Why Mathematicians Hate Good Will Hunting

I still retrieve nan movie nighttime erstwhile I first watched Good Will Hunting pinch my mom. Matt Damon played a janitor astatine nan Massachusetts Institute of Technology. While mopping nan hallways, he walked past a blackboard pinch an precocious mathematics problem written connected it. He stopped and started solving nan problem. I watched, mesmerized, arsenic he created seemingly illegible structures of dots and lines—until abruptly a mathematics professor came retired of a speech hallway and chased him away.

The assemblage was antecedently told that that problem was meant to beryllium incredibly difficult, taking years of master reasoning to resolve, yet it was quickly worked retired by Damon’s insightful janitor successful conscionable moments. At nan time, I was fascinated by nan thought that group could person a hidden talent that nary 1 suspected was there.

As I sewage older and much mathematically savvy, I dismissed nan full point arsenic Hollywood hokum. Good Will Hunting mightiness show a awesome story, but it isn’t very realistic. In fact, nan mathematical situation doesn’t clasp up nether overmuch scrutiny. With nan grant ceremonial for nan Oscars this month, galore group are reasoning backmost connected past winners—including Good Will Hunting. It’s worthy taking a person look astatine nan blackboard successful a movie that, successful 1997, took 9 nominations and won for some original screenplay and character successful a supporting role.

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Based connected Actual Events

The movie was inspired by a existent story—one I personally find acold much compelling than nan fairy communicative type successful Good Will Hunting. The existent communicative centers George Dantzig, who would 1 time go known arsenic nan “father of linear programming.”

Dantzig was not ever a apical student. He claimed to person struggled pinch algebra successful inferior precocious school. But he was not a layperson erstwhile nan arena that inspired nan movie occurred. By that time, he was a postgraduate student successful mathematics. In 1939 he arrived precocious for a speech led by statistic professor Jerzy Neyman astatine nan University of California, Berkeley. Neyman wrote 2 problems connected nan blackboard, and Dantzig assumed they were homework.

Dantzig noted that nan task seemed harder than usual, but he still worked retired some problems and submitted his solutions to Neyman. As it turned out, he had solved what were past 2 of nan astir celebrated unsolved problems successful statistics.

That feat was rather impressive. By contrast, nan mathematical problem utilized successful nan Hollywood movie is very easy to lick erstwhile you study immoderate of nan jargon. In fact, I’ll locomotion you done it. As nan movie presents it, nan situation is this: tie each homeomorphically irreducible trees of size n = 10.

Before we spell immoderate further, I want to constituent retired 2 things. First, nan position of this situation is really nan astir difficult point astir it. It’s rather unrealistic to expect a layperson—regardless of their mathematical talent—to beryllium acquainted pinch nan method connection utilized to formulate nan problem. But that brings maine to nan 2nd point to note: erstwhile you construe nan method terms, nan existent task is simple. With a small patience and guidance, you could moreover delegate it to children.

Solving nan Good Will Hunting problem

Let’s get into nan vocabulary. In mathematics, a character is simply a type of graph—that is, a postulation of points that are connected to 1 another. Trees, notably, cannot incorporate loops, truthful you cannot link nan points successful a measurement that causes them to adjacent into one. The size of nan character is fixed successful position of nan number of points, aliases nodes, successful nan graph. In this case, we cognize we are meant to tie each imaginable character graphs pinch 10 nodes.

The word “homeomorphic” fundamentally refers to nan thought that nan nodes successful this web ever travel a peculiar sequence; nan nonstop style of nan character is not arsenic important arsenic nan series of connections. When I tie a relationship betwixt nodes A and B, I tin make that nexus longer aliases shorter aliases rotated slightly, and it won’t matter truthful agelong arsenic nan wide building of nan web remains nan same. The important portion is that A connects to B.

To deliberation astir that successful a different way, ideate a character shaped for illustration an X pinch 5 nodes and a character shaped for illustration a K pinch 5 nodes. These trees are considered to beryllium nan same character because nan number of nodes and series of connections are unchanged betwixt nan 2 shapes.

And “irreducible,” successful this case, intends that each node successful nan chart must beryllium connected by either 1 statement aliases by 3 aliases much lines specified that nary node is connected by only 2 lines: if a node was connected by only 2 lines, it could beryllium reduced into conscionable a azygous line.

