Here’s a section from the H-group Hanabi strategy page:

LINES

During your turn, part of figuring out the best move involves looking into the future to see what the next player will do. If they discard, will it be okay? Is there some obvious clue that they will do? And so on.

As you get better at Hanabi, you will need to do this prediction not just for the next player, but for an entire go-around of the table. And as you really get good at Hanabi, you will need to do this for as far in the future as you can reasonably predict. (Sometimes, this means 15 moves or more in the future.)

Similar to chess, initiating a move in which you can predict the next sequence of moves is called initiating a “line”.

In post-game reviews, we will often compare and hypothetically “play through” two different lines …

I had a student at MOP ask me something equivalent to “how should I study while at MOP?”There is also a question about whether you should be studying much at MOP at all — you could also spend a lot of time making new friends, for example. That’s a value judgment that I think is better left to individuals and I won’t comment on it further in this post. For those of you that don’t know, MOP is the three-week summer camp for the USA’s team to the IMO.

At first I was going to just link my FAQ. But then I thought about it a bit more, and I was surprised to find that my answer was not the same as the general how-to-study FAQ. The additional condition “while at MOP” was enough to cause me to stay up that night writing an entirely different …

I am always harping on my students to write solutions well rather than aiming for just mathematically correct, and now I have a pair of problems to illustrate why.

Shortlist 2011 N1

Here is Shortlist 2011 N1, proposed by Suhaimi Ramly:

For any integer $d > 0$, let $f(d)$ be the smallest positive integer that has exactly $d$ positive divisors (for example, $f(1)=1$, $f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$, $f(2^k)$ divides $f(2^{k+1})$.

I like this problem, so try it out if you haven’t. This is a problem …

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 …

About five years ago I wrote a blog post warning that I thought it was a bad idea to design math olympiads to be completely untrainable, because I think math olympiads should be about talent development rather than just talent identification, yada yada yada.

So now I want to say the other direction: I also don’t want to design math olympiads so that every problem is 100% required to lie in a fixed, rigid, and arbitrary boundary prescribed by some nonexistent syllabus. From a coach’s perspective, I want to reward “good” studying, and whatever “good” means, I think it should include more than zero flexibility and capacity to deal with slight curveballs.

I was reminded of this because there was a recent contest problem (I won’t say which one to avoid spoilers) that quoted Brianchon’s theorem. Brianchon’s theorem, for those of you that don’t …

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 …

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 …

Sometimes my OTIS students suggest features or things for the OTIS website, and I reply “submit a pull request”. I’m usually half-joking when I say this, because I acknowledge that I’m essentially saying “please do the work for me”.

But part of me isn’t joking. Because, one of the things I’ve grown to most value in gifted education is developing self-agency for my students.

If you’re reading this blog post, you’re likely to have good thinking abilities. You have the capacity to go from point A to point B, to teach yourself geometry from online resources (or a certain print textbook, I suppose), to put two and two together unsupervised, and so on. This gift is rarer than you think.

So let me tell you a secret: if you don’t know how to submit a pull request, you can teach yourself.

I have …

信言不美，美言不信。

I get a lot of questions that are so general that there is no useful answer I can give, e.g., “how do I get better at geometry?”. What do you want from me? Go do more problems, sheesh.

These days, in my instructions for contacting me, I tell people to be as specific as possible e.g. including specific problems they recently tried and couldn’t solve. Unsurpisingly the same kind of people who ask me a question like that are also not the kind of people who read instructions, so it hasn’t helped much. 😛

But it’s occurred to me it’s possible to take this too far. Or maybe more accurately, it’s always better to ask a specific question, but sometimes the best answer will still be “go do more problems, sheesh”.

Here’s a metaphorical example …