Some Thoughts on Olympiad Material Design

(This is a bit of a follow-up to the solution reading post last month. Spoiler warnings: USAMO 2014/6, USAMO 2012/2, TSTST 2016/4, and hints for ELMO 2013/1, IMO 2016/2.)

I want to say a little about the process which I use to design my olympiad handouts and classes these days (and thus by extension the way I personally think about problems). The short summary is that my teaching style is centered around showing connections and recurring themes between problems.

Now let me explain this in more detail.

1. Main ideas

Solutions to olympiad problems can look quite different from one another at a surface level, but typically they center around one or two main ideas, as I describe in my post on reading solutions. Because details are easy to work out once you have the main idea, as far as learning is concerned you can more or less throw away the details and pay most of your attention to main ideas.

Thus whenever I solve an olympiad problem, I make a deliberate effort to summarize the solution in a few sentences, such that I basically know how to do it from there. I also make a deliberate effort, whenever I write up a solution in my notes, to structure it so that my future self can see all the key ideas at a glance and thus be able to understand the general path of the solution immediately.

The example I’ve previously mentioned is USAMO 2014/6.

Example 1 (USAMO 2014, Gabriel Dospinescu)

Prove that there is a constant {c>0} with the following property: If {a, b, n} are positive integers such that {\gcd(a+i, b+j)>1} for all {i, j \in \{0, 1, \dots, n\}}, then

\displaystyle  \min\{a, b\}> (cn)^n.

If you look at any complete solution to the problem, you will see a lot of technical estimates involving {\zeta(2)} and the like. But the main idea is very simple: “consider an {N \times N} table of primes and note the small primes cannot adequately cover the board, since {\sum p^{-2} < \frac{1}{2}}”. Once you have this main idea the technical estimates are just the grunt work that you force yourself to do if you’re a contestant (and don’t do if you’re retired like me).

Thus the study of olympiad problems is reduced to the study of main ideas behind these problems.

2. Taxonomy

So how do we come up with the main ideas? Of course I won’t be able to answer this question completely, because therein lies most of the difficulty of olympiads.

But I do have some progress in this way. It comes down to seeing how main ideas are similar to each other. I spend a lot of time trying to classify the main ideas into categories or themes, based on how similar they feel to one another. If I see one theme pop up over and over, then I can make it into a class.

I think olympiad taxonomy is severely underrated, and generally not done correctly. The status quo is that people do bucket sorts based on the particular technical details which are present in the problem. This is correlated with the main ideas, but the two do not always coincide.

An example where technical sort works okay is Euclidean geometry. Here is a simple example: harmonic bundles in projective geometry. As I explain in my book, there are a few “basic” configurations involved:

  • Midpoints and parallel lines
  • The Ceva / Menelaus configuration
  • Harmonic quadrilateral / symmedian configuration
  • Apollonian circle (right angle and bisectors)

(For a reference, see Lemmas 2, 4, 5 and Exercise 0 here.) Thus from experience, any time I see one of these pictures inside the current diagram, I think to myself that “this problem feels projective”; and if there is a way to do so I try to use harmonic bundles on it.

An example where technical sort fails is the “pigeonhole principle”. A typical problem in such a class looks something like USAMO 2012/2.

Example 2 (USAMO 2012, Gregory Galperin)

A circle is divided into congruent arcs by {432} points. The points are colored in four colors such that some {108} points are colored Red, some {108} points are colored Green, some {108} points are colored Blue, and the remaining {108} points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

It’s true that the official solution uses the words “pigeonhole principle” but that is not really the heart of the matter; the key idea is that you consider all possible rotations and count the number of incidences. (In any case, such calculations are better done using expected value anyways.)

Now why is taxonomy a good thing for learning and teaching? The reason is that building connections and seeing similarities is most easily done by simultaneously presenting several related problems. I’ve actually mentioned this already in a different blog post, but let me give the demonstration again.

Suppose I wrote down the following:

\displaystyle  \begin{array}{lll} A1 & B11 & C8 \\ A9 & B44 & C27 \\ A49 & B33 & C343 \\ A16 & B99 & C1 \\ A25 & B22 & C125 \end{array}

You can tell what each of the {A}‘s, {B}‘s, {C}‘s have in common by looking for a few moments. But what happens if I intertwine them?

\displaystyle  \begin{array}{lllll} B11 & C27 & C343 & A1 & A9 \\ C125 & B33 & A49 & B44 & A25 \\ A16 & B99 & B22 & C8 & C1 \end{array}

This is the same information, but now you have to work much harder to notice the association between the letters and the numbers they’re next to.

