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.

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