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Monday, March 23, 2009

The Putnam Exam: Some Thoughts

The Putnam Exam is a Math Competition which began in 1938. The current form is to have 6 problems in a 3-hour morning session and 6 problems in a 3-hour afternoon session. (Some of the older exams have 7 problems or have you pick 6 out of 7). Some observations.
  1. I went through all of the exams and classified which problems were in physics and which were in combinatorics (to confirm some suspicions that I had- see next two points). The results are in this document. I am sure that if you were to do a similar classification you might disagree with me on some problems, but our tallies would not differ by much. (You can find the problems from 1985 until now here. Most years the problems and their solutions appear in the The American Mathematics Monthly.)
  2. The older exams asked more about physics. To be more precise the exams have had 18 problem on physics: 14 of them before 1960, and only 4 of them since then (1973,1974,1975,1981). Why this trend? I suspect that before 1960 it was just assumed that a math major knew basic classical mechanics--- now it's not as universal an assumption. Any other ideas?
  3. Combinatorics has been a relatively recent popular topic for problems. There have been 32 problems in Combinatorics. 21 since 1978. In my lifetime I have seen combinatorics become more part of the ugrad curriculum, so this may be the reason.
  4. In 1953 one of the problems was to show (in today's terms) that if you 2-color the edges of K6 then there will be a monochromatic triangle. This is now so well known that I doubt they would ask it.
  5. What is better for a young math major to do: study for competitions like the Putnam exam, or do a research project? Both are certainly good. I've asked around and got the following responses.
    1. Several students have told me that taking the exam got them interested in some parts of math they had not heard of, so then they wanted to (and did) do research.
    2. Some other students told me that the exam is good since its a finite well defined goal that they can focus on. By contrast, for a younger (perhaps immature) student doing research is uncomfortably vague.
    3. Another student likened math competition people to people who know lots of words for SCRABBLE but don't know what they mean. I disagree. To understand answers to old exams and to generate answers to new exams you need to understand some real math. (I think that in World Champion Scrabble they should only give you 1/2 of the points if you don't know the meaning of the word.)
    4. There are many Putnam winners who went on to become research mathematicians.
    5. There are many Putnam winners who did not go on to become research mathematicians. I've heard it said of some of them that they could do a problem given to them but not come up with problems on their own. I am skeptical of this as an explanation since there are all kinds of people who and good and bad at all kinds of things. Certainly being a good problem solver does not decrease your problem-creation ability.
  6. Personal note: I took the Putnam exam three times. My best score was a 33 (basically 3 problems right out of 12) in 1980. For one of the problems I felt a bit odd getting it right. I had a year long course in combinatorics (rare in those days) so I knew how to do it from knowledge not cleverness. How good is a 33? Respectable, but not worth putting in my essay to grad school. It was 125th in the country (I think out of around 2000) and my school (SUNY Stonybrook) gave me a copy of Godel-Escher-Bach since it was the highest score at the school. Joel Spencer proctored the exam and finished it, I suspect with a perfect score, in half the time.

16 comments:

  1. I think that for people going into any field with a lot of complex Math would benefit by preparing for and participating in the Putnam competition. Far too often in the class setting, you're not presented with enough diversity of problem type to train you and get you used to making comprehensive use of all that you know and have been exposed too. And bering better able to pull from all that you've done before is a valuable asset.

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  2. I know that, if nothing else, it helps you keep your thoughts on math, particularly if you're spending a semester fulfilling other requirements...

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  3. I spent a good portion of my childhood participating in various math competitions, including Putnam. I agree that preparation for such competitions requires love/understanding of math, but, after you reach amateur level, it becomes much more like "professional training": you read different problems, memorize techniques, and generally start working on speed rather than creativity. You also need a good coach. Indeed, without a coach I was 33rd in my junior year. Then we had a great coach at NYU, and I won it the following year. Then I did not practice for a year, went to proctor the Putnam at MIT next year, and I barely solved half of the problems (no training for a year kills your speed).

    To sum up, after some basic training you reach the level of being able to solve most problems, but slowly (say, between an hour and a day depending on the difficulty). At which point you only train for the speed.

    Regarding the use for research, I found the sets of skills required largely orthogonal. In a competition, you know there is a relatively short solution, and you try to find it as soon as you can. In research, you do not even know what is the right question, let alone if it has short/elegant solution. I struggled a lot during my first few years at MIT, thinking that the love for puzzles would somehow be useful for me. Only when i completely stopped "wasting" time on the puzzles, the research somehow picked up.