This is why, if you are an olympiad student, I strongly encourage you to keep a journal or blog of the problems you’ve done. Solving olympiad problems takes lots of time and so it’s worth it to spend at least a few minutes jotting down the main ideas. And once you have enough of these, you can start to see new connections between problems you haven’t seen before, rather than being confined to thinking about individual problems in isolation. (Additionally, it means you will never have redo problems to which you forgot the solution — learn from my mistake here.)

3. Ten buckets of geometry

I want to elaborate more on geometry in general. These days, if I see a solution to a Euclidean geometry problem, then I mentally store the problem and solution into one (or more) buckets. I can even tell you what my buckets are:

  1. Direct angle chasing
  2. Power of a point / radical axis
  3. Homothety, similar triangles, ratios
  4. Recognizing some standard configuration (see Yufei for a list)
  5. Doing some length calculations
  6. Complex numbers
  7. Barycentric coordinates
  8. Inversion
  9. Harmonic bundles or pole/polar and homography
  10. Spiral similarity, Miquel points

which my dedicated fans probably recognize as the ten chapters of my textbook. (Problems may also fall in more than one bucket if for example they are difficult and require multiple key ideas, or if there are multiple solutions.)

Now whenever I see a new geometry problem, the diagram will often “feel” similar to problems in a certain bucket. Exactly what I mean by “feel” is hard to formalize — it’s a certain gut feeling that you pick up by doing enough examples. There are some things you can say, such as “problems which feature a central circle and feet of altitudes tend to fall in bucket 6”, or “problems which only involve incidence always fall in bucket 9”. But it seems hard to come up with an exhaustive list of hard rules that will do better than human intuition.

4. How do problems feel?

But as I said in my post on reading solutions, there are deeper lessons to teach than just technical details.

For examples of themes on opposite ends of the spectrum, let’s move on to combinatorics. Geometry is quite structured and so the themes in the main ideas tend to translate to specific theorems used in the solution. Combinatorics is much less structured and many of the themes I use in combinatorics cannot really be formalized. (Consequently, since everyone else seems to mostly teach technical themes, several of the combinatorics themes I teach are idiosyncratic, and to my knowledge are not taught by anyone else.)

For example, one of the unusual themes I teach is called Global. It’s about the idea that to solve a problem, you can just kind of “add up everything at once”, for example using linearity of expectation, or by double-counting, or whatever. In particular these kinds of approach ignore the “local” details of the problem. It’s hard to make this precise, so I’ll just give two recent examples.

Example 3 (ELMO 2013, Ray Li)

Let {a_1,a_2,\dots,a_9} be nine real numbers, not necessarily distinct, with average {m}. Let {A} denote the number of triples {1 \le i < j < k \le 9} for which {a_i + a_j + a_k \ge 3m}. What is the minimum possible value of {A}?

Example 4 (IMO 2016)

Find all integers {n} for which each cell of {n \times n} table can be filled with one of the letters {I}, {M} and {O} in such a way that:

  • In each row and column, one third of the entries are {I}, one third are {M} and one third are {O}; and
  • in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are {I}, one third are {M} and one third are {O}.

If you look at the solutions to these problems, they have the same “feeling” of adding everything up, even though the specific techniques are somewhat different (double-counting for the former, diagonals modulo {3} for the latter). Nonetheless, my experience with problems similar to the former was immensely helpful for the latter, and it’s why I was able to solve the IMO problem.

5. Gaps

This perspective also explains why I’m relatively bad at functional equations. There are some things I can say that may be useful (see my handouts), but much of the time these are just technical tricks. (When sorting functional equations in my head, I have a bucket called “standard fare” meaning that you “just do work”; as far I can tell this bucket is pretty useless.) I always feel stupid teaching functional equations, because I never have many good insights to say.

Part of the reason is that functional equations often don’t have a main idea at all. Consequently it’s hard for me to do useful taxonomy on them.

Then sometimes you run into something like the windmill problem, the solution of which is fairly “novel”, not being similar to problems that come up in training. I have yet to figure out a good way to train students to be able to solve windmill-like problems.

6. Surprise

I’ll close by mentioning one common way I come up with a theme.

Sometimes I will run across an olympiad problem {P} which I solve quickly, and think should be very easy, and yet once I start grading {P} I find that the scores are much lower than I expected. Since the way I solve problems is by drawing experience from similar previous problems, this must mean that I’ve subconsciously found a general framework to solve problems like {P}, which is not obvious to my students yet. So if I can put my finger on what that framework is, then I have something new to say.

The most recent example I can think of when this happened was TSTST 2016/4 which was given last June (and was also a very elegant problem, at least in my opinion).

Example 5 (TSTST 2016, Linus Hamilton)

Let {n > 1} be a positive integers. Prove that we must apply the Euler {\varphi} function at least {\log_3 n} times before reaching {1}.

I solved this problem very quickly when we were drafting the TSTST exam, figuring out the solution while walking to dinner. So I was quite surprised when I looked at the scores for the problem and found out that empirically it was not that easy.

