Video: the “hardest” geometry problem in the world

Rushmore is a rather quirky but interesting movie that came out in 1998.

One of the scenes I particularly enjoyed is about the main character doing a mathematical proof. During class, Max Fischer asks the teacher about a math problem written on the board.

The teacher explains the problem is a joke: it is too hard for anyone to solve, which gets Max even more curious about the problem.

Here’s the clip after the jump.

Video: Hardest Geometry Problem in the World

Rough transcript of the clip

Teacher: Good, except when the value of the x-coordinate is equal to or less than value of the y-coordinate. Yes, Isaac?

Isaac: What about that problem?

Teacher: That, oh don’t worry about that problem.

Isaac: Why?

Teacher: I just put that up as a joke. That’s probably the hardest geometry equation in the world.

Isaac: How much extra credit is it worth?

Teacher: Well, considering I’ve never seen anyone get it right, including my mentor at MIT…I guess if anyone here could solve that problem, I’d see to it that none of you would ever have to open another math book again for the rest of your lives. Max, are you going to try it?

Max: I’m sorry, did someone say my name?

Max solves the problem on the board

Teacher: You got it.

Class cheers for Max Fischer. Max then awakes and we learn this is all a DAYDREAM

The problem that Max solved

If you watch closely, you can see the problem that Max was working on. It is not the hardest geometry in the world, nor it is even that hard of a problem at all. That was just part of Max’s fantasy that he solved an unsolved problem and became revered by his classmates.

The geometry problem may not be easy for a high school student, but it is a relatively easy one in the grand scheme of math. The problem is to derive the area of an ellipse using calculus.

I will give the movie credit because the proof is correct and includes all the necessary steps.

So I’ll conclude this post with a proof of the problem presented in the movie. Here it is:



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  • Erik

    I guess you should also put “geometry” in quotation marks.

  • http://mvallamp.blogspot.com Mahesh

    Presh
    A couple of things.

    1.) A circle is a special case of an ellipse where a=b=r and a-b=0. The formula is then pi*r*r which is the classic formula for the area of the circle. In other words, an ellipse is a circle with two centers or foci to be more precise.

    2.) Drawing an ellipse was really hard before the 1860′s and it was a complex math assignment in schools of mathematics. James Clerk Maxwell, the great English Mathematical Physicist at a very young age (14 years), demonstrated it could be done easliy with two nails nailed to thr ground and a thread of twine and when attached to a pencil and firmly moved around would create a close to perfect ellipse.

    3.) Speaking of areas, the area of a circle is the integral over 0 to r for 2*pi*x which would be the length of the rectangle and dx the width of the rectangle for a length x from the centre of the circle.

    Good stuff.

    Mahesh

  • @dilefante

    When I was in school my math teacher sometimes gave me “extra work” (with no grades) just to keep me from fooling around during the lessons. At one point he asked me to derive the area of the ellipse *without using calculus*, just through geometry. Needless to say, I failed. Only years later did I learn that this is (provably) not possible. I still wonder whether he knew this.

  • Lucas

    It depends on what you call “not using calculus”, but what I did to find the area of an ellipse is :
    -the area of a disc of radius a is Pi*a^2,
    -an ellipse is a just a circle dilated in on direction by a factor b/a,
    -then the area of an ellipse is Pi*a^2*b/a=Pi ab

    But I guess that you can easily say that any of my 3 points comes from calculus.





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