On choosing exercises

Posted on June 14, 2020 by Evan Chen (陳誼廷)

Finally, if you attempt to read this without working through a significant number of exercises (see §0.0.1), I will come to your house and pummel you with [Gr-EGA] until you beg for mercy. It is important to not just have a vague sense of what is true, but to be able to actually get your hands dirty. As Mark Kisin has said, “You can wave your hands all you want, but it still won’t make you fly.”

— Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry

When people learn new areas in higher math, they are usually required to do some exercises. I think no one really disputes this: you have to actually do math to make any progress.

However, from the teacher’s side, I want to make the case that there is some art to picking exercises, too. In the process of writing my Napkin as well as taking way too many math classes I began to see some patterns in which exercises or problems I tended to add to the Napkin, or which exercises I found helpful when learning myself. So, I want to explicitly record some of these thoughts here.

1. How not to do it

So in my usual cynicism I’ll start by saying what I think people typically do, and why I don’t think it works well. As far as I can tell, the criteria used in most classes is:

The student is reasonably able to (at least in theory) eventually solve it.
A student with a solid understanding of the material should be able to do it.
(Optional) The result itself is worth knowing.

Both of these criteria are good. My problem is that I don’t think they are sufficient.

To explain why, let me give a concrete example of something that is definitely assigned in many measure theory classes.

Okay example (completion of a measure space). Let ${(X, \mathcal A, \mu)}$ be a measure space. Let ${\overline{\mathcal A}}$ denote all subsets of ${X}$ which are the union of a set in ${\mathcal A}$ and a null set. Show that ${\overline{\mathcal A}}$ is a sigma-algebra there is a unique extension of the measure ${\mu}$ to it.

I can see why it’s tempting to give this as an exercise. It is a very fundamental result that the student should know. The proof is not too difficult, and the student will understand it better if they do it themselves than if they passively read it. And, if a student really understands measures well, they should find the exercise quite straightforward. For this reason I think this is an okay choice.

But I think we can do better.

In many classes I’ve taken, nearly all the exercises looked like this one. I think when you do this, there are a couple blind spots that sometimes get missed:

There’s a difference between “things you should be able to do after learning Z well” and “things you should be able to do when first learning Z“. I would argue that the above example is the former category, but not the latter one — if a student is learning about measures for the first time, my first priority would be to make sure they get a good conceptual understanding first, and in particular can understand why the statement should be true. Then we can worry about actually proving it.
Assigning an exercise which checks if you understand X is not the same as actually teaching it. Okay exercises can verify if you understand something, great exercises will actively help you understand it.

2. An example that I found enlightening

In contrast, this year I was given an exercise which I thought was so instructive that I’ll post it here. It comes from algebraic geometry.

Exercise: The punctured gyrotop is the open subset ${U}$ of ${X = \mathrm{Spec} \mathbb C[x,y,z] / (xy, z)}$ obtained by deleting the origin ${(x,y,z)}$ from ${X}$ . Compute ${\mathcal O_X(U)}$ .

It was after I did this exercise that I finally felt like I understood why distinguished open sets are so important when defining an affine scheme. For that matter, it finally clicked why sheaves on a base are worth caring about.

I had read lots and lots of words and pushed symbols around all day. I had even proved, on paper already, that ${\mathcal O(U \sqcup V) = \mathcal O(U) \times \mathcal O(V)}$ . But I never really felt it. This exercise changed that for me, because suddenly I had an example in front of me that I could actually see.

3. Some suggested additional criteria

So here are a few suggested guidelines which I think can help pick exercises like that one.

A. They should be as concrete as possible.

This is me yelling at people to use more examples, once again. But I think having students work through examples as an exercise is just as important (if not more) than reading them aloud in lecture.

One other benefit of using concrete examples is that you can avoid the risk of students solving the exercise by “symbol pushing”. I think many of us know the feeling of solving some textbook exercise by just unwinding a definition and doing a manipulation, or black-boxing some theorem and blindly applying it. In this way one ends up with correct but unenlightening proofs. The issue is that nothing written down resonates with System 1, and so the result doesn’t get internalized.

When you give a concrete exercise with a specific group/scheme/whatever, there is much less chance of something like that happening. You almost have to see the example in order to work with it. I really think internalizing theorems and definitions is better done in this concrete way, rather than the more abstract or general manipulations.

