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 …

One of my favorite Djikstra programming quotes is about thinking via “lines of code spent” rather than “lines of code produced”. I started using this as a philosophy in my writing too: words spent.

Background

One of the things that’s surprised me about student writing is how poorly words are spent. You’ll have a solution where the trivial boilerplate steps are painfully verbose, and then the actually important parts are missing all the critical details.

I wonder how much of this is because of crummy writing advice. In school essays, even when you have nothing meaningful to say, teachers often impose a minimum word countIn ninth grade, my English teacher preferred the euphemism “develop your ideas” for “write more words”. It wasn’t until halfway through the year I realized why she kept writing that on all my essays. as a “proof of work”. The implied conclusion …

1. Glazed carrots

Okay. Imagine you’re, like, trying to make glazed carrots or something.

Maybe a really simplified recipe looks something like:

1. Cut your carrots into suitably sized pieces with a knife.
2. Use a measuring spoon to get the right amount of oil, sugar, salt, etc.
3. Throw the carrots and other ingredients into a frying pan.
4. Serve the carrots on a plate.

You’ll notice that there were a bunch of different tools you used. The knife was used to cut the carrots into pieces. The measuring spoon was used to get the right amounts of other ingredients. And the plates are just there for the presentation of your dish. All these tools are things you see in any kitchen, but they do a single, completely unrelated thing.

Now imagine someone asks you:

I’m confused, why do people use a measuring spoon for cooking? Why not just use …

don’t ask why it just came in my head

Quandary

So you have a fair coin that you found on the ground,
or at least that’s what everyone says.
But on each of N times that you’ve tossed it around,
you see every flip has been heads.

For which value of N should you start to suspect
that the coin isn’t actually fair?
For which values of N can you firmly declare
that the tails side is not even there?

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 …

(Ed Note: This was earlier posted under the incorrect title “On Designing Olympiad Training”. How I managed to mess that up is a long story involving some incompetence with Python scripts, but this is fixed now.)

Spoiler warnings: USAMO 2014/1, and hints for Putnam 2014 A4 and B2. You may want to work on these problems yourself before reading this post.

1. An Apology

At last year’s USA IMO training camp, I prepared a handout on writing/style for the students at MOP. One of the things I talked about was the “ocean-crossing point”, which for our purposes you can think of as the discrete jump from a problem being “essentially not solved” ($0+$) to “essentially solved” ($7-$). The name comes from a Scott Aaronson post:

Suppose your friend in Boston blindfolded you, drove you around for twenty minutes, then took the blindfold off …

[EDIT 2018/03/05: This description seems significantly less accurate to me now than it did a few years ago, both because my views/values have changed substantially, and because SPARC has changed direction substantially since I attended as a junior counselor in 2015. I’ll leave it here as a reference, but should be taken with a grain of salt.]

I often get asked about what I learned from the SPARC summer camp. This is hard to describe and I never manage to give as a good of an answer as I want, so I want to take the time to write down something concrete now. For context: I attended SPARC in 2013 and 2014 and again as a counselor in 2015, so this post is long overdue (but better late than never).

(For those of you still in high school: applications for 2016 are now open, due March …

In high school, I hated English class and thought it was a waste of time. Now I’m in college, and I still hate English class and think it’s a waste of time. (Nothing on my teachers, they were all nice people, and I hope they’re not reading this.)

However, I no longer think writing itself is a waste of time. Otherwise, I wouldn’t be blogging, even about math. This post explains why I changed my mind.

1. Guts

My impression is that teachers in high school got it all wrong.

In high school, students are told to learn algebra because “we all use math every day”. This is obviously false, and somehow the students eventually are led to believe it.

You can’t actually be serious. Do people really think that knowing the Pythagorean Theorem will help in your daily life? I sure don’t, and …