Do mathematics researchers regularly solve problems like the ones from Project Euler?

Question

I'm not sure that this is the right exchange for this question. It asks about the possibilities of research in mathematics and computer science.

Extended Background

I am very interested in mathematics and theoretical computer science. Over the past few years I've gained lots of experience from math competitions, programming competitions, software development internships, and talking to professors, and I've learned that I do not enjoy writing code; I enjoy writing "beautiful" and elegant code in practice, just as I enjoy beautiful and elegant proofs in mathematics, but I do not enjoy writing hundreds of trivial imperative lines of code to make some company money. I've learned that I like to think critically about problems, and I like finding the quickest, most efficient, most elegant solutions. I enjoy the mathematical (or theoretical) side of computer science.

So, recently, after coding only with imperative languages my whole life, I discovered Haskell, a functional programming language. It is extremely close to mathematics: Everything is a function. Ideas are defined rather than executed; instead of giving a computer step-by-step instructions, the computer is given a definition. This discovery reaffirmed my passion for mathematics and the mathematical and theoretical side of computer science.

Minimal Background/Question

I've also been frequenting a website called Project Euler, which is filled with 450+ mathematics-based programming problems. This is essentially the epitome of my passion. I love solving these problems with Haskell (and sometimes, when I'm clever enough, with merely a pen and paper).

I know that many of the questions (if not all of them) have already been solved and researched in both the field of mathematics and the field of computer science, and hundreds of their variants have been as well. However, many of these problems were initially proposed and solved hundreds of years ago by mathematicians (such as Euler and Gauss), and most of the relatively newer problems were all proposed and solved by the 1980s.

So my question is, are there still problems like this that I can think about for a living? If so, which field and sub-field is closest to this type of research? If not, would I be involved in these problems by teaching and then introducing them to students of mine, or hosting competitions?

TL;DR

• Can I solve problems like this one as a career?
• If this research isn't viable, would I be more involved with this type of problems by teaching and hosting competitions, etc?

Answer 1

This may be a better question for Math.SE. That said...

These sorts of problems fall in the category of contest math. Sometimes they are related to active areas of mathematical research (most often number theory, combinatorics, and geometry), but more often, current research deals with problems that are considerably more complex and can't be solved (or even stated) in a page or two.

Note also that, at least in the US, there are very few people who get paid to do mathematics research full-time and exclusively. Most professional mathematics researchers are professors at colleges and universities, and their duties also include teaching and administration (to varying degrees).

There is a recognized subculture of mathematicians interested in contest math. They may get involved in creating problems, organizing contests, and coaching students. At some universities, such activities may be considered a significant part of their research or "scholarship" duties, but they would normally be teaching regular math classes as well.

Edit: Actually, there is a category of careers I had forgotten: intelligence. The NSA and its counterparts (GCHQ, etc) employ many thousands of mathematicians. Of course, it's hard for an outsider to know what goes on there, but it could be that their activities have more of a problem-solving flavor. At least they are not bound by the requirement that their work be publishable! The NSA has a well-known (and highly competitive) summer internship program for undergrads, the Director's Summer Program, so that could be one way of testing those waters relatively early on.

Answer 2

I believe @Mangara's comment is very relevant here:

Number theory, combinatorics and geometry (the areas Project Euler mainly draws from) actually have lots of accessible problems that are still unresolved.

Accessible here means that the problem can be easily explained. However, unresolved problems that anyone can understand tend to be extremely difficult. If they weren't, then someone would have solved them!

Get a copy of Richard Guy's Unsolved Problems in Number Theory. There you can find some problems that constitute "real mathematical research" but are more or less in the vein you describe.

You can usually only (get paid to) do this research in a university, you have to be very talented to get a tenured job in these areas, and it's a long road. Also, due to the highly interconnected nature of mathematics, the techniques used to solve these problems are often vastly different from what you might expect -- for a famous example, see the proof of Fermat's Last Theorem via algebraic curves. So if you want to work in (say) number theory, you'll still need to learn deeply about and use other areas of mathematics.

Answer 3

As @NateEldredge already explained, such easily-apprisable questions, most often arising in math contests, or as puzzles, are mostly not what could allow a person to make a living as an academic mathematician, except in cases of ultimate-extreme talent, perhaps.

At least as idealized, academic mathematics aims to develop and validate viewpoints that clarify fundamental issues. "Problem solving" is significant, but is at the extreme end of the phenomenological aspects of the business. As Nate E. noted, most "active" questions in mathematics will not be easily understood by amateurs, or perhaps even by non-specialists. These questions are mostly not of the easily-apprisable sort...

Answer 4

Here's the view from my vantage point:

• The kind of interesting problems you can solve in 30 minutes to 7 days each (as found in mathematics olympiads, competitive programming contests, and which, by the way, have tremendous value by being fun for whoever gets into them and being accessible to large numbers of high-school students, as well as developing their mental capacities and practical skills), are typically not of interest to most professional mathematicians because they are considered too trivial. Research publications require more substance than a couple of observations and a direct application of some 100-year-old theorem. Also, you would drown in the task of citing the relevant literature.

• Therefore, as a professional research mathematician, if you want to keep your job, you will be effectively forced to find a frontier somewhere in some field and publish results which are very likely to be novel and well-received by journals. You can expect to meet lots of interesting sub-problems, but also lots of questions that frustratingly don't have easy answers. Also, the chances of this type of work making impact on some field external to mathematics are not great.

• Interesting sub-problems come up all the time in all sorts of other research, not just in mathematics. Even in programming jobs that are not considered research — sometimes a requirement says that certain aspects have to behave in a certain way and meet certain constraints, and naïve solutions turn out to not be good enough.

Putting it all together

To alter the question a bit, I think there is a wide selection of jobs where 10-30% of your time will be spent on solving interesting sub-problems. These are not restricted to research in mathematics. Very few jobs where this figure will be 100% (people do this on a volunteering basis, verifying problems for Codeforces, TopCoder, etc., not to mention actually competing just for the fun of it).