After I thought about this, I have a new tentative idea. You see, when doing this problem I really was thinking about “what does this {\varphi} operation do?”. You can think of {n} as an infinite tuple

\displaystyle  \left(\nu_2(n), \nu_3(n), \nu_5(n), \nu_7(n), \dots \right)

of prime exponents. Then the {\varphi} can be thought of as an operation which takes each nonzero component, decreases it by one, and then adds some particular vector back. For example, if {\nu_7(n) > 0} then {\nu_7} is decreased by one and each of {\nu_2(n)} and {\nu_3(n)} are increased by one. In any case, if you look at this behavior for long enough you will see that the {\nu_2} coordinate is a natural way to “track time” in successive {\varphi} operations; once you figure this out, getting the bound of {\log_3 n} is quite natural. (Details left as exercise to reader.)

Now when I read through the solutions, I found that many of them had not really tried to think of the problem in such a “structured” way, and had tried to directly solve it by for example trying to prove {\varphi(n) \ge n/3} (which is false) or something similar to this. I realized that had the students just ignored the task “prove {n \le 3^k}” and spent some time getting a better understanding of the {\varphi} structure, they would have had a much better chance at solving the problem. Why had I known that structural thinking would be helpful? I couldn’t quite explain it, but it had something to do with the fact that the “main object” of the question was “set in stone”; there was no “degrees of freedom” in it, and it was concrete enough that I felt like I could understand it. Once I understood how multiple {\varphi} operations behaved, the bit about {\log_3 n} almost served as an “answer extraction” mechanism.

These thoughts led to the recent development of a class which I named Rigid, which is all about problems where the point is not to immediately try to prove what the question asks for, but to first step back and understand completely how a particular rigid structure (like the {\varphi} in this problem) behaves, and to then solve the problem using this understanding.

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Against Hook-Length on USAMO 2016/2

A recent USAMO problem asked the contestant to prove that

\displaystyle  (k^2)! \cdot \prod_{j=0}^{k-1} \frac{j!}{(j+k)!}

is an integer for every {k \in \mathbb N}. Unfortunately, it appears that this is a special case of the so-called hook-length formula, applied to a {k \times k} Young tableau, and several students appealed to this fact without proof to produce one-line solutions lacking any substance. This has led to a major controversy about how such solutions should be graded, in particular whether they should receive the {7^-} treatment for “essentially correct solutions”, or the {0^+} treatment for “essentially not solved”.

In this post I want to argue that I think that these solutions deserve a score of {1}.

1. Disclaimers

However, before I do so, I would like to make some disclaimers:

  • This issue is apparently extremely polarized: everyone seems to strongly believe one side or the other.
  • This was an extremely poor choice of a USAMO problem, and so there is no “good” way to grade the HL solutions, only a least bad way. The correct solution to the dilemma is to not have used the problem at all. Yet here’s the bloodied patient, and here we are in the emergency room.
  • While I am a grader for the USAMO, I am one of many graders, and what I say in this post does not necessarily reflect the viewpoints of other USAMO graders or what score the problem will actually receive. In other words, this is my own view and not official in any way.

One last remark is that I do not consider the hook-length formula to be a “well-known” result like so many contestants seem to want to pretend it is. However, this results in the danger of what constitutes “well-known” or not. So in what follows I’ll pretend that the HL formula is about as well-known as, say, the Pascal or Zsigmondy theorem, even though I personally don’t think that this is the case.

One final disclaimer: I am unlikely to respond to further comments about an issue this polarized, since I have already spent many hours doing so, and I’ve done enough of my duty. So if I don’t respond to a comment of yours, please don’t take it personally.

2. Rule for citations

Here is the policy I use for citations when grading:

  • You can cite any named result as long as it does not trivialize the problem.
  • If the result trivializes the problem, then you are required to prove the result (or otherwise demonstrate you understand the proof) in order to use it.

This is what I’ve heard every time I have asked or answered this question; I have never heard anything to the contrary.

Some people apparently want to nit-pick about how “trivialize” is not objective. I think this is silly. If you really want a definition of “trivialize”, you can take “equivalent to the problem or a generalization of the problem” as a rule of thumb.

Clearly it follows from my rule above that the hook-length formula deserves {0^+} grading, so the remainder of the post is dedicated to justifying why I think this is the correct rule.

3. Subjective grading

I would rather have an accurate subjective criteria than a poor objective one.

In an ideal world, grading would be completely objective: a solution which solves the problem earns {7^-} points and a solution which does not solve the problem earns {0^+} points. But in practice this is of course not possible, unless we expect our contestants to write their solutions in a formal language like Coq. Since this is totally infeasible, we instead use informal proofs: students write their solutions in English and human graders assign a score based on whether the solution could in principle be compiled into a formal language proof.