So successful plain language, nan task is to tie each trees pinch nan specified properties that each person 10 nodes. There are respective approaches to this. For example, you could constitute a machine programme that solves nan task successful a fraction of a second. Or you could commencement drafting each nan graphs that fulfill these criteria by hand. It turns retired that you whitethorn only request a fewer minutes of doodling if you determine to spell pinch nan second route.

To show that, you tin first tie a character consisting of 1 cardinal node that radiates retired pinch 9 connections, giving america a full of 10 nodes. That creation meets nan required criteria—it’s 1 of our homeomorphically irreducible trees of size n = 10. Good work!

Next, tie a character pinch 8 connections—you’ll find this creation leads to a dormant extremity because you won’t beryllium capable to adhd a node without either re-creating nan erstwhile character aliases introducing a reducible line. Move connected to drafting a character that originates pinch a node that has 7 connections. You will still request to spot 2 much nodes, but you tin ideate adding them to 1 of nan 7 you’ve conscionable drawn. At this point, you should beryllium capable to support doodling done nan possibilities.

If you for illustration an moreover much systematic approach—though it whitethorn return you a spot much time, depending connected your comfortableness with chart theory—one clever solution involves considering which mathematical conditions nan trees must fulfill and representing them pinch equations.

For this approach, we tin specify nk arsenic nan number of nodes n pinch k connections. Because nan character should beryllium irreducible, location is nary condition wherever n2 tin exist, truthful n2 = 0. Furthermore, we cognize nan character must person 10 nodes total—that intends you’ll ne'er person n10 aliases n11, and truthful on. The maximum is n9.

We tin past correspond what we cognize pinch a mathematical formula:

n1 + n3 + n4 + n5 + n6 + n7 + n8 + n9 = 10

Note that we skipped n2 because we cognize that would adjacent 0.

There’s different constraint that we tin express. Our character pinch 10 nodes will yet person 18 lines, aliases connections, betwixt them if we count successful specified a measurement that nan nexus betwixt node A and node B counts twice, pinch 1 being A-B, and nan different being B-A. We tin usage that to build an equation wherever we correspond each relationship and node individually. For example, if a node links to 1 different node, it creates 1 connection: 1n1. If a azygous node links to 3 different nodes, location will beryllium 3 connections created, truthful 3n3, etcetera. This leads america to nan adjacent equation:

n1 + 3n3 + 4n4 + 5n5 + 6n6 + 7n7 + 8n8 + 9n9 = 18

Now you’ve created 2 equations that corral and constrain our tree-drawing options. But we request to harvester them to place nan position astir applicable for our task. You tin subtract nan first equation from nan 2nd to produce:

2n3 + 3n4 + 4n5 + 5n6 + 6n7 + 7n8 + 8n9 = 8

This equation serves arsenic a reference for drafting your various trees. The thought is to return position that, together, will adjacent 8 erstwhile you sum their first integer, aliases coefficient. Look astatine 8n9 for example. That tells america we only request 1 n9 to build our tree, which corresponds to nan drafting successful which a azygous node has 9 connections.

If you effort to tie n8, you’ll deed nan dead-end scenario, pinch nary character that meets our criteria. If you were utilizing our equation for reference, you wouldn’t moreover fuss trying to tie it because you’d spot you couldn’t harvester 7n8 pinch different word specified that nan first number successful each would adjacent 8.

But a node pinch 7 connections, n7,can activity if you harvester it pinch n3,meaning you tin harvester a character pinch 7 connections (represented by 6n7 successful nan equation) and a character pinch 3 connections (2n3) to find different solution to nan problem. And you tin transportation connected with nan process from there!

Better Examples Exist

I tin understand why Good Will Hunting’s filmmakers shied distant from Dantzig’s existent work. The solution he devised was not short—and nan trees are astir apt much visually appealing for a cinematographer.

But I still deliberation nan filmmakers chose this peculiar mathematics problem poorly, moreover for a Hollywood film. The history of mathematics has galore astonishing stories, including existent stories of existent laypeople solving an unfastened problem, that could beryllium awesome fodder for films.

In nan section of geometry, for example, galore breakthroughs regarding tiling nan level person been achieved by eager group who hadn’t studied mathematics aliases thing similar. One of my individual favorites occurred successful 2022, erstwhile retired people technician David Smith yet recovered the long-sought “einstein tile,” a polygon that tin capable a level wholly without immoderate gaps and without nan resulting shape ever repeating itself.

This article primitively appeared successful Spektrum der Wissenschaft and was reproduced pinch permission. It was translated from nan original German type pinch nan assistance of artificial intelligence and reviewed by our editors.