The problem in Good Will Hunting – Numberphile

DR. JAMES GRIME: Yeah, but like all mathematicians, he’s tall, blond, and handsome. Yeah? Yeah? -Well, he’s not tall or blond or– is he? DR. JAMES GRIME: Yeah. -Is he? He’s not tall. DR. JAMES GRIME: I don’t know. I wasn’t counting. I guess the choice of maths was arbitrary. They wanted to do a film about a troubled genius. And originally they were going to do physics genius. And then they were advised to pick a mathematical genius because it might work better with the script. It’s a really good film. And the maths is less important. Should we talk about what they did in the film, the maths? -Yeah. DR. JAMES GRIME: All right, so we’ll talk about that first. Near the beginning of the film, the MIT professor sets his students a challenge. Who can solve this? I put it on the blackboard in the corridor, and who can solve this problem? And it’s taken MIT professors two years to solve this problem.

Can you do it? Now, is it as hard as he made out? So what was the problem? If I say it first of all– I’m going to say the problem that he gave the students. It might sound like Greek to you, because some of it is Greek. Right so then I’ll say the problem. Then I’ll tell you how it works. And it’s a problem we can all do at home. I promise. So the problem is, draw all homeomorphically irreducible trees of size n equals 10. What does that mean? All right, let’s try this out. These things are called trees. So I have trees. Instead of that, they are networks of dots and lines.

So these are called graphs. It’s like the London Underground map. So a network of dots and lines. So this is called a tree. And what’s not allowed, what is banned is something like this. This is banned. This has a cycle in it, and cycles are banned. Now what was that other big word I said? Homeomorphically. That’s the worst one. That’s actually not so bad. That means if I did this, these two are the same picture. Can you see what I’ve done. I’ve just moved the dots slightly.

So you can rotate them and reflect them. Or you could move them around slightly. But those two pictures would count as the same thing. And there was another clever word. There was the word irreducible in there. So that’s another banned thing. Here this time what is banned is something like this. This is banned. Only two lines go into that dot. Which means pretty much nothing happens. You go in and you go out again. Nothing happens. There’s no change. Nothing interesting here. So this is banned as well. Those are the rules. We want to do it for 10 dots. This is the problem Will Hunting had. 10 dots, how many ways are there to do it? I can tell you, there are 10 ways to do it. What I thought might be fun is if I did a couple of them, and maybe leave some for people to try and work out what I’ve left. No? -I want to see all of them. DR. JAMES GRIME: You want to see all of them? All right. So the first one is. So it’s there. So if you can do that in less than two years, then you’re better apparently than MIT professors.

Or if you prefer, these are all the trees of size 10. Or this is a spider with nine legs and that’s a guy with a funky Afro. I don’t think that’s a particularly– I think people can do this at home. But the problem isn’t the important thing. What I really would like to talk about is who was the real good Will Hunting. The story is, well– so Will Hunting solves this problem. There is an urban legend that’s similar of a student who ran into his exam late. And he copied down the problems from the board. And he went and solved them. And the last one seemed really hard. But he kept working on it. And he managed to solve it. And he handed in his exam paper. And then the professor rings him that night saying, you were only meant to do the first few problems.

The last one was an unsolvable problem. Ah, you solved it..

As found on Youtube