The twisty history and surprise ending of the four-color theorem saga | Analysis
In 1852 a South African mathematician asked a seemingly simple question that triggered endless dispute, left a trail of overturned publications in its wake and culminated in a resolution that has stretched the very tenets of math.
Naturally, we don’t want neighboring states to have the same color, because that would make the boundaries more confusing. Under this constraint, we used four colors to fill in the map above. Could we have done it with only three? Might other maps require five or six? Francis Guthrie first conjectured the four-color theorem in 1852 when he noticed that the counties of England only needed four colors to properly fill. He suspected this rule would generalize to any map, but although any kindergartner could understand the question, neither he nor his colleagues could prove it. It was clear that three colors wouldn’t always hack it, as evidenced by the diagram below, where every region neighbors every other one.
Nobody could find a map requiring five colors, but ruling out the possibility of one altogether remained elusive. Because there are an infinite number of maps, one could never check each of them individually. A key technique toward a solution involved reducing the problem to a finite set of cases that could be checked individually. The leap from infinite to finite seems vast, but the monstrous number of cases to check still far exceeded what any person could manually process.
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