What this means is that in grading, there are subjective decisions made all of the time. A good example of this is omitting steps. Suppose a student writes “case {B} is similar [to case {A}]”. Then the grader has to decide whether or not the student actually knows that the other case is similar or not, or is just bluffing. One one extreme, if {A} and {B} really are identical, then the grader would probably accept the claim. On the other extreme, if {A} and {B} have some substantial differences, then the grader would almost certainly reject the claim. This implies that there is some intermediate “gray area” which cannot be objectively defined.

Fortunately, most of these decisions are clear, thus USAMO grading appears externally to be mostly objective. In other words, accurate grading and objective grading are correlated (yay math!), but they are not exactly the same. All scores are ultimately discretion by human graders, and not some set of rigid guidelines.

Citations are a special case of this. By citing a theorem, a student is taking advantage of convention that a well-known proof to both the student and grader can be omitted from the write-up.

4. Citing the problem

In light of this I don’t think you should get points for citing the problem. I think we all agree that if a student writes “this is a special case of IMO Shortlist 1999 G8”, they shouldn’t get very many points.

The issue with citing HL in lieu of solving the problem is that the hook-length formula is very hard to prove, and it is not reasonable to do so in an olympiad setting. Consequently, it is near certain that these students have essentially zero understanding of the solution to the problem; they have not solved it, they have only named it.

Similarly, I think you should not earn points for trivializing a problem using Dirichlet, Prime Number Theorem, Zsigmondy, etc. This is also historically consistent with the way that grading has been done in the past (hi Palmer!).

5. Citing intermediate steps

Now consider the usage of difficult theorems such as Dirichlet, Prime Number Theorem, etc. on a solution in which they are merely an intermediate step rather than the entire problem. The consensus here is that citing these results is okay (though it is not unanimous; some super harsh graders also want to take off points here, but they are very few in my experience).

I think this is acceptable, because in this case the contestant has done 90% of the solution, and cannot do the remaining 10% but recognize it as well-known. In my eyes this is considered as essentially solving the problem, because the last missing bit is a standard fact. But somehow if you cannot do 100% of the problem, I don’t think that counts as solving the problem.

What I mean is there is a subjective dependence both on how much of the solution the student actually understands, and how accessible the result is. Unfortunately, the HL solutions earn the worst possible rank in both of the categories: the student understands 0% of the solution and moreover the result is completely inaccessible.

6. Common complaints

Here are the various complaints that people have made to me.

  • “HL is well-known.”
    Well, I don’t think it is, but in any case that’s not the point; you cannot cite a result which is (a) more general than the problem, and (b) to which you don’t understand the proof.

  • “Your criteria is subjective!”
    So what? I would rather have an accurate subjective criteria than a poor objective one.

  • “It’s the problem writer’s fault, so students should get {7}.”
    This is not an ethics issue. It is not appropriate to award points of sympathy on a serious contest such as the USAMO; score inflation is not going to help anyone.

  • “It’s elitist for the graders to decide what counts as trivialized.”
    That’s the grader’s job. Again I would rather have an accurate subjective criteria than a poor objective one. In practice I think the graders are very reasonable about decisions.

  • “I don’t think anyone disputes that the HL solution is a correct one, so certainly the math dictates a {7^-}.”
    I dispute it: I don’t think citing HL is a solution at all.

  • “Why do we let students use Pascal / Cauchy / etc?”
    Because these results are much more reasonable to prove, and the “one-line” solutions using Pascal and Cauchy are not completely trivial; the length of a solution is not the same thing as its difficulty. Of course this is ultimately subjective, but I would rather have an accurate subjective criteria than a poor objective one.

  • “HL solutions have substance, because we made an observation that the quantity is actually the number of ways to do something.”
    That’s why I wish to award {1} instead of {0}.

  • “Your rule isn’t written anywhere.”
    Unfortunately, none of the rules governing USAMO are written anywhere. I agree this is bad and the USAMO should publish some more clear description of the grading mechanics.

  • “The proof of the HLF isn’t even that complicated.”
    Are you joking me?

In summary, I don’t think it is appropriate to give full marks to a student who cannot solve a problem just because they can name it.

Stop Paying Me Per Hour

Occasionally I am approached by parents who ask me if I am available to teach their child in olympiad math. This is flattering enough that I’ve even said yes a few times, but I’m always confused why the question is “can you tutor my child?” instead of “do you think tutoring would help, and if so, can you tutor my child?”.

Here are my thoughts on the latter question.

Charging by Salt

I’m going to start by clearing up the big misconception which inspired the title of this post.

The way tutoring works is very roughly like the following: I meet with the student once every week, with custom-made materials. Then I give them some practice problems to work on (“homework”), which I also grade. I throw in some mock olympiads. I strongly encourage my students to email me with questions as they come up. Rinse and repeat.

