## Good Will Hunting Maths Problem

Honor Stringer

The film ‘Good Will Hunting’ (1997) is about an undiscovered mathematical genius who works as a janitor at MIT. One of the most crucial moments in the film is when he is able to solve a graduate level math problem, intended for the students at MIT, that had been left on a blackboard as a challenge by a professor. I wanted to see whether the problem was as complicated as it is portrayed to be and how accurate the math was in the movie, although because I haven’t studied this particular math topic, I was unable to solve it completely by myself and had to do some research to help me.

The problem:

Draw all homeomorphically irreducible trees of size n=10

Trees - ‘Graphs’ made up of connected dots (vertices) and lines (the number of lines from each dot is the number of degrees). These dots and lines cannot be connected in a cycle.

eg.

Homeomorphically - Means that there are different versions of the same tree that look different but are still the same tree.

eg.

Irreducible - Trees can’t have a dot where only two lines are connected to it.

eg.

So the question is asking for 10 dots (n=10), how many different trees can you make that are homeomorphically irreducible?

What I learned:

Overall the problem was easier than I expected once I understood it, although I definitely wouldn’t have been able to solve it without researching what the question meant. I think that it was also easier than the movie made you believe, since the film led you to believe that only an MIT professor would be able to solve it. The movie probably relied more on the fact that most people who watched it, as well as Will’s character, who hadn’t had mathematical training, wouldn’t be able understand the math terms that the question used. This did work, however, because before approaching the problem, I had no idea what “homeomorphically irreducible trees” were and so believed that the problem was as impossible as the movie wanted me to think. Although this particular problem was less difficult after I understood the question, there was more complicated math behind trees and graphs that I found through research, which is harder to understand, so I only really fully understand the problem on a visual level. However, even learning to understand the problem just this way was very rewarding, as I initially thought I wouldn’t be able to comprehend it at all. It has also made me more interested in broadening my math interest to topics I haven’t yet covered in school, and I would like to learn more in the future about graphs and trees.

Websites used:

Grime, James. “The problem in Good Will Hunting - Numberphile.” YouTube, YouTube, 4 Mar. 2013, www.youtube.com/watch?v=iW_LkYiuTKE.

Blatter, Christian. “Proof Involving a Problem from "Good Will Hunting".” Graph theory - Proof Involving a Problem from "Good Will Hunting" - Mathematics Stack Exchange, 17 Nov. 2013, math.stackexchange.com/questions/570240/proof-involving-a-problem-from-good-will-hunting.

The problem:

Draw all homeomorphically irreducible trees of size n=10

Trees - ‘Graphs’ made up of connected dots (vertices) and lines (the number of lines from each dot is the number of degrees). These dots and lines cannot be connected in a cycle.

eg.

Homeomorphically - Means that there are different versions of the same tree that look different but are still the same tree.

eg.

Irreducible - Trees can’t have a dot where only two lines are connected to it.

eg.

So the question is asking for 10 dots (n=10), how many different trees can you make that are homeomorphically irreducible?

What I learned:

Overall the problem was easier than I expected once I understood it, although I definitely wouldn’t have been able to solve it without researching what the question meant. I think that it was also easier than the movie made you believe, since the film led you to believe that only an MIT professor would be able to solve it. The movie probably relied more on the fact that most people who watched it, as well as Will’s character, who hadn’t had mathematical training, wouldn’t be able understand the math terms that the question used. This did work, however, because before approaching the problem, I had no idea what “homeomorphically irreducible trees” were and so believed that the problem was as impossible as the movie wanted me to think. Although this particular problem was less difficult after I understood the question, there was more complicated math behind trees and graphs that I found through research, which is harder to understand, so I only really fully understand the problem on a visual level. However, even learning to understand the problem just this way was very rewarding, as I initially thought I wouldn’t be able to comprehend it at all. It has also made me more interested in broadening my math interest to topics I haven’t yet covered in school, and I would like to learn more in the future about graphs and trees.

Websites used:

Grime, James. “The problem in Good Will Hunting - Numberphile.” YouTube, YouTube, 4 Mar. 2013, www.youtube.com/watch?v=iW_LkYiuTKE.

Blatter, Christian. “Proof Involving a Problem from "Good Will Hunting".” Graph theory - Proof Involving a Problem from "Good Will Hunting" - Mathematics Stack Exchange, 17 Nov. 2013, math.stackexchange.com/questions/570240/proof-involving-a-problem-from-good-will-hunting.