B. They should be enjoyable.

Math majors are humans too. If a whole page of exercises looks boring, students are less likely to do them.

This is one place where I think people could really learn from the math contest community. When designing exams like IMO or USAMO, people fight over which problems they think are the prettiest. The nicest and most instructive exam problems are passed down from generation to generation like prized heirlooms. (Conveniently, the problems are even named, e.g. “IMO 2008/3”, which I privately think helps a ton; it gives the problems a name and face. The most enthusiastic students will often be able to recall where a good problem was from if shown the statement again.) Imagine if the average textbook exercises had even a tenth of that enthusiasm put into crafting them.

Incidentally, I think being concrete helps a lot with this. Part of the reason I enjoyed the punctured gyrotop so much was that I could immediately draw a picture of it, and I had a sense that I should be able to compute the answer, even though I wasn’t experienced enough yet to see what it was. So it was as if the exercise was leading me on the whole way.

For an example of how not to do it, here’s what I think my geometry book would look like if done wrong.

C. They should not be too tricky.

People are always dumber than you think when they first learn a subject; things which should be obvious often are not. So difficulty should be used in moderation: if you assign a hard exercise, you should assume by default the student will not solve it, so there better be some reason you’re adding some extra frustration.

I should at this point also mention some advice most people won’t be able to take (because it is so time-consuming): I think it’s valuable to write full solutions for students, especially on difficult problems. When someone is learning something for the first time, that is the most important time for the students to be able to read the full details of solutions, precisely because they are not yet able to do it themselves.

In math contests, the ideal feedback cycle is something like: a student works on a problem P, makes some progress (possibly solving it), then they look at the solution and see what they were missing or where they could have cleaned up their solution or what they could have done differently, et cetera. This lets them update their intuition or toolkit before going on. If you cut out this last step by not providing solutions, you lose the only real chance you had to give feedback to the student.

4. Memorability

I have, on more occasions than I’m willing to admit, run into the following situation. I solve some exercise in a textbook. Sometime later, I am reading about some other result, and I need some intermediate result, which looks like it could be true but I don’t how to prove it immediately. So I look it up, and then find out it was the exercise I did (and then have to re-do the exercise again because I didn’t write up the solution).

I think you can argue that if you don’t even recognize the statement later, you didn’t learn anything from it. So I think the following is a good summarizing test: how likely is the student to actually remember it later?

USEMO sign-ups are open

Posted on April 21, 2020 by Evan Chen (陳誼廷)

I’m happy to announce that sign-ups for my new olympiad style contest, the United States Ersatz Math Olympiad (USEMO), are open now! The webpage for the USEMO is https://web.evanchen.cc/usemo.html (where sign-ups are posted).

The US Ersatz Math Olympiad is a proof-based competition open to all US middle and high school students. Like many competitions, its goals are to develop interest and ability in mathematics (rather than measure it). However, it is one of few proof-based contests open to all US middle and high school students. You can see more about the goals of this contest in the mission statement.

The contest will run over Memorial day weekend:

Day 1 is Saturday May 23 2020, from 12:30pm ET — 5:00pm ET.
Day 2 is Sunday May 24 2020, from 12:30pm ET — 5:00pm ET.

In the future, assuming continued interest, I hope to make the USEMO into an annual tradition run in the fall.

Circular optimization

Posted on April 15, 2020 by Evan Chen (陳誼廷)

This post will mostly be focused on construction-type problems in which you’re asked to construct something satisfying property ${P}$ .

Minor spoilers for USAMO 2011/4, IMO 2014/5.

1. What is a leap of faith?

Usually, a good thing to do whenever you can is to make “safe moves” which are implied by the property ${P}$ . Here’s a simple example.

Example 1 (USAMO 2011)

Find an integer ${n}$ such that the remainder when ${2^n}$ is divided by ${n}$ is odd.

It is easy to see, for example, that ${n}$ itself must be odd for this to be true, and so we can make our life easier without incurring any worries by restricting our search to odd ${n}$ . You might therefore call this an “optimization”: a kind of move that makes the problem easier, essentially for free.