    Nowadays, I try to stay away from Putnam-like problems at all cost: they are addictive and take time from research :).

    Yevgeniy Dodis

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  4. The older exams asked more about physics. [...] Why this trend?

    It might be because of changing attitudes toward physics and about mathematics itself. V.I. Arnold starts this interesting talk with "Mathematics is a part of physics", and blames Bourbaki (etc.) for the change. :-)

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  5. Many of the people who participate and win the Putnam exam are the same people who've been doing math competitions in high school, and are simply applying the problem-solving skills they already have rather than spending large amounts of time preparing for Putnam specifically. So the choice between studying for Putnam versus doing research is not one most people have to make -- for the first two years, one can do well based on what you already know and minimal practice, and afterwards, as one's competitive skills deteriorate due to insufficient "professional training", one is usually more ready and acquainted with mathematics to shift to research.

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  6. My best score on the Putnam was as a freshman after having done math contests throughout high school. It declined steadily each subsequent year as the stultifying nature of writing down layered definitions in the typical advanced math course took its toll. There is little real problem-solving cleverness required in most such courses, but a lot of pushing around definitions.

    CS theory in my senior year became a welcome return to some of the aspects of math that I had so enjoyed in high school math competitions.

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  7. To Anon 6:

    You do realize that those courses are not "pushing around definitions" for the sake of pushing around definitions, don't you? Definitions are supposed to help you see clearer and better, allowing you to soar higher like a bird.

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  8. Personal Opinion: The reason for shift from Physics to Combinatorics is the problems in Physics needs some fixed methods to solve them, either you have learned it, or it is nearly impossible to solve the problem, where as in combinatorics, you always have very simple looking problems which need real creativity, and I think IMO and IMC are better than Putnam, maybe due to attendance of students from old Eastern Block. I have checked the list of Gold Medal winers in IMO and saw many Fields Medal winners among them. I think problem solving is orthogonal to theory/concept building, but the norm is bounded.

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  9. Certainly being a good problem solver does not decrease your problem-creation ability.


    Sure, but neither does it greatly increase it. The correlation is decent but far from perfect.

    Based on people I've known, here are some reasons why Putnam winners may not become researchers:

    (1) Some just have many interests and are good at many things. Eventually they are going to specialize in something, and there's no reason it has to be mathematics.

    (2) Some people love the puzzle aspect of contests, the fact that you know there's always a beautiful solution and you just have to figure out the trick. They may just not enjoy working on research problems, which might (and often do) turn out to be unsolvable or ugly or not the right question after all. It's a very different experience.

    (3) Some people love the competitive aspect more than the subject area. When the opportunity to win disappears, they lose interest (or move on to another area with a well-defined notion of winning, such as making money on Wall Street).

    (4) Even when someone does love the subject area, it can be hard to adjust to not being the best. If you are the top contest problem solver in your year in the whole country (or even world), but turn out to be only the tenth best researcher, it's tough. Instead of feeling happy about being among the world's best researchers, you can end up feeling sad about no longer being the very best. It's irrational but all too human, and sometimes it is psychologically easier just to do something else.

    (5) Attention span: some people are really good at focusing intently on one thing for an hour or two but have trouble maintaining their enthusiasm throughout a six-month research project. It's easy to get frustrated or impatient.

    (6) Opportunity: becoming either a contest winner or a great researcher is correlated with certain opportunities, such as having a great coach or mentor. This isn't necessary, but it helps a lot. Most people never have such luck, and those who do usually have it in one area but not the other (otherwise they'd be doubly lucky). So if you had a fantastic olympiad coach but inadvertently choose a lousy advisor, you're less likely to become a top researcher.

    (7) A few people win contests through raw talent, but most Putnam fellows combine talent with considerable training and practice. This training is relatively straightforward in its approach. It's not at all easy (you have to be very smart and very hard working), but at least you basically know what to do to improve. By contrast, what should you do to become a good researcher? It's a lot less clear, so some Putnam fellows will lose some of the advantage they got from diligent practice. (Hard work still pays off, but it pays off the most when it's clear what you should be working on.)

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  10. To Anon 6:

    You do realize that those courses are not "pushing around definitions" for the sake of pushing around definitions, don't you? Definitions are supposed to help you see clearer and better, allowing you to soar higher like a bird.


