Thoughts on Math Competitions

Why they exist, and how to write a good one

MMATHS is a math contest for high-schoolers run (primarily) by Yale’s math competition club.

I’ll be talking about why students do math competitions, why students do undergrad-run math competitions, why MMATHS exists, and more!

---

zero: what’s the point of running a high school math contest?

Some think it’s to determine who’s the best at math; some think it’s to help high-schoolers stack their résumés. (This is a common viewpoint among parents.)

Some think it’s to show off the cool problems the contest organizers wrote, difficulty be damned. (This appears to be the guiding principle behind HMMT’s February contest.)

The largest competition series in the United States is the AMC series, which carries the most legitimacy and prestige.

Each year, thousands of students take the AMC 10/12, top scorers qualify for the AIME, and from these two contests a small subset qualify for the USA(J)MO. The point of the AMC series is to raise interest in math, and specifically problem-solving — at least, that’s what the website says.

Some “hardcore” students don’t make this their reason for taking the contests — their intent is to qualify for the highest round possible.

There also exist undergraduate-run math contests. Students from across the country (or world) drive or fly to college campuses; the contests are organized by volunteer-run clubs, and the problems are written by previous math competitors who’ve graduated. Popular contests include HMMT, CMIMC, PUMaC, and my own contest — MMATHS.

The people volunteering for these contests, as problem writers or organizers, are the people who enjoyed high school math contests. Many of us have great memories associated with college contests — finally meeting online friends, organizing reunions with other PROMYS or Mathcamp alumni — memories not necessarily related to the actual math.

one: what’s the point of an online college-run contest?

In high school, I took the AMC series because I enjoyed problem-solving and I wanted to prove I was good at it. But the reason I attended contests like HMMT and PUMaC was to meet up with friends; the math was an afterthought.

With regards to MMATHS this year (forced online due to the pandemic): it seems hard to justify the existence of the contest socially.

In particular, we didn’t have planned social events — this is my fault, as I had intended to organize some before the contest, but had been sidelined by an eye injury. If students wanted to “meet up,” they’d do so only outside of our scheduled Zooms; our meetings were for proctoring students taking the exams.

So, moreso than before, it was important to justify MMATHS’s existence by achieving the other goal that (some) high school math contests try to achieve: raising interest in math.

two: how do math contests actually generate interest in math?

In fact, the reverse is often true: many look at a math contest, solve the first question or two, then stare down something that looks impossible — something they’ve never seen before. “I’ll never be good at math; how can I solve this? I never learned this in school!”

This reaction comes from schooling that emphasizes the linear “learn method A, then apply method A to problems A*, A**, and A***.” But then how do people who aren’t already into math competitions get into them after taking a contest?

The key is those problems after the first few which are approachable, but not trivial. (Almost) everyone in high school has learned basic algebra. So questions like “If strawberries cost $3 and blueberries cost $2, and Deb wants 20 strawberries and 21 blueberries, how much money does she need?” should be solved by every test-taker. Those questions are trivial.

(This is what the first question on the AMC often looks like.)

On the other hand, questions like these often pique newer students’ interest. They’re quite approachable: students understand what it means to delete a digit from a number. They can experiment: what happens when we delete the units digit from 100? 570? 2020?

Doing so is equivalent to dividing the number by 10 — a conclusion that many high-schoolers can draw. The next step is to realize that one can set up an equation: if x is the number with a 0 in the units place, then x/10 is the other number, and their sum is 17402. The rest is algebra!

This problem is simple, but problems like these — problems that require trying things out, taking multiple steps, thinking through steps with certain depth — appeal to some high-schoolers, who might decide: hey, this math competition thing is kind of cool.

three: did MMATHS generate interest in math through previous years?

Upon becoming head problem editor for MMATHS and Girls in Math at Yale (GiM), I asked the previous head to send me data from past years’ contests. I’m no Excel whiz, but I can compute the mean of a list of scores. In 2019, our 500 test-takers had averaged slightly above four problems out of twelve total on the individual round. More importantly, almost 100 had solved one problem or less — a demoralizing experience.