But often times such “safe moves” or not enough to solve the problem, and you have to eventually make “leap-of-faith moves”. For example, maybe in the above problem, we might try to focus our attention on numbers ${n = pq}$ for primes ${p}$ and ${q}$ . This does make our life easier, because we’ve zoomed in on a special type of ${n}$ which is easy to compute. But it runs the risk that maybe there is no such example of ${n}$ , or that the smallest one is difficult to find.

2. Circular reasoning can sometimes save the day

However, a strange type of circular reasoning can sometimes happen, in which a move that would otherwise be a leap-of-faith is actually known to be safe because you also know that the problem statement you are trying to prove is true. I can hardly do better than to give the most famous example:

Example 2 (IMO 2014)

For every positive integer ${n}$ , the Bank of Cape Town issues coins of denomination ${\frac 1n}$ . Given a finite collection of such coins (of not necessarily different denominations) with total value at most ${99 + \frac12}$ , prove that it is possible to split this collection into ${100}$ or fewer groups, such that each group has total value at most ${1}$ .

Let’s say in this problem we find ourselves holding two coins of weight ${1/6}$ . Perhaps we wish to put these coins in the same group, so that we have one less decision to make. However, this could rightly be viewed as a “leap-of-faith”, because there’s no logical reason why the task must remain possible after making this first move.

Except there is a non-logical reason: this is the same as trading the two coins of weight ${1/6}$ for a single coin of weight ${1/3}$ . Why is the task still possible? Because the problem says so: the very problem we are trying to solve includes this case, too. If the problem is going to be true, then it had better be true after we make this trade.

Thus by a perverse circular reasoning we can rest assured that our leap-of-faith here will not come back to bite us. (And in fact, this optimization is a major step of the solution.)

3. More examples of circular optimization

Here’s some more examples of problems you can try that I think have a similar idea.

Problem 1

Prove that in any connected graph ${G}$ on ${2004}$ vertices one can delete some edges to obtain a graph (also with ${2004}$ vertices) whose degrees are all odd.

Problem 2 (USA TST 2017)

In a sports league, each team uses a set of at most ${t}$ signature colors. A set ${S}$ of teams is color-identifiable if one can assign each team in ${S}$ one of their signature colors, such that no team in ${S}$ is assigned any signature color of a different team in ${S}$ . For all positive integers ${n}$ and ${t}$ , determine the maximum integer ${g(n,t)}$ such that: In any sports league with exactly ${n}$ distinct colors present over all teams, one can always find a color-identifiable set of size at least ${g(n,t)}$ .

Feel free to post more examples in the comments.

Meritocracy is the worst form of admissions except for all the other ones

Posted on January 13, 2020 by Evan Chen (陳誼廷)

I’m now going to say something explicitly that I hinted at in June: I don’t think a student deserves to make MOP more because they had a higher score than another student.

I think it’s easy to get this impression because the selection for MOP is done by score cutoffs. So it sure looks that way.

But I don’t think MOP admissions (or contests in general) are meant to be a form of judgment. My primary agenda is to run a summer program that is good for its participants, and we get funding for N of them. For that, it’s not important which N students make it, as long as they are enthusiastic and adequately prepared. (Admittedly, for a camp like MOP, “adequately prepared” is a tall order). If anything, what I would hope to select for is the people who would get the most out of attending. This is correlated with, but not exactly the same as, score.

Two corollaries:

I support the requirement for full attendance at MOP. I know, it sucks for those star students who qualify for two conflicting and then have to choose. You have my apologies (and congratulations). But if you only come for 2 of 3 weeks, you took away a spot from someone who would have attended the whole time.
I am grateful to the European Girl’s MO for giving MOP an opportunity to balance the gender ratio somewhat; empirically, it seems to improve the camp atmosphere if the gender ratio is not 79:1.

Anyways, given my mixed feelings on meritocracy, I sometimes wonder whether MOP should do what every other summer camp does and have an application, or even a lottery. I think the answer is no, but I’m not sure. Some reasons I can think of behind using score only:

MOP does have a (secondary) goal of IMO training, and as a result the program is almost insane in difficulty. For this reason you really do need students with significant existing background and ability. I think very few summer camps should explicitly have this level of achievement as a goal, even secondarily. But I think there should be at least one such camp, and it seems to be MOP.
Selection by score is transparent and fair. There is little risk of favoritism, nepotism, etc. This matters a lot to me because, basically no matter how much I try to convince them otherwise, people will take any admissions decision as some sort of judgment, so better make it impersonal. (More cynically, I honestly think if MOP switched to a less transparent admissions process, we would be dealing with lawsuits within 15 years.)
For better or worse, qualifying for MOP ends up being sort of a reward, so I want to set the incentives right and put the goalpost at “do maximally well on USAMO”. I think we design the USAMO well enough that preparation teaches you valuable lessons (math and otherwise). For an example of how not to set the goalpost, take most college admissions processes.

