4 Simple Math Problems No One Has Solved Yet

Mathematics can be really confusing sometimes so you might be forgiven for thinking that the only math problems that can’t be solved are the complicated ones. Yet, here are some problems that are so simple that although anyone with some basic knowledge of math can understand them, nobody has been able to solve these yet.

1. Collatz Conjecture


This is arguable one of the simplest and most famous of all unsolved math problems.  Try it yourself: Pick a number. If your number is even divide it by 2, if it’s odd multiply it by 3 and add 1. Repeat this process with your answer and eventually you will end up with 1. This process will always eventually reach the number 1, regardless of which positive integer is chosen initially.

This is the Collatz Conjecture. Mathematicians have tried this process with millions of numbers but never found one that didn’t end up in 1. The challenge is to find such a number.

2. The Moving Sofa Problem

Hammersley’s sofa

This problem is something that was born straight out of our everyday experiences. Imagine that you’re moving your sofa from one room to another through a hallway. All’s well till you encounter a corner and you aren’t sure if your sofa will now fit around it. If it’s a small sofa it might, but if it’s too big it will definitely get stuck, so there must be some limit to how large the sofa can be to fit the corner.

This is the Moving Sofa Problem. What is the largest sofa that you could fit around the corner? You can be as creative as you want with the shape as well, it doesn’t just have to be our usual rectangular sofa!

Over the years there have been various types of sofas that mathematicians have come up with, like the one in the above image. But the problem compels us to find the largest possible solution, and it’s extremely difficult to show that any one possible shape is the largest possible.

Fun Fact: The solution to this problem (the unknown maximum area A) is known as the sofa constant!

3. Twin Prime Conjecture

Twin prime conjecture says that that there are infinite numbers of twin primes (a pairs of primes that differ by 2). For example, 3 and 5, 5 and 7 or 41 and 43 are all twin primes. But as the numbers get larger, prime numbers become less frequent and twin primes become even rarer. The question is do these twins ever run out?

We know that primes go on infinitely, as Greek mathematician Euclid proved almost 2,300 years ago, but we don’t have proof that twin primes go on forever as well.

4. The Inscribed Square Problem


As a math student you must have observed how you can easily inscribe squares within shapes like triangles, rectangles and circles.

But what if the shape is shapeless?

Take a closed loop for example. The loop doesn’t have to be a circle, it can be whatever shape you want it to be but it must of course be closed and the loop can’t cross itself. Now try to draw a square within the loop, with the four corners of the square touching some point of the loop. Now according to the inscribed square hypothesis, every closed loop (specifically every plane simple closed curve) should contain all four corners or vertices of a square inscribed within it. This has already been proved for a number of other shapes, such as triangles and rectangles but it hasn’t yet been proven for squares!