Some people are very resilient. But most people who spend a day attending a math contest and solve but one problem — or none — during a 75-minute contest round will feel rather bad, and probably won’t want to continue learning more math.

Of course, there are other contests where many students don’t solve any problems — like the USAMO — but the point of the USAMO is not to raise interest in math. (Anyone taking this exam is a top-500 math student in the nation, by some metric.)

I won’t say that MMATHS didn’t raise interest in math in previous years, but I do claim that MMATHS could have done better.

four: it is possible to write a contest that serves many different groups of people

In this case, the groups of people are students new to math contests — impressionable kids who might choose to continue studying math based on how they feel after MMATHS — and veterans — kids who’ve done math contests for years, who’ve studied books and past exams.

An important thing to keep in mind is Goodhart's Law. That is, if I wanted to just make sure the average score on MMATHS was higher, and that students would solve more than one problem, the solution would be quite simple: make the first question 1 + 1 = ?, the second 2 + 2 = ?, and boom! — the average increases, and everyone scores a minimum of two problems. But that’s not really adapting MMATHS to serve the newer students.

Rather, the idea is to make the first few questions easy-but-gettable. The target feeling for a student to have after solving a problem is “That’s cool!”, not “Oh, another simple algebra problem.” So as head problem editor for MMATHS and GiM, I organized sets of early-contest problems that required little background: those included a digit problem with the key observation that every third number is the same; a geometry puzzle with the property that the actual points chosen don’t matter; a riddle based on two pairs of numbers with the same sum.

So that handles the newer competitors — how do you cater to experienced competitors at the same time? Firstly, I’d argue that it’s possible to make even easy problems cater to veterans — I think the aforementioned geometry problem and the riddle are cool (or at least interesting) results no matter how much math you’ve done. More importantly, though, I believe the correct strategy involves the difficulty gradient looking like an exponential curve. Something like this:

This is because the distribution of student strengths usually looks like a bell curve (normal, as it were.) The number of very strong competitors is quite small, and therefore the contest doesn’t need many difficult questions to differentiate between them.

I decided to write more easy and medium-difficulty problems because I figured that the last two to four questions would differentiate top places from one another, and that if that didn’t work, we had a tiebreaker round for a reason.

I think it’s better to err on the side of writing easier problems, provided that they are as interesting as harder ones. Especially given that MMATHS and GiM were online this year, the point of these contests should be to generate interest in math and problem solving — possible only if students can reasonably attempt the problems, if the problems are not too unapproachable.

My philosophy of math contests is that they should strive to be more inclusive. I believe that giving people problems they can try examples, generate hypotheses, find mini-results, and establish patterns for is the best way to build interest in problem-solving.

five: mmaths was not a perfect example of the contest described in the previous section

MMATHS did quite well. In my first year as head problem editor, the number of students who solved a problem or less dropped to around 25 (of 600), and the average score increased by almost two points. I reiterate that these metrics are imperfect — but I do think they reflect a conscious and successful effort to make the contest more accessible to the average participant.

Moreover, no one got a perfect score — quite a few people solved problem 12; quite a few people solved problem 11; but no one solved them both. This was also good: it meant that no one was sitting around for 10 minutes at the end of the contest wondering why it’d been so easy.

There was a large tie, and that is where MMATHS went wrong. Our tiebreaker round was not difficult enough. The system works as follows: high scorers (in this case, 10/12 and 11/12) took the tiebreaker round to determine final placings; unfortunately, many people perfected the tiebreaker round because I’d underestimated the skill level of the average top MMATHS contestant.

In the future, we’ll create a speed-based tiebreaker round instead. That eliminates the possibilities of ties.

six: recapitulation

Designing a contest to perfectly satisfy the varying abilities of 600 students is impossible.

