What is a math question that looks easy but is difficult?

The [Collatz Conjecture](https://en.wikipedia.org/wiki/Collatz_conjecture). Basically, it's a game anyone with a basic knowledge of maths can play. Pick a positive whole number -- any number -- and then do the following to it: * If it's **even**, divide it by 2. * If it's **odd**, multiply it by 3 and add 1. * Repeat with your new number. So start with, say, 6. It's even, so divide by 2 to get 3. That's odd, so multiply by 3 and add 1 to get 10. Even, so divide to get 5. Odd, so multiply and add to get 16. Even, so divide to get 8. Even, so divide to get 4. Even, so divide to get 2. Even, so divide to get 1. Odd, so multiply and add to get 4... and you're trapped in a loop -- no matter how many times you do it now, you'll circle between 4, 2, and 1. The Collatz conjecture states that **every positive number** will eventually lead you to one -- that is, there are no other loops, right the way up to infinity. The only issue is, despite it being such a simple set-up -- and a great many mathematicians believing it to be true -- no one knows how to *prove* it. Paul Erdős claimed that 'Mathematics may not be ready for such problems', and in 2010 Jeffrey Lagarias stated that 'this is an extraordinarily difficult problem, completely out of reach of present day mathematics.' In other words, no matter how simple it appears at first glance, if you can manage to find a proof you can pretty much pick your Fields medal up at the door.
— Portarossa

I can't find an article as I'm on mobile bit the basic question is "what is the biggest couch you can fit around a corner". The question is still unresolved afaik. To give a bit more detail; the corner is defined as two corridors of x width meeting at 90 degrees and a "couch" (meaning any 2d shape) can not leave the ground. "bigness" is defined as the greatest surface area.
— Sq33KER

Fermat's theorem: no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. This was postulated by Perre de Fermat in the 17th century. He famously claimed that he could prove it, but lacked space for the proof in the book page margin where he wrote it. It seems very simple, but mathematicians have been working hard to prove it for more than 300 years. It was finally proven in a series of papers published between 1994 and 2001.
— wordserious

Goldbach's conjecture: Every even integer greater than 2 can be written as the sum of two primes. Unsolved since 1742.
— Perrrson

[Squaring the circle](https://en.wikipedia.org/wiki/Squaring_the_circle), proved unsolvable in 1882 though long before the problem had already been deemed untreatable.
— Munninnu

According to facebook posts apparently it's something as stupid as "What's 3+2*5". "only 1% of people get this right"
— egnards

Find positive integer solutions to a/(b+c) + b/(a+c) + c/(a+b) = 4 [Solution](https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y%2Bz-%2B-frac-y-z%2Bx-%2B-frac-z-x%2By-4) Edit: solution is a=154476802108746166441951315019919837485664325669565431700026634898253202035277999, b=36875131794129999827197811565225474825492979968971970996283137471637224634055579, c=4373612677928697257861252602371390152816537558161613618621437993378423467772036
— chillwombat

Everyone knows the [quadratic formula] (https://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Quadratic_formula.svg/220px-Quadratic_formula.svg.png). It can be used generally to solve every [quadratic equation](https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e). You also may have encountered the [cubic formula](https://math.vanderbilt.edu/schectex/courses/cubic/cubic.gif) which is the same, but for polynomials of degree three. There is also a [quartic formula] (https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/Quartic_Formula.svg/1920px-Quartic_Formula.svg.png) which as you may have guessed, can solve degree 4 polynomials generally (warning though it's very long). It seems like there should be another formula for quintic equations, but no one was ever able to find one. Only until very recently when Galois theory came around, were mathematicians able to show that it is impossible to find a quintic formula or a general formula for any polynomial degree 5 or higher.
— PM_ME_YOUR_JOKES

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