Honestly, the core issue might really be cultural, rather than an admissions problem. I wish there was a way we could do the MOP selection as we do now without also implicitly sending the (unintentional and undesirable) message that we value students based on how highly they scored.

MOHS hardness scale

Posted on November 26, 2019 by Evan Chen (陳誼廷)

There’s a new addition to my olympiad problems and solutions archive: I created an index of many past IMO/USAMO/USA TST(ST) problems by what my opinions on their difficulties are. You can grab the direct link to the file below:

https://evanchen.cc/upload/MOHS-hardness.pdf

In short, the scale runs from 0M to 50M in increments of 5M, and every USAMO / IMO problem on my archive now has a rating too.

My hope is that this can be useful in a couple ways. One is that I hope it’s a nice reference for students, so that they can better make choices about what practice problems would be most useful for them to work on. The other is that the hardness scale contains a very long discussion about how I judge the difficulty of problems. While this is my own personal opinion, obviously, I hope it might still be useful for coaches or at least interesting to read about.

As long as I’m here, I should express some concern that it’s possible this document does more harm than good, too. (I held off on posting this for a few months, but eventually decided to at least try it and see for myself, and just learn from it if it turns out to be a mistake.) I think there’s something special about solving your first IMO problem or USAMO problem or whatever and suddenly realizing that these problems are actually doable — I hope it would not be diminished by me rating the problem as 0M. Maybe more information isn’t always a good thing!

Understanding with System 1

Posted on October 26, 2019 by Evan Chen (陳誼廷)

Math must be presented for System 1 to absorb and only incidentally for System 2 to verify.

I finally have a sort-of formalizable guideline for teaching and writing math, and what it means to “understand” math. I’ve been unconsciously following this for years and only now managed to write down explicitly what it is that I’ve been doing.

(This post is written from a math-centric perspective, because that’s the domain where my concrete object-level examples from. But I suspect much of it applies to communicating hard ideas in general.)

S1 and S2

The quote above refers to the System 1 and System 2 framework from Thinking, Fast and Slow. Roughly it divides the brain’s thoughts into two categories:

S1 is the part of the brain characterized by fast, intuitive, automatic, instinctive, emotional responses, For example, when you read the text “2+2=?”, S1 tells you (without any effort) that this equals 4.
S2 is the part of the brain characterized by slow, deliberative, effortful, logical responses; for example, S2 is used to count the number of words in this sentence.

(The link above gives some more examples.)

The premise of this post is that understanding math well is largely about having the concept resonate with your S1, rather than your S2. For example, let’s take groups from abstract algebra. Then I claim that

$G = \{ a/b \mid a,b \text{ odd integers} \}$

is a group under the usual multiplication. Now, if you have a student who’s learning group theory for the first time, the only way they could see this is a group is to compare it against a list of the group axioms, and have their S2 verify them one by one. But experienced people don’t do this: their S1 automatically tells them that $G$ “feels” like a group (because e.g. it’s closed and doesn’t have division-by-zero issues).

I think this S1-level understanding is what it means to “get it”. Verifying a solution to a hard olympiad problem by having S2 check each individual step is straightforward in principle, albeit time-consuming. The tricky part is to get this solution to resonate with S1. Hence my advice to never read a solution line by line.

Writing for S1

What this means is that if you’re trying to teach someone an idea, then you should be focusing on trying to get their S1 to grasp it, rather than just their S2. For example, in math it’s not enough to just give a sequence of logical steps which implies the result: give it life.

Here are some examples of ways I (try to) do this.

First, giving good concrete examples. S1 reacts well when it “sees” a concrete object like $G$ above, and can see some intuitive properties about it right away. Abstract “symbol-pushing” is usually left to S2 instead.