The actual logistics vary; for example, for small in-person groups I prefer to do every other week for 3 hours. But the thing that never changes is how the parents pay me. It’s always the same: I get N \gg 0 dollars per hour for the actual in-person meeting, and 0 dollars per hour for preparing materials, grading homework, responding to questions, and writing the mock olympiads.

Now I’m not complaining because N is embarrassingly large. But one day I realized that this pricing system is giving parents the wrong impression. They now think the bulk of the work is from me taking the time to meet with their child, and that the homework is to help reinforce what I talk about in class. After all, this is what high school does, right?

I’m here to tell you that this is completely wrong.

It’s the other way around: the class is meant to supplement the homework. Think salt: for most dishes you can’t get away with having zero salt, but you still don’t price a dish based on how much salt is in it. Similarly, you can’t excise the in-person meeting altogether, but the dirty secret is that the classtime isn’t the core component.

I mean, here’s the thing.

  • When you take the IMO, you are alone with a sheet of paper that says “Problem 1”, “Problem 2”, “Problem 3”.
  • When you do my homework, you are alone with a sheet of paper that says “Problem 1”, “Problem 2”, “Problem 3”.
  • When you’re in my class, you get to see my beautiful smiling face plus a sheet of paper that says “Theorem 1”, “Example 2”, “Example 3”.

Which of these is not like the other?

Active Ingredients

So we’ve established that the main active ingredient is actually you working on problems alone in your room. If so, why do you need a teacher at all?

The answer depends on what the word “need” means. No USA IMO contestant in my recent memory has had a coach, so you don’t need a coach. But there are some good reasons why one might be helpful.

Some obvious reasons are social:

  • Forces you to work regularly; though most top students don’t really have a problem with self-motivation
  • You have a person to talk to. This can be nice if you are relatively isolated from the rest of the math community (e.g. due to geography).
  • You have someone who will answer your questions. (I can’t tell you how jealous I am right now.)
  • Feedback on solutions to problems. This includes student’s written solutions (stylistic remarks, or things like “this lemma you proved in your solution is actually just a special case of X” and so on) as well as explaining solutions to problems the student fails to solve.

In short, it’s much more engaging to study math with a real person.

Those reasons don’t depend so much on the instructor’s actual ability. Here are some reasons which do:

  • Guidance. An instructor can tell you what things to learn or work on based on their own experience in the past, and can often point you to things that you didn’t know existed.
  • It’s a big plus if the instructor has a good taste in problems. Some problems are bad and don’t teach you anything; some (old) problems don’t resemble the flavor of problems that actually appear on olympiads. On the flip side, some problems are very instructive or very pretty, and it’s great if your teacher knows what these are.
  • Ideally, also a good taste in topics. For example, I strongly object to classes titled “collinearity and concurrence” because this may as well be called “geometry”, and I think that such global classes are too broad to do anything useful. Conversely, examples of topics I think should be classes but aren’t: “looking at equality cases”, “explicit constructions”, “Hall’s marriage theorem”, “greedy algorithms”. I make this point a lot more explicitly in Section 2 of this blog post of mine.

In short, you’re also paying for the material and expertise. Past IMO medalists know how the contest scene works. Parents and (beginning) students less so.

Lastly, the reason which I personally think is most important:

  • Conveys strong intuition/heuristics, both globally and for specific problems. It’s hard to give concrete examples of this, but a few global ones I know were particularly helpful for me: “look at maximal things” (Po-Shen Loh on greedy algorithms), “DURR WE WANT STUFF TO CANCEL” (David Yang on FE’s), “use obvious inequalities” (Gabriel Dospinescu on analytic NT), which are take-aways that have gotten me a lot of points. This is also my biggest criteria for evaluating my own written exposition.

You guys know this feeling, I’m sure: when your English teacher assigned you an passage to read, the fastest way to understand it is to not read the passage but to ask the person sitting next to you what it’s saying. I think this is in part because most people are awful at writing and don’t even know how to write for other human beings.

The situation in olympiads is the same. I estimate listening to me explain a solution is maybe 4 to 10 times faster than reading the official solution. Turns out that writing up official solutions for contests is a huge chore, so most people just throw a sequence of steps at the reader without even bothering to identify the main ideas. (As a contest organizer, I’m certainly guilty of this laziness too!)

Aside: I think this is a large part of why my olympiad handouts and other writings have been so well-received. Disclaimer: this was supposed to be a list of what makes a good instructor, but due to narcissism it ended up being a list of things I focus on when teaching.

Caveat Emptor

And now I explain why the top IMO candidates still got by without teachers.

It turns out that the amount of math preparation time that students put in doesn’t seem to be a normal distribution. It’s a log normal distribution. And the reason is this: it’s hard to do a really good job on anything you don’t think about in the shower.

