The 2001 movie “A Beautiful Mind” focused on game theorist John F. Nash and was a huge success, winning four Academy Awards. It is based on Sylvia Nasar's unauthorized biography with the same title, published in 1998. There is an amazing passage in the book, which unfortunately was left out of the movie. It is where Nash visited John von Neumann in his office to discuss the main idea and result of the Ph.D. thesis he was writing. At the time Nash was a talented and promising mathematics student at Princeton University and von Neumann was a professor at the Institute for Advanced Study in Princeton, a member of a handful of mathematicians of the highest distinction. Albert Einstein was a colleague. According to the book's description of the meeting, based on an interview with Harold Kuhn, the visit was short and ended by von Neumann exclaiming: “That's trivial, you know. That's just a fixed point theorem” [N1998].

Reading the scene, I see the two men, one eager to impress with his theorem, the other seeming to try to not be impressed. Both men were competitive, more than what may be normal for mathematicians. Although few can answer difficult mathematical questions, many can tell right from wrong answers. For instance, solving an equation may be hard, but checking whether a number is a solution is easy. For this reason, contests are much more frequent in mathematics than in other academic areas. In the 16th century, mathematicians challenged each other, posing questions that they hoped would show their superiority. Nowadays, there are many mathematical contests for students, Putnam Exams, national and international Mathematical Olympiads, and others. For adult mathematicians, there is the game of challenging each other with conjectures, and winning by being the first to prove or disprove one.