*For example, let's use our numbers with the common prime factor of 5 from before....5This one requires a little drawing.*On a piece of paper, draw a loop - it doesn't have to be any set shape, just a closed loop that doesn't cross itself.According to the inscribed square hypothesis, inside that loop, you should be able to draw a square that has all four corners touching the loop, just like in the diagram above. but mathematically speaking, there are a whole lot of possible loop shapes out there - and it's currently impossible to say whether a square will be able to touch all of them."This has already been solved for a number of other shapes, such as triangles and rectangles," writes Thompson, "But squares are tricky, and so far a formal proof has eluded mathematicians."Goldbach's conjecture Similar to the Twin Prime conjecture, Goldbach's conjecture is another seemingly simple question about primes and is famous for how deceptively easy it is.

The Poincare conjecture was solved by Grigori Perelman.

So hard, in fact, that there's literally a whole Wikipedia page dedicated to unsolved mathematical problems, despite some of the greatest minds in the world working on them around the clock.

(This is no slight to him—I have students bring me problems I can’t solve, too! ” We ask hard questions because so many of the problems worth solving in life are hard.

If they were easy, someone else would have solved them before you got to them.

We also have some sofas that don't work, so it has to be smaller than those. If AAnd A, B, C, x, y, and z are all positive integers (whole numbers greater than 0), then A, B, and C should all have a common prime factor.

All together, we know the sofa constant has to be between 2.2195 and 2.8284." The Collatz conjecture is one of the most famous unsolved mathematical problems, because it's so simple, you can explain it to a primary-school-aged kid, and they'll probably be intrigued enough to try and find the answer for themselves. A common prime factor means that each of the numbers needs to be divisible by the same prime number.The problems are the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Poincaré conjecture, Riemann hypothesis, and Yang–Mills existence and mass gap.Only one of these problems has been solved and it is the Poincaré conjecture, which states that if every loop in a three dimensional manifold can be shrunk to a point, then the manifold can be deformed into a three-dimensional sphere.It sounds obvious that the answer would be yes, after all, 3 1 = 4, 5 1 = 6 and so on.But, again, no one's been able to prove that this will always be the case, despite years of trying.You can’t learn how to do that without fighting with problems you can’t solve.If you are consistently getting every problem in a class correct, you shouldn’t be too happy—it means you aren’t learning efficiently enough. The problem with not being challenged sufficiently goes well beyond not learning math (or whatever) as quickly as you can.But, of course, you have to maneuver it around a corner before you can get comfy on it in your living room.Rather than giving up and just buying a beanbag, at this point, mathematicians want to know: what's the largest sofa you could possible fit around a 90 degree corner, regardless of shape, without it bending?(Although they're looking at the whole thing from a two-dimensional perspective.)Thompson explains: "The largest area that can fit around a corner is called - I kid you not - the sofa constant. Eventually, if you keep going, you'll eventually end up at 1 every single time (try it for yourself, we'll wait). But the problem is that even though mathematicians have shown this is the case with millions of numbers, they haven't found any numbers out there that won't stick to the rules."It's possible that there's some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1," explains Thompson.Nobody knows for sure how big it is, but we have some pretty big sofas that do work, so we know it has to be at least as big as them. "But no one has ever been able to prove that for certain."The Beal conjecture The Beal conjecture basically goes like this...

## Comments Hardest Math Problem Ever Solved

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