- 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.)
- 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?
- 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.
- 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.
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
- 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.
- 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.
- 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.)
- There are many Putnam winners who went on to become research mathematicians.
- 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.
- 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.
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.