Curiosity; a desire to know the answer is usually enough for me, for the problems I enjoy working on anyway.

Wouldn't one good usage be enough?

Wouldn't the existence of the puzzle be enough?

If we wanted to argue against, we can appeal to priorities.

Despite the popular adage, human ingenuity is limited. We can distribute more hard puzzles than puzzle solver can solve.

All else equal, if more puzzles exist than we can solve, we should solve puzzles that help advance our goals.

If we can show more important problems exist, it could be possible to make a good faith attempt at trying to see from the point of view of someone who thinks some puzzles should be ordered after other puzzles.

In that frame, the mere existence of all puzzles might be enough to justify many problems (even some "suboptimal" problems lower down the list), but there would also exist problems so uninteresting that their mere existence is not enough.

(All that being said, I'm not entirely sure why I bothered typing all of that in response to a rethorical question... but I suppose you could see arguing as a puzzle, and isn't the existence of that puzzle enough? :P)

"more important problems", very slippery slope Edit: I also have the feeling that quite often "the shoulders of giants" things stand on looked like less important problems

The flaw in your reasoning is that you can’t know a priori the value of all discovery. For example, you might make a discovery today that is key to unlocking some critical application that triples the GDP in 100 years. When is the right time to focus on it? How will you know?

The way to think about mathematics is that it’s charting what’s possible and developing new techniques to explore the various frontiers of mathematics. Figuring out practical applications happens later, once you’ve explored. Both are useful but you don’t get the latter without the former. Additionally the skill sets may differ because some people are driven by researching novelty while others are motivated by figuring out novel applications.

There’s so many stories of mathematicians thinking they were working on something useless only for it to become critically valuable. I vaguely recall a story from college about a discrete mathematician that intentionally worked on a “useless” problem space only for that to become the basic research of some branch of digital cryptography decades later.

Predicting the minimum-energy configuration of any physical system that can be modeled with a cluster of bubbles.

Motivation? Minimization is a large and old problem. In physics, some problems can be solved by noting that energy, momentum are conserved and maybe also minimized or action is minimized.

In operations research, minimization is a central theme -- linear programming, Kuhn-Tucker conditions in nonlinear programming, dynamic programming, integer linear programming (early source of NP completness theory), etc.

But more generally, it is nagging that something as simple as a soap bubble a child with some wire can create is so difficult to analyze with math. So, a guess would be that math needs some new techniques.

However, none of that would motivate me to get involved with soap bubbles. I heard Almgren lecture, and then and to now I still am not interested in investing time in soap bubbles. Instead, I want a more visible and greater need.

Pure mathematics doesn't always have an immediately known use, but it has historically been at the core of science. In the words of Carl Friedrich Gauss:

"Mathematics is the queen of the sciences."

While there are often cash prizes and prestige for solving great mathematical problems, it is also studied simply for the beauty of mathematics and for the desire to make new discoveries.

For example, algorithms eventually found their way into computers, number theory became essential to cryptography, set theory gave us fields and therefore quantum mechanics and modern electronics. Mathematicians create theories and proofs just for the sake of mathematics.

Pythagoras might have been seen as strange by his ancient contemporaries for spending his time calculating the area of squares along the sides of right triangles, but his theorems have proven essential to the progress of humanity

Strangely enough, soap bubble geometry has been a subject of interest in analog computing. Some suggest that soap films are more efficient than computers in some cases for finding proofs about surface systems. Topology has wide-reaching application from computing and electronics to physics and game theory.

https://www.americanscientist.org/article/the-soap-film-an-a...

> Tangentially, what drives the authors into solving them?

Why does a climber go up the mountain? Why does the spelunker go down the cave? Because it’s there.