Were the world’s most “difficult” math problems constructed to be unsolvable?
In: Mathematics
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No, not intentionally. Usually.
Most of the “hardest” math problems are such that the phrasing of the problem is rather easy, but the solution remains frustratingly out of reach. On the other hand, there are innumerably many insanely difficult, yet solvable, math problems that you will never hear about. These types of problems are typically just very dense with calculations that are hard to perform. For that reason, let’s just stick with the millennium prize problems.
Almost all of the millennium prize problems came from a very intuitive observation of facts. Some are a bit less obvious, but all of them could be understood in a few days of intense study. It is for this reason that they are so tantalizing – a layman could understand the problem – and the likely solution – in a couple of days; yet the brightest mathematical minds of the last century have only been able to crack a couple of them. You can look into it the history of some of these problems on your own, and you might be surprised just how simple the problems are to understand. The genius required to identify the problem is clear, yet the solution is counterintuitively impossible to divine.
Personally, I think the Navier-Stokes equations are the best example. The equations describe how fluids work. Scientists use the equations to model real world systems all the time, and we have analytical solutions to some systems. But there is no general solution that we can find, and stranger still, we can’t prove that a general solution doesn’t exist. The icing here is that this is a mathematical model for the real world! The solution must exist. But here we are.
The solution must exists. We know it must. But we can’t prove it. The difficulty isn’t intentional. Our math just hasn’t caught up. Or something. We just don’t know.
There is no singular “most difficult” math problem. And as time goes on, some clever people find new ways to solve some problems, while other people think up new problems. And no. The most difficult math problems would be amazing if someone figured out a way to solve them. People have earned doctorates for being able to prove the statement “a solution exists” without being able to compute the answer.
I donʼt think so. Especially how many of them intuitively _seem_ easy. Fermatʼs Last Theorem is like finding pythagorean triplets (we know a lot since ancient times) for powers greater than two. (this one got solved after 3 and a half centuries)
Collatz Conjecture is basic arithmetics. One can imagine playing with numbers like this to kill time.
P/NP problem stems from very practical questions of computer science. It seems like obviously if P equaled NP then someone should have found an algorithm that solves some NP problem in P. Nobody did, but proving that it is impossible is very tricky.
Compare this with problems that are designed to be hard and convoluted. Like the three aliens that use two words https://youtu.be/LKvjIsyYng8?si=iVydk3i0AewpdTQ2
The werent constructed as much as they were discovered. Discovered while trying to figure out the workings of the universe.
Fir example we have a simple solvable problem to convert metric to imperial measurements. On the other hand gravity governing all the matter in the universe in a well understood way dosent work, cant be converted or translated to the quantium level. This results in math problems that cannot yet be solved. This problem wasnt constructed. It was drilled down to through many solvable problems until found and they got stuck.
They’re not designed to be unsolvable on purpose. It’s rather the opposite.
The problems are *basically* unsolvable using *today’s techniques*, but some kind of solution *probably* exists. The hope is that by the time we solve them, we will have invented new clever techniques and interesting theories of math.
Just knowing the yes/no answer to every problem wouldn’t be very useful to us, so e.g. chugging the problem into a supercomputer and having it make 1000-page proof that e.g. P=NP would actually not advance our understanding of mathematics that much. Rather, the problems are a milestone that indicate that our general math intelligence (achieved through new theories and new ways of looking at the world) is now much more powerful than before. Being able to solve those problems has much greater practical use than merely having the answer.
No, maybe your math teacher did intend for it. However, at the scholar level, it’s a practice of “I figured something out that I don’t know the answer to” when you have all of the available pieces of the puzzle.
There’s something else as well. The most difficult math problems are more phylisophical than equations. That means, you can’t write them with numbers, you write them in sentences. Here’s some examples
There was a theory that you could always color in a map if you have 4 colors. There was a theory that if you take a knot and picture it from the other side, it is the same knot. There was a theory that this one way was the densest way to pack spheres.
All of these were math problems. Because you can’t really solve them with like philosophy or science.
It’s not hard to construct a math problem that’s unsolvable. For example, 3x + 5 = 3x + 4 has no solution. There’s no value of x where that will make that equation valid. In a way “there’s no solution” is a solution because that’s the point when a mathematician will stop thinking about a problem
For an example of difficult unsolved problem, take the collatz conjecture. Let’s say you have some positive number. If it’s even, divide it by 2, and if it’s odd, multiply it by 3 and add 1. If you repeat these steps over and over again, most numbers will eventually turn into a 1. In fact, we have not found a number that does not eventually lead to 1. The question is, is this the case for all positive numbers? We don’t know. To solve this, we would either need to find a number that does not eventually reach 1(kind of like finding a value for X), or show that no such number could exist(showing that no value of x could possibly exist). Either answer would be good enough to solve the problem, but we don’t have either.
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