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I was rereading the book Littlewood's Miscellany and this passage struck me:

It used to be said that the discipline in 'manipulative skill' bore later fruit in original work. I should deny this almost absolutely - such skill is very short-winded. My actual experience has been that after a few years nothing remained to show for it all except the knack, which has lasted, of throwing off a set of (modern) Tripos questions both suitable and with the silly little touch of distinction we still feel is called for; this never bothers me as it does my juniors. (I said 'almost' absolutely; there could be rare exceptions. If Herman had been put on to some of the more elusive elementary inequalities at the right moment I can imagine his anticipating some of the latest and slickest proofs, perhaps even making new discoveries.)

I would like to ask a question to former math olympiad students who now are actively involved in math research. Do you find the training for olympiads useful in later research career as a mathematician?

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    $\begingroup$ Olympiad training in Germany 15 years ago (and probably still to these days) included active classes in several subjects that are commonly done in university. We learned quadratic residues, max-flow-min-cut with applications (including Hall), some basic convexity and majorization, elementary linear algebra including the $\mathbb{F}_p$ case, generating functions and discrete Fourier transforms IIRC. And this all was supported with better exercises than in most university classes. This alone makes it useful, nevermind the olympiad-specific methods some of which are more and some less helpful. $\endgroup$ – darij grinberg Apr 21 at 9:07
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    $\begingroup$ When I was an undergraduate I knew several of the best contest takers in the country. At least half of them dropped out of grad school before getting a PhD, although those that stayed succeeded brilliantly (Jeremy Kahn, Bjorn Poonen). It seemed to me that having put a great deal of effort into developing a skill and then discovering that it wasn't really that helpful in doing research must have been very disappointing. $\endgroup$ – Nik Weaver Apr 21 at 9:15
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    $\begingroup$ One of my professors said that according to him, problem solving and doing research required very different aptitudes and skill sets. He said he had known an IMO gold medallist who had failed quite spectacularly as a researcher (well, maybe what made it so spectacular was the fact that he had won a gold medal). Anyway, when I was a mathematician, on occasion I found it useful to switch over to "problem-solving mode", and hack my way to results which I could not obtain by (let's say) "pure thought". But I was only ever a postdoc... $\endgroup$ – RP_ Apr 21 at 10:03
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    $\begingroup$ John Rickard represented Britain in the IMO three times. He went up to Cambridge (Trinity) at the age of 15 and skipped Part 1A. But he never submitted his thesis and became a programmer. He died from liver cancer at the age of 41. I write this not to answer the question but as a memorial to a friend who should have had better pastoral care. I hope that his brother @JeremyRickard will excuse me. $\endgroup$ – Paul Taylor Apr 21 at 13:24
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    $\begingroup$ Not getting through the preliminary selection was humbling. Handling disappointments is crucial in research. $\endgroup$ – Wilberd van der Kallen Apr 23 at 8:16
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It's my belief that a large part of mathematical research, perhaps more than we would like to admit, comes down to finding clever elementary arguments. This is particularly true in my own area (combinatorics and related fields) but it is true of many other fields as well. Of course there's usually some machinery to be mastered, but at the end of the day, you're trying to come up with something new, and it's not so often that you're building some gigantic new machine out of whole cloth. Typically you're taking various known ideas and trying to figure out how to adapt them and put them together in a new way to prove something new. When the pieces of the puzzle finally fall into place, I find the experience to be not unlike the process of solving an Olympiad problem. The Olympiad training is useful for building a sense of confidence that something nontrivial can emerge with a bit of persistence and cleverness. I find that some of my colleagues without this kind of problem-solving background will sometimes give up too quickly, because they are at a loss as to how to proceed when their usual box of tools doesn't apply.

A second way in which I find Olympiad training useful is that when I am confronted with a new and difficult problem that seems too hard to tackle directly, I can often find a way to invent a toy version of the problem whose solution may give some insight. Experience with Olympiad problems has given me a sense of what a "bite-sized" problem looks like—subtle enough to be nontrivial, but simple enough to be tractable. Interestingly, I often find that some of my colleagues who are better than I am at solving Olympiad problems can often solve my bite-sized problems when I can't; at the same time, I often seem to be better than those same colleagues at coming up with the bite-sized problems in the first place. This may partly explain why some Olympiad stars don't become good mathematical researchers. Research requires several skills, and those who only know how to solve tractable problems and don't know how to formulate them in the first place may not do so well at research. But I think that experience with Olympiad problems can help with the formulation process as well.

