Getting to know problems

Power Overwhelming 2025-02-15

I recently had a student writing to me asking for advice on problem-solving. The student gave a few examples of problems they didn’t solve (like I tell people to). One of the things that struck me about the message was their description of their work on USAMO 2021/4, whose statement reads:

A finite set ${S}$ of positive integers has the property that, for each ${s\in S}$ , and each positive integer divisor ${d}$ of ${s}$ , there exists a unique element ${t\in S}$ satisfying ${\gcd(s,t) = d}$ . (The elements ${s}$ and ${t}$ could be equal.) Given this information, find all possible values for the number of elements of ${S}$ .

Roughly (for privacy reasons, this isn’t exactly what the student said), the student talked about having done small cases on the size ${|S|}$ , and figuring out that ${|S|}$ was even, but then running out of ideas. They gave a few other examples of problems that had a similar issue where they tried some stuff and then just ran out of ideas.

I ended up writing a long email in response that I thought might be interesting to the high schoolers reading this blog, so here it is.

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I get the feeling that you are rushing to try to guess the solution to the problem rather than getting to know the problem better. For example, for the sets problem, trying to do cases by the number of elements in ${S}$ is actually extremely unnatural. You really should be much more interested in the question “what’s a set satisfying this look like? Can I give any examples at all? Can I find a way to generate big examples?” and so on. However, you got so latched on to what the extraction was (the number of elements in the set) that it feels like you stopped trying to engage your curiosity (“all the while, I did not think about the condition of the problem”). I don’t know how to explain it but trying to do ${|S|=1,2,3,4,5}$ in order just feels so obviously wrong to me that it wouldn’t even occur to me consider this. When I see a condition like this, my reaction is more like “is that even usually possible” and “what would that even look like”, both because it’s a pretty abstract condition (I immediately want to see concrete examples) and because it’s requiring so much (it’s a requirement on every ${s}$ and then every ${d}$ dividing ${s}$ : double for-all).

Here’s a silly analogy I just thought of (experimental analogy, so don’t take it too seriously — also, as I said I don’t actually know how to make this actionable advice, it’s just a description).

You can imagine maybe you’re going on a first date with someone. Sure, there’s eventually a checklist of things that you probably should know, like what major they are, where they’re from, what their phone number is, etc. But it’s not a criminal investigation where you go off a list of questions and write the answers in a list (see 1:00-1:40 of this scene from SVTFOE). It’s supposed to be a conversation that flows naturally back and forth where you get to know the other person better because you’re genuinely interested in them.

Problem-solving is like that. It should follow your natural curiosity (“I want to know what’s going on” not “I want the solution asap”). And it should go back and forth (when you try something, you get more information than just “did it work”). “Promptly running out of ideas” is like promptly running out of conversation during a first date. It might sometimes happen if you and the other person just don’t have chemistry (for example if you’re trying to solve USA TST 2025/3 without the right background knowledge), but if it happens all the time you’re not doing first dates correctly.

(TSTST 2013/7 is an anti-example where actually the way to solve the problem is to pay the bill and leave ASAP. Don’t even try to count ${T_n}$ because it’s so obviously impossible. Just find a method that gives you groups of ${n}$ .)

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