Solve a simpler version of the problem.

For some reason that I can’t remember, I was recently motivated to take a look at the classic book on mathematical heuristics, How to Solve It, by George Pólya. When searching for it at the library, I came across a more recent book of the same title by Zbigniew Michalewicz and David Fogel. I checked the two books out of the library and skimmed them. Both books were readable and interesting, and I think if I’d been given them as a student, I would’ve read them cover to cover and enjoyed them. As it is, we all have shorter attention spans and less free time than before, also I’m already familiar with lots of the material in the two books, so I couldn’t quite bring myself to read them completely carefully. But I liked the material.

Books are long. Even Pólya’s book, which is short for a book, is long compared to a blog post. It contains lots of advice.

I wanted to make a post that’s useful to all of you, and I thought that, instead of giving a list of tips, it would be best to give just one tip. And, here it is:

Solve a simpler version of the problem.

Here’s the scenario:

You have a problem to solve, a math problem or maybe a math-like problem such as a coding problem or a negotiation problem or a challenge in getting a plot element to work in a story or just about anything.

And you’re stuck, not sure how to start.

Here’s the solution:

Construct a simpler version of the problem, a special case of the problem you’re given, but with less complexity. Set all the parameters to zero, or set all but one of the parameters to zero, then solve that simpler problem.

This sort of simplification has been a useful research tool for me (for example here and here), and it works for applied problems too.

P.S. Pólya’s How to Solve It is a classic. Why hadn’t I read it before? Here’s why. Back when I was in math olympiad, the coaches were always pushing that How to Solve It book.

And, usually, most of the things they were pushing were things that I resisted.

What were they pushing? – The idea that analytic geometry and calculus were kinda cheating, and that it’s better to solve geometry problems without using those tools. – The “nine-point circle.” – Disdain and lack of interest in applied problems. – The history of math as a history of heroes. – Math as competition. All sorts of attitudes that I didn’t like and still don’t like. I’m not saying that these attitudes are inherently bad, just that they weren’t for me, and they’re not for me now. The people pushing those attitudes were also pushing How to Solve It. So I wasn’t really interested in reading it, even though it came up a lot. My loss (until now).