This post is advice for problem-solving beginners in math. It’s halfway between Getting to know problems and Imperative statements in geometry don’t matter, but now I got it down to three words that are easy to remember: don’t be GPT-3.

What does that mean?

Basically, I see a failure case in a lot of beginners who get fixated on some (often superficial) detail or wording in the problem statement, and then try to pattern-match it to something they read somewhere before. In doing so, they stop actually thinking, and that’s a disaster recipe.

The reason I call this “don’t be GPT-3” is in reference to this toy problem:

Prove that if $\theta$ is a real number for which $\cos \theta$ is an integer, then $\sin \theta$ is an integer.

For a good high school math student, this problem could be called a test of “common sense”. If $\cos \theta$ is an integer, then it’s $0$ or $\pm 1$ and then $\theta$ must be a multiple of $90^\circ$.

However, this problem gave 2020-era AI a lot of trouble. LLM’s would often start by writing $\sin^2 \theta = 1 - \cos^2 \theta$ and start going down some Pythagorean-triples rabbit hole. The solver got so fixated on trying to pattern-match the trig functions that it stopped thinking.

Actual examples

OK, enough AI. Here are some examples that I’ve actually seen.

• My whining in Imperative statements in geometry don’t matter is a special case of this.

• There are non-geometry examples too: I get emails sometimes that say “I have difficulty with problems involving sequences”, for example, and in my head I’m always thinking “what does that even mean?”. Because problems that just happen to be about sequences have almost nothing in common.

• “I tried induction” is a similar pet peeve.I’ve had two other people independently complain about this particular example to me; one is David Yang’s post. Similarly for contradiction sometimes, since a lot of uses of contradiction are just cosmetic rewriting of the goal and not the key idea of the solution.

• I also see this sometimes with people working on olympiad inequalities trying to blindly pattern-match specific shapes of expressions. Example: just because a problem has $a^2+b^2+c^2 = 1$ as a hypothesis does not mean the solution is guaranteed to be Cauchy-Schwarz, etc.

• USA TSTST 2016/4 was one of the first examples that really struck me in my teaching career. It was staggering to me how many students tried to prove $\varphi(n) \ge n/3$, despite the fact this statement isn’t even true.