Most of you have probably seen Good Will Hunting.  If you haven't seen it, go watch it.  It's a really good movie.

One problem that movies like Good Will Hunting often have is that they have to either make up a new math problem, use an existing math problem, or be really vague about what the problem actually is.  The movie Proof takes this last approach.  Gwyneth Paltrow's character solves "a really important problem."  For mathematicians like me, this approach is kind of agonizing.  I want to know what the problem is!  But the other approaches can be worse.

The book Uncle Petros and Goldbach's Conjecture actually names the proof.  Goldbach's Conjecture is one of the big unsolved conjectures in mathematics.  If anyone actually solved it, it would be a huge deal.  So right from the outset you know one of two things about the outcome of the book: either he doesn't solve it (a bit of a letdown) or he solves it but for some reason never tells anyone his proof (even more of a letdown).  I'll tell you this much: it ends in one of these two ways, and it is a letdown.  

Good Will Hunting takes an approach somewhere between making up a math problem and being really vague.  The first time I saw it, I was really impressed by the idea that Matt Damon's character could solve such a difficult problem.  But, upon closer inspection, my admiration dwindled.  On the very first day of graph theory class, my professor put in this movie and said, "Now pay attention."

When the big math problem is first introduced in the film, the professor says, "I also put an advanced Fourier system on the main hallway chalkboard..."

I happen to know what a Fourier system looks like (I wrote my thesis about Fourier series).  Fourier series are a neat concept that only prove useful in a very specific setting.  A Fourier series is a sum of sines and cosines used to approximate some function (like f(x) = x).  The formula should look something like this:
Picture
At the very least, it should have sines and cosines in it.  The graph for a Fourier series should look something like this: 
Picture
It's got a bunch of squiggly lines that look like they're trying to be some other function.  When Will looks at the hallway chalkboard (mere seconds after the professor says it's a Fourier system), this is the problem he sees written there:
Picture
Do you see any sines or cosines?  Squiggly lines?  Me neither.  This is a graph theory problem, and this type of graph is a completely different kind from the Fourier series graph I shared above.  What's more, this is an easy graph theory problem!  We solved the first half of it on our first day of class, and you could solve it too.  I'll walk you through the solution next time.  For now, enjoy your math films (I certainly do!) but don't believe everything you hear in them.
WILL
5/14/2013 07:38:10 am

Kelly,
My name is will collins. I have some questions about part #3 and #4.

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Elizabeth
5/27/2017 09:58:32 pm

Will,

Per your question re: 3 and 4 those are effectively the Fourier part of the graph theory problem - essentially the idea of Graph Fourier Transform. See http://digitalassets.lib.berkeley.edu/techreports/ucb/text/EECS-2013-209.pdf - this gives the idea behind a generating function and GFT generally.

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Ryan
11/15/2017 06:31:44 am

Kelly,

Did you ever post the walk through to this problem, I saw the movie for the first time yesterday and I'm curious on the explanation of the walk through.

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Greg Haskins
1/4/2018 04:50:55 pm

Please explain all this to me as if I was a 5 year old..

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    Kelly Patton has somehow completed 20 years of formal mathematical education with her love of math intact.  She wishes every person were so lucky, so that's why she writes this blog.