What is the Putnam exam?

The Putnam exam is the preeminent undergraduate mathematics exam in North America. It's organized by the Mathematical Association of America and is taken by over 4,000 participants at more than 500 colleges and universities. The most common motivation for taking the Putnam exam is “because it's there”: participants enjoy the challenge of attacking extremely difficult math problems that wouldn't come up in the typical math curriculum.

The exam is given on the first Saturday of December every year; it begins at 8 AM (here on the west coast) and lasts until 4 PM. (Alternate arrangements can be made for participants with religious reasons for avoiding that time.) Participants work on one set of six problems for the first three hours, then there is a lunch break, and participants work on a second set of six problems for the last three hours.

Am I eligible to take the Putnam exam?

Any student enrolled at UBC, who does yet not have an undergraduate degree, can take the Putnam exam up to four times. So unless you've already taken the Putnam exam four times or have received a bachelor's degree, you are eligible. You don't have to have a particular major, grade point average, or anything.

What should I bring to the Putnam exam when I take it?

The only thing you have to bring is something to write with. The answer sheets and scratch paper are provided at the exam. You are also allowed to bring a straightedge (not a ruler!) and compass if you wish. Calculators, notes, and other aids are not permitted.

Can I make a profit from this?

The Putnam awards cash prizes for the 25 top-scoring participants and the members of the top 5 teams. UBC also awards the more accessible Lawrence Roberts Putnam Prize, funded through a bequest by Frances Roberts in honour of her son Lawrence Roberts. The $250 awards are offered to undergraduate students who finish within the top 200 participants in a Putnam competition. (Recipients are eligible for this award only once.)

How do I register to take the Putnam exam? Is there a fee? How do I sign up for UBC's practice sessions?

If you are considering taking the Putnam exam or attending the practice sessions (you may do either or both), please register online by the first week of October. There's no fee for taking the Putnam or attending the practice sessions. You can come to the organizational meeting and start coming to the practice sessions even if you haven't registered yet; conversely, the registration is not a vow of participation, so feel free to register even if you're not sure yet whether you'll be participating.

What happens at UBC's practice sessions?

I hand out practice problems to students the week before (or they can be downloaded from here). Students try to solve the problems at home, and then we have students present solutions to the problems in the practice sessions; audience participation is also important for making sure that solutions really are airtight. The UBC math department also provides pizza and pop for Putnam practice participants!

What is the Quarter Putnam?

The Quarter Putnam is a practice exam given during one of the regular practice session times, usually the last Tuesady in October or the first Tuesday in November. Students get 1½ hours to solve 3 Putnam-like problems. This is an excellent opportunity to practice not only solving difficult problems under some time pressure and other exam conditions (including no pizza ...) but also writing up completely rigorous solutions.

How is the Putnam exam graded? What is the “Gap of Death”?

Each of the twelve problems is marked out of 10 points. Partial credit is given; however, the graders are instructed to assign only 0, 1, 2, 8, 9, or 10 as possible scores for each problem. As a result, only virtually perfect solutions receive 8 or more points; flawed solutions fall below the “Gap of Death” between 3 and 7 points, ending up with at most 2 points. Furthermore, what constitutes partial progress towards a solution is judged much more strictly than in most mathematics courses. To illustrate the difficulty of the problems and the strictness of the grading, note that the median score each year is usually 1 out of 120.

Because how clearly and completely a solution is written has such a large impact on a participant's score, I am willing to give feedback on written solutions during the term. Write up your solutions to a Putnam practice problem(s) and give them to me; I'll have feedback on your solution writing by the next practice session.

One of my courses conflicts with the practice sessions. Can I still take the Putnam exam itself? Can I participate in other ways?

Yes, you can take the Putnam exam without attending the practice sessions, or for that matter attend the practice sessions without taking the Putnam exam. (Please register in either case.) Practice materials will be posted on this web site, for you to use even if you can't make the practice sessions. You are also welcome to write up your solutions to a Putnam practice problem(s), as described above.

What topics do I need to know for the Putnam exam? What's the best way to prepare for it?

The Putnam exam doesn't test encyclopedic knowledge of advanced mathematics; rather, it tests problem solving, “thinking outside the box”, and the ability to find unexpected ways to interpret questions. Therefore the best way to prepare for the Putnam exam is to work on as many Putnam-like problems as possible. Problems from similar contests, like the Canadian/American/International Math Olympiads, can also be helpful. Be aware that every solution must be fully justified, and so Putnam problems are proof problems, as opposed to calculation problems.

As for topics, I would make the following lists:

  • Essentials: All of high-school mathematics (algebra, geometry, trigonometry), some calculus (derivatives, integrals, limits, basic differential equations), and foundations (sets, induction, roots of polynomials, binomial coefficients, the AM/GM inequality)
  • Appear often: Number theory (primes, congruences/modular arithmetic), how to set up probability problems (by counting or by integrals), linear algebra (matrix multplication, determinants), recursively defined sequences, groups
  • Appear occasionally, or can be helpful behind the scenes: Complex numbers, permutations, game theory, finite fields, abstract algebra, generating functions
Note that the topics that tend to be included on Putnam exams have drifted over the decades.

How can I get better at proofs?

Why do some people get better quickly when they work hard, while others don't seem to progress as fast? One answer is that deliberate practice is much more effective than going through the motions. From a Freakonomics blog post (boldface is my emphasis): “For example, in school and college, to develop mathematics and science expertise, we must somehow think deeply about the problems and reflect on what did and did not work. One method comes from the physicist John Wheeler (the PhD advisor of Richard Feynman). Wheeler recommended that, after we solve any problem, we think of one sentence that we could tell our earlier self that would have ‘cracked’ the problem. This kind of thinking turns each problem and its solution into an opportunity for reflection and for developing transferable reasoning tools.”

Tim Gowers, a Fields Medalist and world-class mathematical expositor as well, has written a series of essays on logic, mathematical foundations, and constructing proofs (the oldest entries are the most general and therefore probably the most helpful). Anyone who takes the trouble to thoughtfully read all these essays will definitely become better able to write and speak the language of mathematics, and their written solutions to Putnam problems will surely improve.

Where is the official Putnam web site?

There is an official Putnam web page, although it's not as good as it used to be. Other than a list of top-five individuals and teams, there's only very general information there.

Where can I get old Putnam exams and solutions?

Old Putnam exams, solutions, and results, going back to about 1995, can be found at the privately maintained website of Prof. Kiran Kedlaya.

Three books that collect old Putnam problems and solutions and three books on mathematical problem solving in general are available in the Barber Learning Centre:

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