I once watched Good Will Hunting with a math student, and she scoffed at the so-called impossibility of the problems on the hallway blackboard.
Her skepticism is validated by Professor Robin Wilson of Gresham College:
That’s right, homeomorphically irreducible trees of degree ten have nothing to do with function analysis. And this particular problem isn’t that hard.
However, when the film was released, some were simply impressed that they actually used real math.
On NPR’s Weekend Edition back on April 4, 1998, host Scott Simon spoke with mathematician Keith Devlin about the plausibility of the math in the film. Devlin’s opinion is that “they got the math right,” and describes the blackboard problem:
What they did that was very smart was… they had to make sure that it was a problem that someone like Will Hunting, who was innately a genius but had no mathematical training, someone like him had to have been able to solve the problem… and graph theory is one of the few areas of mathematics where that can happen. Someone could literally come out of the streets — or come along the corridor at night with a mop and a bucket, which is what the Will Hunting character does — and if they’ve got the ability, they don’t need the training, and they can just solve it. They have just got to be smart.
The Weekend Edition clip is definitely worth a listen in its entirety; they go on to discuss the real life story of self-taught mathematician Srinivasa Ramanujan, part of the inspiration for the Will Hunting character, as well as what the filmmakers get not-so-right.