So don’t think about it as designing a contest to satisfy everyone. Design subsets of problems to challenge subsets of students. The kids who’re new to math can grind their teeth on a tricky-but-elementary sequences problem; the kids who’ve done math all their life can hack-and-slash their way through polynomial problems involving the triple-angle identity.

This might seem an absurdly obvious realization. But in practice, almost every mock contest written on AoPS, and quite a few “official” contests, suffer from a combination of the problems described above.

It’s hard to write interesting problems of the correct difficulty level, because as a problem writer you have to imagine yourself as a kid who’s just getting into contests, and figure out what would interest you, and what would turn you off to the whole thing.

It’s easy to fall into the temptation of “this problem I wrote is super cool, I’m going to force it onto the contest whether people can solve it or not!” It’s easy to think that our job as problem writers is to design the most interesting contest possible — even if our definition of “interesting” is skewed by our own math ability, and doesn’t consider how a new contestant might react. It’s easy to make these contests about ourselves rather than about the students taking the exams. It’s easy to cater only to the tiny minority of contestants around our skill level.

And if your intent isn’t to get people interested in math, go off. Even I wrote a bunch of contests like that. But if you want to write and design a good contest, or handout, or admissions test — try to remove yourself from the equation, really think about what groups of people you’re trying to serve, figure out what’s the point of all this. You’ll do better that way.

addendum 1: a little bit of data

It’s possible to believe this is not a large problem, so here’s some recent data:

• 30 people scored 0/8 on PUMaC’s Combinatorics B round in 2020, out of 112 total. Another 31 of them solved only one problem. That’s more than half the competitors spending an hour solving a single problem or less.

• 144 people scored 0 points total at HMMT February in 2021, out of 766 total. I didn’t even try and count the number of students who solved one or two problems — keep in mind there are 30 problems total across HMMT’s three individual rounds, and about 20% of the attendees aren’t solving any of them.

• I’m not cherry-picking: on PUMaC’s Algebra B and Geometry B round, approximately 25% of competitors solved one or less problems out of eight. MMATHS had these problems in the past; I’ve already talked about that data in the rest of the post.

addendum 2: a bit more specific advice

I’ll just append a collection of tips that might help problem writers and contest organizers write better contests:

• If you think your problem is the correct difficulty, it’s probably still a bit too hard. We all write problems we can solve, and often solve easily. Our perspectives are always skewed by our knowledge of the concept or method necessary for the problem.

• Analyze the data. The first thing I did as MMATHS head problem editor was ask for the data from previous years’ MMATHS and GiM contests. I calculated average scores, numbers of low and high scores, easiest and hardest problems. Figure out what the strengths and weaknesses of your normal contest-takers are. A lot of students taking GiM are pretty young, and I didn’t fully realize this until I saw the statistics for what I thought was a relatively easy similar-triangles problem — half of these kids haven’t even taken high school geometry.

• Ask for student feedback. MMATHS didn’t really execute this very well this year (again, our surveys got messed up), but next year we’ll ask students what problems they enjoyed, what problems they felt were misplaced, and more.

• Get testsolvers who aren’t math geniuses. If anyone who’s running HMMT is reading this, go find an MIT student who scored an 80-90 on the AMC 10/12. (They exist.) Ask them to take the exam, or attempt the easier problems you set. Get their feedback.

• Teach an intro-level math class. And by intro-level, I mean middle school MATHCOUNTS to an average team, not an intense AMC 10 prep class. You’ll get to know how students at lower levels think about math, what they think about doing, what they don’t think about doing.

• Don’t be afraid of putting “stupid” problems on contests. Think about it this way: if you think a problem is truly pretty easy or dumb, it’s probably that way for 90%, maybe 95% of contestants — but perhaps 5-10% of contestants will find the problem nontrivial, and something they’d like to spend some time solving. This sort of problem is isomorphic to those last one or two problems in a contest which 90%, 95% of contestants stand no chance of solving — but perhaps 5-10% of the contestants have the background and ability to solve those problems, and no one ever has a problem with putting problems that hard on the exam.

I hope this is helpful!