Key is early on in the video with the ”mathematics isn’t ripe enough yet”. Since brute forcing the problem seems unlikely, could be that a solution to this problem takes mathematics a step to a new direction. Many useful things like modern cryptography are ultimately based on mathematical research eons ago that were once ridiculed to be useless and unnecessary by some

> Why does this need to be solved?

Obviously it "doesn't". Equally obviously this is mathematics, and it's worth doing if it's interesting.

But a somewhat more serious answer is this: the fact that we don't know the answer to this relatively simple question implies strongly that we don't know how numbers work. And if there are obvious gaps in our understanding of number theory that show themselves in trivial ways like this, there are probably more serious questions that we could answer if we had a better framework.

And that's sort of how hard problems in mathematics work. They aren't solved by amazing insight within their own realm, they turn out to be evidence that a new branch of theory needs to be developed first.

This is a simple bottle universe.

What hope do we have to make sense of our own universe if we can't solve this simple case.

In the video, Derek shows that you can see the problem as a turing machine. But if you view it as operation on strings in basis 6, you observe that the carry doesn't propagate which means that you can compute using a 1d cellular automaton (with local rules).

https://demonstrations.wolfram.com/CollatzProblemAsACellular...

This way of viewing the Collatz conjecture as a cellular automaton mean that the Collatz conjecture is like a simpler version of Conway's game of life. You can then use some memoization tricks like hashlife to compute it more efficiently.

In the game of life, some interesting patterns emerge, whereas in Collatz it is conjectured that they all end in the same pattern of desolation.

Are the Collatz cells doomed ? Is the Collatz automaton turing-complete ? Can life emerge in the Collatz universe ? Would this change anything for these small little cells ?

Why does one climb a mountain?

Decades ago the problem was considered a "deadend" because it was hard to solve but seemed unrelated to any other fields of mathematics and solving it wouldn't yield any techniques or partial solutions that could crack other well-known problems.

I wonder if that thinking has changed in the age of Computer Science since solving this kind of recursive/iterative problem might open up new ways to analyze algorithms or tackle the Halting Problem for some classes of programs.

I think the answer is we _want_ to solve it.

Now the important question is why human beings want to solve this, and what exactly do human beings want.

That's not what number theory is about. You might as well ask the same question of any art.

As for why: it’s interesting, and it is very hard, even though it looks easy.

Would it change anything? Who knows—we have yet to solve it using our current mathematical knowledge, so who knows what new mathematical knowledge will be developed in order to solve it?

From reading other comments here, we need it because it inspires some people to become (better) programmers. :-)

It's so someone can use it as a coding challenge in a technical interview, of course.