Officially, when I was a contestant I spent maybe 20 hours a week doing math contest preparation. But the actual amount of time is higher. The reason is that I would think about math contests more like 24/7. During English class, I would often be daydreaming about the inequality I worked on last night. On the car ride home, I would idly think about what I was going to teach my middle school students the next week. To say nothing of showers: during my showers I would draw geometry diagrams on the wall with water on my finger.

So spiritually, I maybe spent 10 times as much time on math olympiads compared to an average USA(J)MO qualifier.

And that factor of 10 is enormous. Even if I as a coach can cause you to learn two or three or four times more efficiently, you will still lose to that factor of 10. I’d guess my actual multiplier is somewhere between 2 and 3, so there you go. (Edit: this used to say 3 to 4, I think that’s too high now.)

The best I can do is hope that, in addition to making my student’s training more efficient, I also cause my students to like math more.

On Problem Sets

(It appears to be May 7 — good luck to all the national MathCounts competitors tomorrow!)

1. An 8.044 Problem

Recently I saw a 8.044 physics problem set which contained the problem

Consider a system of {N} almost independent harmonic oscillators whose energy in a microcanonical ensemble is given by {E = \frac 12 \hbar \omega N + \hbar \omega M}. Show that this energy can be obtained is {\frac{(M+N-1)!}{M!(N-1)!}}.

Once you remove the physics fluff, it immediately reduces to

Show the number of nonnegative integer solutions to {M = \sum_{i=1}^N n_i} is {\frac{(M+N-1)!}{M!(N-1)!}}.

And as anyone who has done lots of math contests knows, this is the famous stars and bars problem (also known as balls and urns).

This made me really upset when I saw it, for two reasons. One, the main difficulty of the question isn’t related to the physics at hand at all. Once you plug in the definition you get a fairly elegant combinatorics problem, not a physics problem. And secondly, although the solution to the (unrelated) combinatorics is nice, it’s very tricky. I don’t think I could have come up with it easily if I hadn’t seen it before. Either you’ve seen the stars-and-bars trick before and the problem is trivial, or you haven’t seen the trick, and you could easily spend a couple hours trying to come up with a solution — and none of that two hours is teaching you any physics.

You can see why a physics instructor might give this as a homework problem. The solution is short and elementary, something that a undergraduate student could understand and write down. But somewhere at MIT, some poor non-mathematician just spent a good chunk of their evening struggling with this one-trick classic and probably not learning much from it.

2. Don’t I Like Hard Problems?

Well, “not learning much from it” is not entirely accurate\dots

Something that bothered me (and which I hope also bothers the reader) was I complained that the problem was “tricky”. That seems off, because as you might already know, I like hard problems; in fact, in high school I was well despised for helping teachers find hard extra credit problems to pose. (“Hard” isn’t quite the same as “tricky”, but that’s a different direction altogether.) After all, hard problems from math contests taught me to think, isn’t that right?

Well, maybe what’s wrong is that there’s no physics in the hard part of the problem; the bonus problems I provided for my teachers were all closely tied to the material at hand. But that doesn’t seem right either. Euclidean geometry might be useless outside of high school, but nonetheless all the time I spent developing barycentric coordinates still made me a smarter person. Similarly, Richard Rusczyk will often tell you that geometry problems trained him for running the business that is now the Art of Problem Solving. For exactly the same reason, thinking about the stars and bars problem is certainly good for the mind, isn’t that right? Why was I upset about it?

Well, I still hold my objection that there’s no physics in the problem. Why? So at this point we’re naturally led to ask: what was the point of the problem set in the first place? And that answer this, you have to ask: what was the point of the class in the first place?

On paper, it’s to learn physics. Is that really all? Maybe the professor thinks it’s important to teach students how to think as well. Does she? And the answer here is I really don’t know, because I have no idea who’s teaching the class. So I’ll instead ask the more idealistic question: should she?

And surprisingly, I think the answer can be very different from place to place.

On one extreme, I think high school math should be mainly about teaching students to think. Virtually none of the students will actually use the specific content being taught in the class. Why does the average high school student need to know what {\int_{[0,1]} x^2 \; dx} is? They don’t, and that shouldn’t be the point of the class; not the least of reasons being that in ten years half of them won’t even remember what {\int} means anymore.

But on the other extreme, if you have a math major trying to learn the undergraduate curriculum the picture can change entirely, just because there is so much math to cover. It’s kind of ridiculous, honestly: take the average incoming freshman and the average senior math major, and the latter will know so much more than the former. So in this case I would be much more worried about the content of the course; assuming for example that I’m hoping to be a math major, the chance that the (main ideas of) the specific content will be useful later on is far higher.

