You have probably come across the 1997 Academy Award-winning movie Good Will Hunting featuring the late Robin Williams, Matt Damon, and Ben Affleck. Quickly summarized, the film centers around the fictional, tortured genius Will Hunting. Despite his intelligence and eidetic memory, Hunting works as a humble janitor at the Massachusetts Institute of Technology in Cambridge, MA.
One day, he spots a mathematics problem on a blackboard in a hallway posed by a Fields Medal award-winning professor named Gerald Lambeau. In the story, this math problem took two MIT professors two years to solve. Will Hunting solves the problem in just one day, anonymously. Eventually, the professor discovers that Hunting authored the solution, and the plot kicks off. This story has been referenced and even memed countless times within the math community. However, has something like this ever happened?
Good Will Hunting: A Math Urban Legend
There is an urban legend out there that is slightly similar to this story. As the story goes, a student shows up late to an exam. In a rush to complete his exam, he copies down the problems written on the classroom blackboard without any question or thought. He makes it through the exam questions, with the last math problem presenting only a slightly bigger challenge than usual, but he pushes through and submits his results. Later that night, he receives a frantic call from his professor, stating that he was only supposed to do the first few problems. The last question on the board was an unsolved math problem.
Though the details are slightly different, this urban legend is based on the story of the young George Bernard Dantzig, the American mathematical scientist who made contributions to industrial engineering, operations research, computer science, economics, and statistics.
As previously mentioned, there are several math problems out there that remain unsolved to this day. Some of these problems look deceptively simple, while others look like an alien language. Regardless, they exist, forever reminding us that there are ideas out there about the nature of our reality that we have yet to grasp.
If you are able to solve any of these math problems, do let us know, as some come attached with a million-dollar prize. This could be your Will Hunting moment.
The Navier-Stokes Equations
You might not know about this math problem. However, you are probably familiar with the principles that it describes. Named after the French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes, the Navier–Stokes equations are a set of partial differential equations that are used to explain the motion of viscous fluid substances. These equations could be used to describe air passing over an aircraft wing or the water flowing out the tap in your kitchen sink. However, there is a problem. The equations fail in certain situations and mathematicians are not exactly sure why.
The Navier–Stokes equations are only valid as long as the representative physical length scale of a given system is much larger than the mean free path of the molecules that make up the fluid. That is, the literal wiggle room afforded to particles in a fluid must be bigger than the box that contains them. There are people out there who have purportedly solved this conundrum only to retract their answers later on. If you feel like you have an idea of how to solve this problem, it might be worth your time. The Navier-Stokes Equation is one of seven Millennium Prize Problems, a list of mathematics problems whose correct solutions carry a prize of $1 million each.
The Collatz Conjecture
This Problem falls under the category of deceptively simple when, in reality, people have pulled their hair out trying to solve it. The funny thing is that you could probably explain it to your little brother or sister. Watch. Pick a number, any number. If you’ve selected an even number, divide it by 2.
If your number is odd, divide it by three and add 1. With your new number, repeat those same steps. Interestingly, no matter the path, you eventually get the number 1 . Mathematicians have proven that the Collatz conjecture holds true over and over again. They have not found any number out there that won't break the rules. What has eluded them is an explanation why. This year, Marijn Heule, a computer scientist at Carnegie Mellon University, announced that he plans to solve this unsolvable math problem using a computerized proof technique called SAT solving. Good Luck!
In the world of mathematics, prime numbers are oddities and the source of inspiration for two major unsolved mathematical problems. Goldbach's conjecture is one of them. Much like the Collatz conjecture, this problem is simple to explain: Is every even number greater than 2 the sum of two primes? You can try testing this conjecture right now. If you add 3 + 1, what do you get? Or what about 5 + 1? Though the answer may seem obvious, it's not. Mathematicians have found numbers that break the rules, defying all logic.
The Beal Conjecture
This Math problem looks unassuming at first, but just wait. Dubbed the Beal conjecture, this unsolved math problem centers around the formula A^x + B^y = C^z. If all of the values, including the exponents, are all positive integers, then they should all have a common prime factor. A quick reminder: factors are numbers that you multiply to generate another number.
For example, the numbers 15, 10, and 5 share the factor 5. But things quickly fall apart when your exponents are greater than 2. Going back to our example 5^1 + 10^1 = 15^1 works with no problem, but 5^2 + 10^2 ≠ 15^2 is a no-go. The answer to this mathematical dilemma will also earn you a prize of $1 million.
The Moving Sofa Problem
Yes, we are talking about that same old sofa sitting in your living room right now. The process of moving furniture around directly inspires this math problem. Whether you're moving in or moving out, you need to find a way to get your sofa through a corridor. This Unsolved geometry problem asks a straightforward question: What's the largest sofa you could possibly fit around a 90-degree corner, regardless of shape, without it bending?
It is essential to know that mathematicians are only looking at this problem through the lens of 2 dimensions. Interestingly, to this day, mathematicians have no idea about the bounds of the sofa constant, the largest area that can fit around a corner. Think of that next time your roommate says that they will not be able to get that Ikea sofa in your apartment.
Math still has a lot to show us.
Math is fascinating if only for the simple fact that once something is proven true, it is set in stone for all eternity. Of course, you can play around with the new concept, expand upon it, or even manipulate it, but the core idea never changes. This is the "romance of mathematics," says theoretical physicist, mathematician, and string theorist Brian Greene in his book Until the End of Time. Greene states that math is "Creativity constrained by logic, and a set of axioms dictates how ideas can be manipulated and combined to reveal unshakable truths."
If our study of the universe has taught us one thing, it is the fact that there are some unshakable truths out there that have yet to be discovered. Will you be the one to solve them?