I don't think it is accurate to say that Zeno's paradoxes are resolved by summing infinite series, and in any case Zeno's paradox is related more to physics and philosophy than mathematics.

With the Collatz conjecture there's a surface level notion of difficulty, namely that many people have tried to attack it without success. For a more technical account, Terry Tao has a nice blog post here where he gives perspective on "why the conjecture is (a) plausible, and (b) unlikely be proven by current technology".

In addition to what @testaccount said, regarding "since people also thought about Zeno's paradoxes for over a century before they discovered infinite series can be summed", if we're going to take this perspective, then "two millennia" makes more sense than "a century".

You may want to look at the Egg of Columbus. en.wikipedia.org/wiki/Egg_of_Columbus

If a problem is very well known and has been open for a long time, it is hard. Even if later a short/easy proof is found with the correct insight, I would still call it a difficult problem. See also here mathoverflow.net/questions/95837/…

@DaveL.Renfro if I'm not mistaken, Archimedes first started summing infinite series in the 2nd century BCE, although Euler developed the theory more.

@testaccount surely there are more such problems, e.g. consider a water tank with sides 10m x 10m that's currently full. Water flows out of it at a rate of $1t + 0.2$ kg/s. How long does it take before the tank empties? Without knowing calculus the problem is extremely hard; with it, it's quite simple.

To be a great achievement, it does not suffice to be hard; it needs to be important.

@GerryMyerson define important in this context.

@GerryMyerson As long as you expand "it" in "it needs to be important" to include the proof, then I'd agree with you. As long as there's something to be learned from the proof, the theorem itself can be completely useless, yet the proof may be quite a great achievement. The worst possible situation would be for an unimportant theorem to have an ad hoc proof.

@Allure: I guess the issue is when it was felt summing an infinite series resolved the paradox, which off-hand I don't know. In fact, as far as I'm aware, there are still philosophical concerns with "summing an infinite series" being a way of resolving Zeno's Achilles & Tortoise paradox. See the references in this Mathematics Stack Exchange answer.

@DanielDonnelly "Importance" might be hard to determine in the moment, but if we allow enough time to pass, the only meaningful measure of "importance" is arguably how many other authors reference the result.

One might say, since it too, several thousand years for all of humanity to get a resolution to Zeno, and even now, it is not something understood by all people who only take high school math, it is a "difficult" problem. More,interesting are the long-standing problems people tried to solve which, it turned out, were impossible to solve.

See math.stackexchange.com/questions/1733910/…

@Daniel, to be important, in this context, is to have effects beyond the specific result proved. For example, to have far-reaching applications, as the Riemann Hypothesis does, or to give us a new tool for doing mathematics, as the introduction of Galois Theory did

One attempt to make “difficulty” precise would be to prove certain superficially appealing types of solution are doomed to failure. As an extreme case of this, there are independence results, e.g. that the continuum hypothesis is independent of ZFC. But there are less extreme cases in the literature on most “hard problems”, I’d reckon.

One thing I like is when a technique suddenly makes something fundamental to be obvious. The h`omotopy proof of the fundamental theorem of algebra is not something I would call "easier," but it makes the result feel so much more intuitive. The proof relies on some pretty big machinery, but a lot of it has a geometric intuition.

I think to answer this question it would be helpful for us each to pick the well-known open problem we've secretly or not so secretly have been working on in our spare time, and show with first-hand knowledge why the problem is so difficult. Like this tried list of attacks doesn't work, and here's the wall you run into. Hence my edits.

@GerryMyerson, I have a NT attack on several problems. Approach is elementary though, so who's gonna read it, as you seem to imply. The results get into deep areas such as Group Theory, but don't use that much of it.

If you want an example, take a look at my answer here.

Talk by Prof Maria Chudnovsky of Princeton at the 2024 Joint Math Meetings, on the topic, "What Makes a Problem Hard?" youtu.be/cQfBBZ2D3yA?feature=shared