Maths competitions tips

This is just a bunch of random maths competition experience. You may disagree or not find them useful at all. Anyway, I have committed all of the mistakes that I warn against :-)

• Do practice, but don’t work up to the point that you start hating maths, especially right before the exam. Your unconscious will put up a nice strong fight against thinking if you’re bored.

• Relax any way you can before the exam and get a good night’s sleep. There is no need (indeed no benefit) to “revise” anything on the day before. Also eat well.

• Arrive without expectations and especially try not to think about how you’re doing during the exam, be it positive or negative. Just do your stuff as best you can. Empty your mind and practice some good occlumency :-)

• Don’t assume that problems are always in increasing order by difficulty (i.e. number 3 is more difficult than number 2, etc.). Even if this was the selectors’ intention, it’s hard to judge, and there are great personal differences. Some people find it easier to have fewer problems in their brain at one time, but at least have all problems read and understood 1 hour before time is up, so that you can begin work if you have a spark.

• For this reason, never spend too much time on a single problem early on. You can still come back to it if it seems more promising than the other yet unsolved ones. Beware of the sunk cost fallacy.

• On the other hand, try not to jump back and forth between problems too often (obviously).

• Being aware of the time is nice, but don’t obsess with it. Keep fighting until the last minute and try not to panic or give up near the end.

• Don’t obsess with higher (graduate-level and beyond) maths you know. It may occasionally save you a bit of awkward technical details, but every effort is made so that competition problems not be easier for people who know a lot of theory than for those who only have a solid understanding of undergrad maths. If anything, try methods and approaches you have seen, rather than the heavy arsenal itself.

• Try Pólya’s suggestions for hard problems: consider baby versions for them first, make wishful assumptions, roll up your sleeves and check the cases n = 1, 2 by hand, etc. But don’t grow too attached to any of them. Be sure to document them – you may well earn partial credit for something that indeed brings you closer to the solution. Identify them clearly as dead ends if you want to be nice to the grader, though.

• As soon as you solved a problem, check the solution/make a plan and start writing the solution down; never wander to another problem and put off the writing part. I found that writing can be a good rest to your brain before you start thinking full-power again.

• If your argument is long or feels vulnerable to small errors, definitely try to check it before you start writing. (Ideally, “robust” proofs – those that aren’t easily killed by a small error – are preferred, but you don’t need to stick to this with time so tight.) You don’t want to laboriously fill 3 pages and then find out that your proof has an uncorrectable mistake.

• Try to write so that it’s clear to an intelligent person. I’m not familiar with Putnam policies, but I’m quite sure you can omit really trivial calculations – especially if they almost repeat something you did just before – but be sure to fully explain anything that contains ideas.

• Quote theorems if you know they have an established name (e.g. Euler’s polyhedron formula) or you know where to find them (e.g. “This lemma is proved in Niven, Zuckerman: Elementary number theory, Chapter 2”). State the theorem fully (not using your proof’s context) and identify how they apply to the situation at hand if there is any chance the grader might not know it or if the way you use it is not 100% obvious. Don’t quote a theorem without proof if it makes the problem really trivial (which would be a mistake in the problem selection – but you still risk losing your points).

• Write legibly and avoid obscure symbols or abbreviations (take the 10 seconds to clear up any non-standard notations if you need them).

• Check the logical structure of what you did. If there’s a question like, say, “which are the natural numbers n for which property A holds”, then you have 3 things to do: name a set N, check that elements of N satisfy A, and check that other natural numbers do not satisfy A. Usually this is only ever an issue when the thing you forget is trivial – which makes it (and getting the points deducted) ever more annoying. Also, try to organise your paper according to your logic, make it as linear as you can, and give the reader a short clue about what & why you’re doing if necessary. It’s worth the minute of thought that it takes.

• Finally: remember that this is only a game, however great fun it be. Keep good sportsmanship, don’t despair at all if you didn’t make it this time, and work as hard as ever if you did well. There doesn’t seem to be a huge correlation between Putnam performance and later professional success, and you can be sure that employers and grad admission committees are well aware of this.

Back to the Putnam practice page