*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.*

*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.*

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 question here is: is every even number greater than 2 the sum of two primes?

It sounds obvious that the answer would be yes, after all, 3 1 = 4, 5 1 = 6 and so on.

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.

When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed.

When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice. First and foremost, the best way to become better at problem-solving is to try solving lots of problems!

Yet as I persisted with it for a long time it eventually started to click - ‘it’ referring to being able to solve problems much more easily.

This happens because your brain starts to recognise that problems fall into various categories and you subconsciously remember successes and pitfalls of previous ‘similar’ problems." A Problem-solving Heuristic for STEP Below you will find some questions you can ask yourself while you are solving a problem.

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