When mathematicians look at the four color map problem, they often turn the regions into vertices or nodes - here represented by ovals - and draw a line between any two nodes whose regions share a border. This object is called a graph, and because none of the lines connecting nodes crosses any other, the specific type of graph is called planar.
Dr. Kenneth Appel proved that all such planar graphs can be painted with four or less colors such that no two nodes that have a line connecting them directly are the same color. If you think about any graph, you could always add a new node an a few lines from that node to some of the other nodes on the existing graph. The implication of this is that there are infinitely many graphs.
Here's the tricky part of Dr. Appel's proof and I will not try to present an explanation for it. There are ways to make one graph equivalent to another, which means that if one graph can be four-colored, the equivalent graph can be four-colored as well. Even though there are infinitely many graphs, Dr. Appel's method proved there are exactly 1,936 different equivalence classes.
It's not uncommon for mathematical proofs to have to several cases that have to be taken separately, but 1,936 is a lot of cases. Instead of doing each one by hand, Appel decided to write a computer program that looked at every case. When it was finished and it had looked at all the data, it had proved each of the cases could be color with no more than four colors and the proof was complete.
There were many mathematicians who were not satisfied with the new method, but there were many who saw this as a way to use computers to forward the field of mathematical inquiry. While the controversy is not yet over, Appel's proof is accepted and the use of computers in math research is much more common now than it was in the 1970s.
Best wishes to the family and friends of Dr. Kenneth Appel, from a fan.
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