Sometimes I get asked broad advice questions on solving problems, for example questions like:

1. How do I know when to switch or prioritize approaches I come up with?
2. How do I know which points or lines to add in geometry problems?
3. How can I tell if I’m making progress on a problem?
4. How can I guess the answer if “find all” or “find min/max” problems?
5. How can I tell whether a conjecture I made is true or not?
6. What should I do on a problem when I am stuck?

and so on.

I think all of these questions have a certain quality that, for lack of a better name, I’ll dub as being “NP-hard”. This is a bit of abuse of terminology borrowed from complexity theory, but let me explain why I think the name fits.

We know that solving math problems is generally difficult. There’s …

Editorial note: this post was mostly written in February 2023. Any resemblance to contests after that date is therefore coincidental.

Background

A long time ago, rubrics for the IMO and USAMO were fairly strict. Out of seven, the overall meta-rubric looks like:

• 7: Problem solved
• 6: Tiny slip (and contestant could repair)
• 5: Small gap or mistake, but non-central
• 2: Lots of genuine progress
• 1: Significant non-trivial progress
• 0: “Busy work”, special cases, lots of writing

In particular, traditional rubrics were often sublinear. You’d see problems where you could split it into two parts, and solving either part would only give 2 points, whereas solving both was worth 7.

Increasingly, I’ve noticed this is less and less common. Particularly, at the IMOAs far as I know, the IMO rubrics aren’t really available anywhere. (On the other hand, I’ve never been told that rubrics explicitly need …

Here’s a mix of several publicity-related things I’d like to broadcast.

AlphaGeometry

A lot of you have already heard the buzz about the AlphaGeometry news and Nature paper. (I’ve known about this paper for a while now, so I’m glad I can finally talk about it!)

I managed to snag a cameo in the DeepMind post where I wrote

AlphaGeometry’s output is impressive because it’s both verifiable and clean. Past AI solutions to proof-based competition problems have sometimes been hit-or-miss (outputs are only correct sometimes and need human checks). AlphaGeometry doesn’t have this weakness: its solutions have machine-verifiable structure. Yet despite this, its output is still human-readable. One could have imagined a computer program that solved geometry problems by brute-force coordinate systems: think pages and pages of tedious algebra calculation. AlphaGeometry is not that. It uses classical geometry rules with angles and similar …

I remember reading a Paul Graham essay about how people can’t think clearly about parts of their identity. In my students, I have never seen this more clearly than when people argue about the difficulty of problems.

Some years ago I published a chart of my ratings of problem difficulty, using a scale called MOHS. When I wrote this I had two goals in mind. One was that I thought the name “MOHS” for a Math Olympiad Hardness Scale was the best pun of all time, because there’s a geological scale of mineral hardness that coincidentally has the same name. The other was that I thought it would be useful for beginner students, and coaches, to help find problems that are suitable for practice.

I think it did accomplish those goals. The problem is that I also inadvertently helped catalyze an endless, incessant stream of students constantly arguing …

This is a short advertisement announcing that the OTIS Mock AIME 2024 is out. The short version is that I wanted to give my students a chance to try their hand at problem composition, which they took enthusiastically, and from their submissions I chose 15 problems to replicate an AIME.

There’s some really nice problems on here (I have some favorites, but to avoid spoilers for people using this as a practice test, I won’t say which ones yet). You can check it out here:

https://web.evanchen.cc/mockaime.html

I expect a number of students who plan to use this test as practice for the upcoming real AIME, so I’ve set a “deadline” of January 15 and ask to avoid public discussion of spoilers before then.

I remember when I got the central aha, I justified it to my teammates as “it’d be so cool, so it has to be right”. — Nathan Pinsker

This is a post meant to explain what makes puzzle hunts appealing to people who haven’t done them before.

If you do care about the actual mechanical details, Brian’s introduction is great. The one-sentence summary is: you’re (usually) trying to get an English word/phrase as the final answer, there are (usually) no directions or instructions, and I write “usually” everywhere because puzzle hunts love breaking rules.

When I first tell people about puzzle hunts, their initial reaction is usually that the fun must be in the challenge. And it is not untrue that there is a notion of skill, and it’s satisfying to become a stronger solver. However, I think this misses the point: it ignores the …

Note: if you are a prospective OTIS student, read the syllabus instead. More useful, less bragging.

In the unlikely event that I’m a social gathering like a party or family gathering, people will sometimes ask me about my teaching. Invariably they ask, “so do you do like 1:1 meetings or group lessons?”. Then I have to explain, no, I have 400 students, there are no synchronous meetings at all. The core of the program is literally a Python web server that serves PDF files.

Then it sounds less impressive. I guess when people hear I’m a teacher, they expect me to teach classes, and it’s a bit embarrassing to explain that I’m not a teacher in that sense anymore.

But the purpose of OTIS isn’t to make Evan sound cool at parties; the purpose of OTIS to be effective for the students. So this …

This was originally a diary entry, but I showed it to some students who told me I should put it in my blog instead.

Imagine you’ve moved to a new town, and want to explore the local offerings, because there’s a lot to do and see, and you’re expecting to live here a while.

The first few days, it’s really overwhelming. Everything is unfamiliar. You get lost just trying to buy groceries. You constantly have to consult maps to get anywhere. It takes a while to adjust.

But after the first week, you notice you don’t need a map as much. You can walk to the grocery store yourself; you remember which turn to take each crossing. You know the names of the biggest streets and a few landmarks, and you can get around with familiar roads as anchors. Though you’ve only been inside …

So I have an FAQ now for contest-studying advice, but there’s a “frequently used answer” that I want to document now that doesn’t fit in the FAQ format because the question looks different to everyone that asks it.

The questions generally have the same shape: “would it be better to do X or Y when studying?”. Like:

• Is it better to use GeoGebra when practicing geometry?
• Should I work on some new OTIS units or go back through some old ones that I didn’t finish?
• Should I work on hard problems in my strongest subject or medium problems in my weaker subjects?
• Would it be better if I learned this or that first?

and things like this.

And the answer is, for a lot of pairs (X,Y), if you’re so unsure that you’re asking me about it, then you should just do whatever you …

This post is a short chrono-logue about my time with the card game Hanabi, which I play with the H-group. Thus, it’s also implicitly an advertisement for why I enjoy the game Hanabi so much.

I think the progression is a bit interesting because it can be divided into almost discrete “stages”, with each stage feeling really different from the last.

0. Casual in-person play: a memory game

Like many other people in my age group, I first met the card game Hanabi in-person at some summer math camp or other (either MOP or SPARC?). The rules are pretty simple to explain, so it’s popular. But we didn’t have much strategy behind it. We had the idea that we played from left to right, a clue means “play all”, and some form of a Finesse-type blind play.

That meant the game felt kind of like a …