people, and people did work on it, but didn’t come up with a solution in the 1850s. Or 1860s. Or a hundred years after that. It wasn’t until the 1970s that Kenneth Appel and Wolfgang Haken came up with a proof, one that was so computer-intensive that for the next two decades, until there was another shorter proof, there was still question about whether Guthrie’s conjecture had been proven.

So let’s raise a glass/cup of coffee to the power of interesting observations, and the places they lead!

Sources:

• “Francis Guthrie: A Colourful Life” by Pieter Maritz and Sonja Mouton in The Mathematical Intelligencer.
• “9. Note on the Colouring of Maps” by Frederick Guthrie in The Proceedings of the Royal Society of Edinburgh.
• “The Four Color Theorem” in Wolfram MathWorld