Similarly, drawing pictures, so your S1 can actually see the object. On one extreme end, you can write something like “a point $S$ lies on the polar of $T$ if and only if $T$ lies on the polar of $S$”, but it’s much better to just have a picture:

You can even do this for things that aren’t really geometrical in nature. For example, my Napkin features the following picture of cardinal collapse when forcing.

Third, write like you talk, and share your feelings. S1 is emotional. S1 wants to know that compactness is a good property for a space to have, or that non-Noetherian rings are way too big and “only weirdos care about non-Noetherian rings” (just kidding!), or that ramified primes are the “finitely many edge cases” and aren’t worth worrying about. These S1 reactions you get are the things you want to pass on. In particular, avoid standard formal college-textbook-bleed-your-eyes-dry-in-boredom style. (To be fair, not all textbooks do this; this is one reason why I like Pugh’s book so much, for example.)

Even the mechanics on the page can be made to accommodate S1 in this way. S1 can’t read a wall of text; S2 has to put in effort to do that. But S1 can pick out section headers, or bolded phrases like this one, and so on and so forth. That’s why in Napkin all the examples are in separate red boxes and all the big theorems are in blue boxes, and important philosophical points are typeset in bold centered green text. This way S1 naturally puts its attention there.

But do not force it

On the flip side, if you’re trying to learn something, there’s a common failure mode where you try to keep forcing S2 to do something unnatural (rather than trying to have S1 figure it out). This is the kind of thing when you don’t understand what the Chinese Remainder Theorem is trying to say, so you try to fix this by repeatedly reading the proof line by line, and still not really understanding what is going on. Usually this ends up in S2 getting tired and not actually reading the proof after the third or fourth iteration.

(For the Chinese remainder theorem the right thing to do is ask yourself why any arithmetic progression with common difference 7 must contain multiples of 3: credits to Dominic Yeo again for that. I’m not actually sure what you’re supposed to do when stuck on math in general. Usually I just ask my friends what is going on, or give up for now and come back later.)

Actually, I really like the advice that SSC mentions: “develop instincts, then use them”.

MOP should do a better job of supporting its students in not-June

Posted on August 25, 2019 by Evan Chen (陳誼廷)

Up to now I always felt a little saddened when I see people drop out of the IMO or EGMO team selection. But actually, really I should be asking myself what I (as a coach) could do better to make sure the students know we value their effort, even if they ultimately don’t make the team.

Because we sure do an awful job of being supportive of the students, or, well, really doing anything at all. There’s no practice material, no encouragement, or actually no form of contact whatsoever. Just three unreasonably hard problems each month, followed by a score report about a week later, starting in December and dragging in to April.

One of a teacher’s important jobs is to encourage their students. And even though we get the best students in the USA, probably we shouldn’t skip that step entirely, especially given the level of competition we put the students through.

So, what should we do about it? Suggestions welcome.

IMO 2019 Aftermath

Posted on July 23, 2019 by Evan Chen (陳誼廷)

Here is my commentary for the 2019 International Math Olympiad, consisting of pictures and some political statements about the problem. Summary This year’s USA delegation consisted of leader Po-Shen Loh and deputy leader Yang Liu. The USA scored 227 points, … Continue reading →

An opening speech for MOP

Posted on June 1, 2019 by Evan Chen (陳誼廷)

While making preparations for this year’s MOP, I imagined to myself what I would say on orientation night if I was director of the camp, and came up with the following speech. I thought it might be nice to share on this blog. Of course, it represents my own views, not the actual views of MOP or MAA. And since I am not actually director of MOP, the speech was never given.

People sometimes ask me, why do we have international students at MOP? Doesn’t that mean we’re training teams from other countries? So I want to make this clear now: the purpose of MOP is not to train and select future IMO teams.

I know it might seem that way, because we invite by score and grade. But I really think the purpose of MOP is to give each one of you the experience of working hard and meeting new people, among other things. Learn math, face challenges, make friends, the usual good stuff, right? And that’s something you can get no matter what your final rank is, or whether you make IMO or EGMO or even next year’s MOP. The MOP community is an extended family, and you are all part of it now.