This is especially true for, say, students who did math contests extensively in high school, because that ability to solve hard problems is already there; it’s not an interesting use of time to be slowly doing challenging exercises in group theory when there’s still modules, rings, fields, categories, algebraic geometry, homological algebra, all untouched (to say nothing of analysis).

What this boils down to is trying to distinguish between the actual content of the given class (something very local) versus the more general skill of problem-solving or thinking. In high school I focused almost exclusively on the latter; as time passes I’ve been shifting my focus farther and farther to the former.

3. {\text{A} \ge 90\%}

Now suppose that we are interested in teaching how to think on these problem sets. There’s one other difference between the problem sets and math contests. You’re expected to finish your problem sets and you’re not expected to finish math contests.

I want to complain that there seems to be a stigma that you have to do exercises in order to learn math or physics or whatever, and that people who give up on them are somehow lazy or something. It is true in some sense that you can only learn math by doing. It is probably true that thinking about a hard problem will teach you something. What is not true is that you should always stare at a problem until either it or you cracks.

This is obviously true in math contests too. One of the things I was really bad at was giving up on a problem after hours of no progress. In some sense the time limit of contests is kind of nice; it cuts you off from spending too long on any one problem. You can’t be expected to be able to solve all hard problems, or else they’re not hard.

Problem sets fare much more poorly in this respect. The benefit of thinking about the hard problem diminishes over time (e.g. a typical exercise can teach you more in the first hour than it does in the next six) and sometimes you’re just totally dead in the water after a couple hours of staring. The big guy seem to implicitly tell you that you should keep working because it’s supposed to be hard. Is that really true? It certainly wasn’t true in the math contest world, so I don’t see any reason why it’s true here.

In other words, I don’t think our poor physics student would have lost much by giving up on balls and urns after a few hours. And really, for all the warnings that looking up problems online is immoral, is asking your friend to help really that different?

Teaching A* USAMO Camp

In the last week of December I got a position as the morning instructor for the A* USAMO winter camp. Having long lost interest in coaching for short-answer contests, I’d been looking forward to an opportunity to teach an olympiad class for ages, and so I was absolutely psyched for that week. In this post I’ll talk about some of the thoughts I had while teaching, in no particular order.

1. Class Format

Here were the constraints I was working with. After removing guest lectures, exams, and so on I had four days of teaching time, one for each of the four olympiad subjects (algebra, geometry, combinatorics, number theory). I taught the morning session, meaning I had a three-hour block each day (with a 15-minute break). I had a wonderfully small class — just five students.

Here’s the format I used for the class, which seemed to work reasonably well (as in, if I were to teach the class again I would probably not change it very much.)

  • (0:00-0:10) I usually started the class with a quick warm-up problem (something pretty easy), just to soak up time from latecomers and give students a chance to get ready and glance through the handout. (If you give smart students a pretty handout, the first thing they will do is look through it, regardless of what you tell them to actually do.)
  • (0:10-1:30) Afterwards I would go through the lecture, both theory and examples, up until the break. On average this got split up with about half the time for the theory and half the time for the examples. I typically let students try the examples themselves for five minutes (again, smart students will automatically start on the problems regardless of whether you tell them to or not) before I discuss the solution, just so they at least have a feeling for what it is — I consider it immoral to start talking about a solution before students have had a chance to try a problem.
  • (1:30-2:40) After a break, I would give the students a long period (a little over an hour) to try the practice problems in the last section of the handout. Since the class was so small, I would prepare about 5-7 practice problems and then let each of them pick a different problem to start working on. (Once they solved their own problem, they would go on and try other ones.)During this time, I was able to take advantage of the small class size in a pretty great way: throughout the hour I would walk around the room talking to each of the students about the problem they were working on. In particular, I tried to make sure every student at least solved the problem they started with.
  • (2:40-3:00) In the last 20 minutes of class or so, I had each student present the solution to the problem which they worked on. I think the main utility of this is that it forces the present-er to know clearly in their head what their solution is. This was actually possibly more useful for my feedback than for the students: if a student could present the solution to their problem to the class then I knew they understood at least each of the individual steps.Overall I think this format did more or less what I intended it to do, and will definitely be re-using it if I ever teach an olympiad class in this style again. Though I don’t know how well the second half might work in a bigger room: I actually had to do a bit of running to keep up with questions and ideas that the students came up with while working, and of course the presentation time is proportional to the number of students you have (maybe 3-5 minutes each). So if I had, say, 10 students, I would probably re-think how to run the end.

2. Picking Topics

I think it’s general kind of useless to teach a class where you do a mix of unrelated problems. For example, I never really liked “functional equations” as a class. And don’t even get me started on the typical “divisibility” class. That’s what the IMO Shortlist is for, and the students already have that. Anyone competing at this level already knows how to pick up a collection of problems and practice against it. Class needs to something more than that.