    I am not denying the beauty and value of the mathematics involved. Fundamental algebra, Galois theory, Fourier analysis, analytic functions, etc, were all highly appealing. There are great insights involved in coming up with these definitions. However, courses present most definitions as a fait accompli. "We will go to the next definition because it will be useful in the proof of a theorem about a problem that we haven't yet stated." The problem-solving detective game and failed attempts to prove theorems that led to the definitions is buried. Problem sets are designed for students to explore properties of the definitions so that they internalize them. This is the part that is stultifying. Internalizing the definitions is important but it is a step removed from the actual problem-solving.

    Anon 6

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  11. This is the part that is stultifying. Internalizing the definitions is important but it is a step removed from the actual problem-solving.

    There is a deeper issue here. Is mathematics about mere "problem solving" or understanding properties of structures that makes problem solving possible (and indeed trivial) ? If mathematics is to be considered as a serious scientific discipline, and not a mere collection of little tricks, then I believe it should be the latter. If you believe this, then the little tricks you get trained for IMO or the Putnam are indeed orthogonal to mathematics research. The correlation between research mathematicians and IMO winners is just that those who have the temperament to become mathematicans are the natural candidates for math teams at the school or college level. A more serious correlation would be if many of these famous mathematicians who are past IMO/Putnam winners are currently involved in these contests which would testify to their faith in these contests. I believe this number is quite small but I might be mistaken on this.

    As a side not, I think G.H. Hardy devoted a lot of energy trying to (apparently unsuccessfully) abolish the Cambridge Tripos (which could be seen as a British equivalent to Putnam).

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  12. ... stultifying nature of writing down layered definitions in the typical advanced math course ...

    There is a pedagogical issue here. Of course, instead of defining abstract groups a course could meander through the questions about symmetry of roots of polynomials etc. that motivated Galois and others to arrive at the notion of groups even if they did not call it so. Similarly, for analytic functions and L^2 spaces, or for that matter the definition of real numbers not to mention the complex numbers, and so forth. But each of these notions took almost a human life span or more to take shape -- and it would be unrealistic take a Lamarckian approach to math pedagogy. Its important I guess to emphasize this right at the beginning of a math course. Not everything can be or will be motivated.
    So LEARN THIS NOW AS ALL YOUR MATHEMTATICAL FOREFATHERS DID DURING THEIR MATHEMTICAL APPRENTICEHOOD and have faith that it will be USEFUL LATER.

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  13. A more serious correlation would be if many of these famous mathematicians who are past IMO/Putnam winners are currently involved in these contests

    Richard Stanley and Hartley Rogers are the coaches of the MIT Putnam team. Kiran Kedlaya also proctors the exam and is additionally involved in high school math contests. Akamai (read: "Tom Leighton") gives scholarships to math contest winners. These are just a few I know about in my vicinity.

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  14. Is mathematics about mere "problem solving" or understanding properties of structures that makes problem solving possible (and indeed trivial) ?

    Understanding the properties of structures is part and parcel of problem-solving. I never implied that I was distinguishing the two (though I agree that this skill is not exercised by Putnam competitions).

    One does not need to repeat the entire line of false starts to motivate the right path.
    We motivate the group definition by immediately providing a number of examples that group theory unifies. We motivate non-Abelian groups by giving natural examples such as matrices or permutations.

    The issue of the way math courses are frequently presented is reflected in one of the standard styles of mathematical papers vs the usual standard in CS theory papers. In CS theory papers, one frequently sees extensive motivating discussions for definitions, intuitions for proofs, etc included in the write-up. In math, the opposite is often true - people try to cover their tracks and stick to the bare definitions, theorems, and proofs. A great math writer like Timothy Gowers, who gives both motivations and intuitions, is the exception rather than the rule.

    Wouldn't math teaching be much better if it were taught with these motivations and intuitions?

    Anon 6

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  15. In math, the opposite is often true - people try to cover their tracks and stick to the bare definitions, theorems, and proofs.

    This is a common canard and is not true. It is true that referees and editors of math journals will not as a rule tolerate pages of fluff -- as a result math papers often look terse by comparison (to say theoretical CS papers). On the other hand the typical math paper is better written (often adhering to the two months in the drawer rule) with proper motivations/intuitions etc. -- but it is true that they also have a higher expectation on the part of the reader.

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  16. IMO, many math and tcs students would prefer to read failed attacks on problems, not just the successful ones. They want to (at least I did) learn *why*, not just what is true; they want to see research in action. I think books can be written better than they are, to help readers understand things deeper. Maybe some day authors will put exercises like: Try proving the theorem using this other idea/method and explain why it fails.

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