Taking a Putnam (General Questions)

Question

How wide is the learning gap between completing first and second/second and third/etc problems?

Answer 1

1. The Putnam competition is offered every year on the first Saturday in December. It is limited to undergraduates, but not limited to math majors - any undergraduate in the US is eligible to take it. However, if you intend on taking the exam you need to let your math department know, preferably sometime in October or so; the Putnam committee only sends as many exams as necessary to each school.

2. In theory, every problem can be answered by someone who has had some education in Euclidean geometry, linear algebra, definitions in abstract algebra, and calculus (up to multivariable). In practice, some exposure to number theory, basic real analysis, and combinatorics will probably help. You should never need things like complex analysis, topology, advanced group theory, or other advanced topics to solve a problem on the Putnam. Knowledge of more advanced topics is never necessary, and often not particularly helpful either (though many problems have multiple solutions, and sometimes some of these solutions can pull in more advanced material).

3. The Putnam is all about knowing the tricks to solve a problem. There is a sort of standard toolbox for Putnam problems, and for the most part you shouldn't have to deviate outside of this toolbox. Things like the basic inequalities (AM-GM-HM, Cauchy-Schwarz, Holder, etc.), generating functions, pigeonhole principle. Some institutions offer a Putnam seminar or problem-solving session where such a toolbox is developed.

4. The first problems on each set are usually the easiest. The fifth and sixth problems are usually quite difficult for all takers. There are years where no one in the top 200 takers score on one of these problems. For scoring, it's better to provide a very complete solution to one problem than provide incomplete solutions to two or three problems. The scoring committee is very stingy with partial credit; answers usually score 0,1,2, 9, or 10, and rarely in between. There is, strictly speaking, not really much of a learning gap between the problems, only a major difficulty gap after problem 4.

5. To do well on the Putnam, you need practice with competition math problems. These sorts of problems are typically eventually solvable using something in a standard toolbox, but you need to put the problem in a form where you can recognize what tools you need. This sort of intuition and creativity can only be gained with practice, since by design the problems will try to avoid being intuitively approachable.

6. If you decide to take the Putnam, remember not to take it overly seriously. While doing well can be a plus to your CV should you decide to apply to graduate school or some mathematics REUs, it is at best a rather minor plus. There is no such thing as doing poorly on a test whose median score is a zero. It's supposed to be for fun, so you should have fun!