What I mean to say is, the camp is designed with all 80 of you in mind. It made me sad back in 2012 when one of my friends realized he had little chance of making it back next year, and told me that MAA shouldn’t have invited him to begin with. Even if I can only take six students to the IMO each year, I never forget the other 74 of you are part of MOP too.

This means one important thing: everyone who puts in their best shot deserves to be here. (And unfortunately this also means there are many other people who deserve to be here tonight too, and are not. Maybe they solved one or two fewer problems than you did; or maybe they even solved the same number of problems, but they are in 11th grade and you are in 10th grade.)

Therefore, I hope to see all of you put in your best effort. And I should say this is not easy to do, because MOP is brutal in many ways. The classes are mandatory, we have a 4.5-hour test every two days, and you will be constantly graded. You will likely miss problems that others claim are easy. You might find out you know less than you thought you did, and this can be discouraging. Especially in the last week, when we run the TSTST, many of you will suddenly realize just how strong Team USA is.

So I want to tell you now, stay determined in the face of adversity. This struggle is your own, and we promise it’s worth it, no matter the outcome. We are rooting for you, and your friends sitting around you are too. (And if the people around you aren’t your friends yet, change that asap.)

Hard and soft techniques

Posted on May 3, 2019 by Evan Chen (陳誼廷)

In yet another contest-based post, I want to distinguish between two types of thinking: things that could help you solve a problem, and things that could help you understand the problem better. Then I’ll talk a little about how you can use the latter. (I’ve talked about this in my own classes for a while by now, but only recently realized I’ve never gotten the whole thing in writing. So here goes.)

1. More silly terminology

As usual, to make these things easier to talk about, I’m going to introduce some words to describe these two. Taking a page from martial arts, I’m going to run with hard and soft techniques.

A hard technique is something you try in the hopes it will prove something — ideally, solve the problem, but at least give you some intermediate lemma. Perhaps a better definition is “things that will end up in the actual proof”. Examples include:

Angle chasing in geometry, or proving quadrilaterals are cyclic.
Throwing complex numbers at a geometry problem.
Plugging in some values into a functional equation (which gives more equations to work with).
Taking a given Diophantine equation modulo ${p}$ to get some information, or taking ${p}$ -adic evaluations.
Trying to perform an induction, for example by deleting an element.
Trying to write down an inequality that when summed cyclically gives the desired conclusion.
Reducing the problem to one or more equivalent claims.

and so on. I’m sure you can come up with more examples.

In contrast, a soft technique is something you might try to help you understand the problem better — even if it might not prove anything. Perhaps a better definition is “things not written up”. Examples include:

Examining particular small cases of the problem.
Looking at the equality cases of a min/max problem.
Considering variants of the problem (for example, adding or deleting conditions).
Coming up with lots of concrete examples and playing with them.
Trying to come with a counterexample to the problem’s assertion and seeing what the obstructions are.
Drawing pictures, even on non-geometry problems (see JMO2 and JMO5 in my 2019 notes for example).
Deciding whether or not a geometry problem is “purely projective”.
Counting the algebraic degrees of freedom in a geometry problem.
Checking all the linear/polynomial solutions to a functional equation, in order to get a guess what the answer might be.
Blindly trying to guess solutions to an algebraic equation.
Making up an artificial unnatural function in a functional equation, and then trying to see why it doesn’t work (or occasionally being surprised that it does work).
Thinking about why a certain hard technique you tried failed, or even better convincing yourself it cannot work (for example, this Diophantine equation has a solution modulo every prime, so stop trying to one-shot by mods).
Giving a heuristic argument that some claim should be true or false (“probably ${2^n \bmod n}$ is odd infinitely often”), or even easy/hard to prove.

and so on. There is some grey area between these two, some of the examples above might be argued to be in the other category (especially in context of specific problems), but hopefully this gives you a sense of what I’m talking about.

If you look at things I wrote back when I was in high school, you’ll see this referred to as “attacking” and “scouting” instead. This is too silly for me now even by my standards, but back then it was because I played a lot of StarCraft: Brood War (I’ve since switched to StarCraft II). The analogy there is pretty self-explanatory: knowing what your opponent is doing is important because your army composition and gameplay decisions should change in reaction to more information.

2. Using soft techniques: an example

Now after all that blabber, here’s the action item for you all: you should try soft techniques when stuck.

