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Getty/Photo Illustration by PM Jacob Holm was flipping through proofs from an October 2019 research paper he and colleague Eva Rotenberg an associate professor in the department of applied mathematics and computer science at the Technical University of Denmark had published online, when he discovered their findings had unwittingly given away a solution to a centuries-old graph problem. Holm, an assistant professor of computer science at the University of Copenhagen, was relieved no one had caught the solution first. “It was a real ‘Eureka!’ moment,” he says. “It suddenly seemed obvious.” Holm and Rotenberg were trying to find a shortcut for determining whether a graph is “planar” that is, if it could be drawn flat on a surface without any of its lines crossing each other (flat drawings of a graph are also called “embeddings”). ....