No matter how much many of us hate mathematics, it’s a crucial tool in our quest to better understand the world. Almost everything we know abides by its rules, from our everyday lives – as we’re constantly solving equations and dealing with numbers – to distant objects in the cosmos. It’s a universal language not all of us really understand, but still affects everything around us.

As a discipline, mathematics has come a long way in the past few decades, though there are still many equations and problems that we haven’t solved yet. Some of them are older than you’d imagine, yet continue to baffle modern mathematicians and scientists.

## 8. Goldbach’s Conjecture

In mathematics, a conjecture is a proposition that sounds true on paper, but has never been proven to be true – or untrue. Golbach’s conjecture was proposed by a Russian mathematician Christian Goldbach in 1742, though still hasn’t been proven by anyone.

It states that every even number greater than 2 can be expressed as a sum of two prime numbers. It sounds simple enough, but in order to prove it, mathematicians require a working proof that establishes without a doubt that it’s true.

In 1938, a mathematician proved the conjecture for numbers up to 100,000. A computer program later proved it for higher numbers, too, though that still doesn’t provide a general proof. It remains one of the oldest unsolved math problems in history, even though countless mathematicians have tried to solve it since it was proposed.

## 7. Twin Primes Conjecture

Prime numbers – numbers that can’t be divided by any other number than 1 or itself – form the basis of many of the oldest and most perplexing problems in math. The Twin Primes conjecture is one of them. Also sometimes known as Polignac’s conjecture – after the mathematician that first proposed it in 1846, Alphonse de Polignac – it’s another one of the oldest unsolved math problems we know of.

According to it, there should be an infinite number of ‘twin primes’, or prime numbers that differ by 2, like 3 and 5, 5 and 7, and so on. It sounds accurate if you ask us, but mathematically, primes get rarer and much more difficult to calculate once the numbers get higher. To prove that no matter how high you go, there would always be a pair of primes differing by a factor of 2 isn’t an easy task, and has so far proven impossible for some of our best mathematicians.

## 6. The ‘Moving Sofa’ Problem

The ‘Moving Sofa’ problem is a simple, everyday problem that, despite their best efforts, mathematicians can’t seem to solve. It was proposed by Leo Moser in 1966, and many other mathematicians after him have racked their heads over it and come up with nothing.

The problem is rather simple to understand: for someone moving into a new house, what’s the largest possible couch they can turn around a corner in an L-shaped room of any given size? It’s not complicated, but more than half a decade after it was proposed, no one has been able to find a method to calculate it.

There are many ways to come up with a large enough object that could be turned around any L-shaped room, though someone could always come up with a shape of a bigger area, depending on their expertise in geometry. Currently, the largest shape that could fit around the corner is the Gerver sofa, proposed by Joseph Gerver in 1992. While many mathematicians believe it to be the solution, there’s still no way to prove it.

## 5. The Rendezvous Problem

Much like the other most perplexing problems in mathematics, the Rendezvous problem is also fairly easy to understand, but near impossible to actually solve. In its most basic form, it asks that if two people visit a park for the first time and get lost, what should they do to ensure that they’d find each other in the minimum amount of time possible? Should one of them move and the other one stay in one place? Should both of them stay in one place?

Of course, both of them staying in one place can’t be the solution, as they’d never find each other. This problem is expressed in a variety of ways. The room can be a maze, the two people can represent computers, data structures or anything else, and the problem could be applied to a wide variety of fields other than search and rescue operations.

Surprisingly, no one has been able to solve the Rendezvous problem in any of its form since 1976 – when it was first proposed by Steve Alpern – even though many mathematicians have tried.

## 4. Bellman’s ‘Lost In A Forest’ Problem

This one may sound a bit similar to the Rendezvous problem, though that’s only because both of them are expressed in the form of someone trying to escape a forest. The simply-named Bellman’s ‘Lost in a Forest’ problem is a geometrical problem first postulated in 1955 by a mathematician called Richard E. Bellman.

Basically, it aims to find out if there’s a way – or more exactly, an algorithm – for a lost hiker to accurately calculate the most optimal path out of a forest of any given area. It assumes that they know the shape of the forest, though they don’t know their starting point or where they’re facing. To leave the forest every time they try, there must be a calculation that allows them to find out the shortest worst-case distance from wherever they are in the forest to the nearest edge.

The problem remains unsolved, as no such algorithm has been found yet. While solutions exist for a few basic shapes we know of, they don’t work for a forest of any given shape, which is really what makes the problem so hard.

## 3. Fine-Tuned Universe

No matter how far we look, everywhere in the universe seems to be based on a few mathematical constants, like the gravitational constant, speed of light, mass of an electron etc. For life to exist as it is, all of these constants require their value to be the same, and accurate to a perfect degree. If these values were even slightly different, the chemical reactions that allowed life to evolve on Earth may have never happened. If it was too off, the universe may have never existed. It’s almost like everything around us is conveniently ‘fine-tuned’ for life.

This also happens to be an open, unsolved question for scientists: why are the values of these constants what they are? We’ve never been able to come up with a theory that can predict them, and it seems like these values are a part of the fundamental rules of the universe. We still don’t have a working equation or theory that can accurately derive the known value of these constants.

## 2. The Unknotting Problem

Another one of math’s biggest problems that sound ridiculously simple, the unknotting problem is an umbrella class of similar problems that have never been solved. In its most fundamental form, it asks if it’s possible to come up with an algorithm to prove that a random type of knot is actually unknotted. For clarification, a knot is a closed curve of any kind that exists in three dimensions, and it could be proven unknotted if it has no intersecting lines and exists in two dimensions.

To prove that for any knot possible, you’d have to detangle it again and again and ensure there are no intersecting lines. If the resulting figure can perfectly lie in a two-dimensional plane every time this algorithm runs, the solution could be accepted. Except, mathematicians have been trying to do exactly that for a while now, though no such algorithm has ever been found.

## 1. The Unreasonable Applicability Of Mathematics

Mathematics applies to everything we can see around us, from the smallest of objects to the largest celestial bodies in distant space. We can formulate equations in a specific field and apply it to something completely unrelated, and still come up with accurate results. As an example, geometrical constants derived from triangles and circles can show up in something as different as fluid mechanics. It’s like a universal language that doesn’t just work for everything, but works a bit too well.

The ‘unreasonable effectiveness of mathematics’ – named after a 1970 article of the same name by Eugene Wigner – isn’t a new concept, as it has been discussed by mathematicians in some form for a long time. The fact that we can come up with mathematical theories and see them work almost everywhere they’re supposed to work – even in fields not related to math at all – is baffling to them, and there’s still no rational explanation for it.