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    $\begingroup$ A quote from timothychow.net/cv.pdf: "Princeton University Class of 1939 Prize for the senior with the highest academic standing" -- Interesting! $\endgroup$ – Wlod AA Apr 22 at 1:44
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    $\begingroup$ Excellent answer ! $\endgroup$ – Alexander Chervov Apr 24 at 10:14
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I'd point out a trivial thing: often, you just need to sit down and do a calculation, or a case-by-case analysis, etc. I mean something that can't be directly fed to Mathematica, requiring higher-level reasoning, still technical enough so that you need to crunch it with pen and paper.

Olympiad kids are trained to concentrate on such tasks and do them quickly and cleanly. Or course, any research mathematician, knowing the problem boils down to a computation, will be eventually able to do it. But they might spend more time, get distracted, make a mistake, spend even more time because of that, etc. Even more importantly, you often don't know in advance whether the outcome of your computation will solve the problem at hand. The quicker and more confident you are at such things, the more you can try.

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My personal experience and view is that there are certain olympiad type problems that, if you can solve them, demonstrate genuinely useful skills applicable in research.

Often in research you encounter similar questions perhaps as sub-problems to your main problem or in the search for special cases or counterexamples. Unless there is an obvious approach by existing theory progress usually then involves simply making an educated guess and trying to prove it. The ability to "feel" ones way to a good guess in reasonable time say by raw instinct, clever use of heuristics or analogy is vital in this case. Similar skills are needed I believe in solving many Olympiad Questions, especially under time pressure.

For example the "Pentagon Game" discussed on Matt Baker's Math Blog involves a pentagon with integers at the vertices and a rule to evolve those. You have to prove that the game ends in a finite time - the solution involves finding an positive integer invariant that always decreases. Finding this invariant quickly is non-trivial and requires the ability to guess some good options and/or exclude many which won't work. There is no standard theory to fall back on you which is often the case for genuine research problems.

(See this question for a discussion of Olympiad Questions with connections to real mathematics, many of which fit the above criteria and this paper for more mathematical developments from the "Pentagon Game".)

More generally concerning the question of whether problem solving skills are important to learn and practise for budding mathematicians you might be interested in John Hammersley's view which was certainly an outlier amongst mathematicians at the time - he believed that manipulative skills, problem solving skills were much more important than abstraction and theory which often did not help in solving real world problems.- see his article "On the enfeeblement of mathematical skills by "Modern Mathematics" and by similar soft intellectual trash in schools and universities"

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  • $\begingroup$ About the pentagon game, you write: "There is no standard theory to fall back on you which is often the case for genuine research problems." But as Chazelle's solution (sketched in Matt Baker's blog) shows, there is: The problem can be more or less restated as "any element of the affine symmetric group has a reduced expression" (i.e., if you remove its descents one by one through multiplication by $s_i$'s, it eventually becomes the identity). It's not quite about the affine symmetric group, ... $\endgroup$ – darij grinberg May 21 at 22:31
  • $\begingroup$ ... because the $\sum_{i=1}^n \omega\left(i\right) = \dbinom{n+1}{2}$ condition does not hold in general in the pentagon game, but at this point you can take the argument for the affine symmetric group and generalize it straightforwardly. $\endgroup$ – darij grinberg May 21 at 22:31
  • $\begingroup$ @darijgrinberg Thank you for your comment. I was aware of Chazelle's solution but the statement in my answer was comparing the skills required to solve problems solved at an olympiad level against those needed for genuine research questions. It is hard to imagine that any student attempting the "Pentagon Game" would be able to draw on the theory you describe and hence would be forced to guess an invariant and try to prove it. I wasn't saying that this was the only approach to this question and clearly it isn't as you rightly point out. $\endgroup$ – Ivan Meir May 25 at 10:32
  • $\begingroup$ However I think it's also true that most research mathematicians if faced with the Pentagon game during their research would not probably follow a theoretical route like Chazelle but rather guess and prove the invariant. Also I think that in this case coming up with Chazelle's approach seems a lot harder though it does obviously lead to a more general and informative solution. $\endgroup$ – Ivan Meir May 25 at 10:41

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