My idea is that problems in an olympiad class should be linked by some underlying, specific theme. It doesn’t have to be a specific technique, but it can be. The reason is that this way, you can see the theme re-appear over and over again. By the time you see it the fifth time, hopefully things start to click.

Let me phrase this another way. Suppose I gave wrote down the following:

\displaystyle \begin{array}{lll} A1 & B11 & C8 \\ A9 & B44 & C27 \\ A49 & B33 & C343 \\ A16 & B99 & C1 \\ A25 & B22 & C125 \end{array}

You can tell what each of the {A}‘s, {B}‘s, {C}‘s have in common by looking for a few moments. But what happens if I intertwine them?

\displaystyle \begin{array}{lllll} B11 & C27 & C343 & A1 & A9 \\ C125 & B33 & A49 & B44 & A25 \\ A16 & B99 & B22 & C8 & C1 \end{array}

This is the same information, but now you have to work much harder to notice the association between the letters and the numbers they’re next to.

I think the class is kind of the same idea. If you want to draw out the idea of orders, pick a bunch of problems that involve orders in spirit. They don’t have to be exactly the same problem, but they should be reasonably related.

So to produce a good olympiad handout, you need to have something to say. I think my Chinese Remainder Theorem handout is a good example (it was actually something I was considering using for the NT session, but I decided on something else eventually). I want to point out how CRT is used in constructions, so the examples and practice problems are all designed to illustrate this point. There’s a large degree of micro-control throughout the entire thing.

Honestly, I think it’s really easy to teach olympiad math badly: just pick a bunch of unrelated problems, go through the solutions one by one, then give some more unrelated problems for practice. The students will still get better, because they are practicing. But is that all you can do as a teacher?

For the record, here’s the topics I ended up using for the camp.

  • Orders / Lifting the Exponent
  • Irreducibility of Polynomials
  • Projective Geometry
  • Double Summation

3. Narrowing Problems

Something new I tried for this lecture was trimming a lot.

At MOP, I’d often get a handout for a MOP class with something like 30 problems on it. We’d get to pick which ones we worked on, and then we’d see or present some of the solutions in class. The issue is that, well, a class isn’t that long, so I would only be able to work on two or three problems, and these wouldn’t be the same as the two or three problems other people worked on or presented.

I think the hope was that when we went home we’d still have like 20 various problems to work on. The problem is that I couldn’t possibly have worked on the left-over 20 problems from every class even if I wanted to — there were just too many.

I fought this issue at A* by trimming down the practice problems a lot. My handouts essentially had only 5-7 problems to work on. This way, more people had looked at the same problems when it came time for presentations.

The reason I picked 5-7 was so that every student could work on (and hence present) a different problem. In retrospect I’m not sure this was a good idea. If I were to teach again, I might even cut it down to fewer than that, maybe four problems or so. That way, everyone really works on the same problems, and presentations of solutions are infinitely more useful. I would just have to work around the fact that on any given day, not all students would have a chance to present.

4. Things I Did Badly

Finally, here’s a couple things I wish I had fixed.

First, I made a lot of assumptions about what people knew and didn’t knew. I thought I had made the NT lecture too hard because the room was very quiet, but it in fact turned out that it was because the students had actually seen most of the order material before.

The only reason I found out was because after I had finished presenting all the order material, I asked out of curiosity whether anyone had seen this already, not actually expecting anyone to raise their hands. Instead, the entire class did. Students really are too polite — I must have bored them to tears for those first 45 minutes.

The solution to this is really simple: just ask the students if they’ve seen it before. Any teacher knows that students are shy to admitting they don’t understand what you’re talking about, but if you just ask “have any of you seen this before?” the students will in general be pretty honest. (If you phrase it as “have any of you not seen this before” the results are less accurate.) So that’s something I will remember to do much more of later on.

The other thing is that I likely made the practice problems too hard. I felt like I had to give too many hints: at least the students understand the solution, but I’m not sure how helpful it is to only solve the problem because the hints given amounted to an outline of the solution. In my defense, I was guessing in the dark as to the abilities of the students, and erred on the side of hard. (Any of you who do math olympiads know how useless and boring a too-easy class is; in contrast, classes which are a tad too hard can often still be beneficial.) But the point stands that my estimate was wrong.

Finally, I think I wore the students out a bit too much. I was happy with their performance in my class, but apparently they were all pretty tired during afternoon. But I think that might just be because of the way the camp is set up — six hours of class a day is really a lot, even for the very hardcore.

5. Closings

Overall I was quite happy with how the classes turned out, and I think the students were too (either that or they were very generous with my instructor ratings). I can’t wait until I get an opportunity like this again, but that might be a long time in coming — there really aren’t that many USAMO-level students out there as I would like!