**The Sofa Problem**

Say you’re moving into a new house. You’ve just finished U-Hauling your furniture over, and are in the process of setting up your sofa in the living room.

There’s just one problem. The hall to the living room has a 90oo turn to it. No matter how hard you push, you can’t get your sofa past. It’s stuck.

But say you *really *want a big sofa. What’s the largest sofa you can fit past the corner?

The answer is… no joke, the **sofa constant. **It’s seriously called that, in formal mathematics. And what, you ask, is the sofa constant?

We have no idea.

We’ve got some lower and upper bounds, but the specific maximum of the area that can be bent around a 90 degree turn is an open problem in mathematics[1]. As it turns out, the question of “how can an arbitrary shape of arbitrary size be bent around a corner” is a remarkably difficult one.

Like moving wasn’t enough of a pain already.

2.

Question: Divide 3/4th square into four equal parts:

The Answer For This Question is:

(Broke the sentence into lines to make sure that you don’t see the solution while reading the question)

**The**

**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 integer **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.

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