When you first start doing a problem, you will often have some good ideas for what to try. (For example: a wild geometry appeared, let’s scout for cyclic quadrilaterals.) Sometimes if you are lucky enough (especially if the problem is easier) this will be enough to topple the problem, and you can move on. But more often what happens is that eventually you run out of steam, and the problem is still standing. When that happens, my advice is to try doing some soft techniques if you haven’t already done so.

Here’s an example that I like to give.

Example 1 (USA TST 2009)

Find all real numbers ${x}$ , ${y}$ , ${z}$ which satisfy

$\displaystyle \begin{aligned} x^3 &= 3x - 12y + 50,\\ y^3 &= 12y + 3z - 2,\\ z^3 &= 27z + 27x. \end{aligned}$

A common first thing that people will try to do is add the first two equations, since that will cause the ${12y}$ terms to cancel. This gives a factor of ${x+y}$ in the left and an ${x+z}$ in the right, so then maybe you try to submit that into the ${27(x+z)}$ in the last equation, so you get ${z^3 = 9(x^3+y^3-48)}$ , cool, there’s no more linear terms. Then. . .

Usually this doesn’t end well. You add this and subtract that and in the end all you see is equation after equation, and after a while you realize you’re not getting anywhere.

So we’re stuck now. What to do? I’ll now bring in two of the soft techniques I mentioned earlier:

Let’s imagine the problem had ${\mathbb R}$ replaced with ${\mathbb C}$ . In this new problem, you can imagine solving for ${y}$ in terms of ${x}$ using the first equation, then ${z}$ in terms of ${y}$ , and then finally putting everything into the last equation to find a degree ${27}$ polynomial in ${x}$ . I say “imagine” because wow would that be ugly.
But here’s the kicker: it’s a polynomial. It should have exactly ${27}$ complex roots, with multiplicity. That’s a lot. Really?

So here’s a hint you might take: there’s a good reason this is over ${\mathbb R}$ but not ${\mathbb C}$ . Often these kind of things end up being because there’s an inequality going on somewhere, so there will only be a few real solutions even though there might be tons of complex ones.
Okay, but there’s an even more blatant thing we don’t know yet: what is the answer, anyways?
This was more than a little bit embarrassing. We’re half an hour in to the problem and thoroughly stuck, and we don’t even have a single ${(x,y,z)}$ that works? Maybe it’d be a good idea to fix that, like, right now. In the simplest way possible: guess and check.

It’s much easier than it sounds, since if you pick a value of ${z}$ , say, then you get ${x}$ from the third equation, ${y}$ from the first, then check whether it fits the second. If we restrict our search to integer values of ${z}$ , then there aren’t so many that are reasonable.

I won’t spoil what the answer ${(x,y,z)}$ is, other than saying there is an integer triple and it’s not hard to find it as I described. Once you have these two meta-considerations, you suddenly have a much better foothold, and it’s not too hard to solve the problem from here (for a USA TST problem anyways).

I pick this example because it really illustrates how hopeless repeatedly using hard techniques can be if you miss the right foothold (and also because in this problem it’s unusually tempting to just think that more manipulation is enough). It’s not impossible to solve the problem without first realizing what the answer is, but it is certainly way more difficult.

3. Improving at soft techniques

What this also means is that, in the after-math of a problem (when you’ve solved/given up on a problem and are reading and reflecting on the solution), you should also add soft techniques into the list of possible answers to “how might I have thought of that?”. An example of this is at the end of my earlier post On Reading Solutions, in which I describe how you can come up with solutions to two Putnam problems by thinking carefully about what should be the equality case.

Doing this is harder than it sounds, because the soft techniques are the ones that by definition won’t appear in most written solutions, and many people don’t explicitly even recognize them. But soft techniques are the things that tell you which hard techniques to use, which is why they’re so valuable to learn well.

In writing this post, I’m hoping to make the math contest world more aware that these sorts of non-formalizable ideas are things that can (and should) be acknowledged and discussed, the same way that the hard techniques are. In particular, just as there are a plethora of handouts on every hard technique in the olympiad literature, it should also be possible to design handouts aimed at practicing one or more particular soft techniques.

At MOP every year, I’m starting to see more and more classes to this effect (alongside the usual mix of classes called “inversion” or “graph theory” or “induction” or whatnot). I would love to see more! End speech.