Tips to be a better problem solver [Last live lecture] | Ep. 10 Lockdown live math

Welcome back everybody. It's hard to define exactly what mathematicians mean when they use the phrase problem solving. However you go about it, it's going to involve some notion of approaching puzzles that you've never seen before and still being able to systematically and creatively find some solution to them. But that's a weird thing when you think about it because it makes it a very hard thing to teach.

I mean, you can teach someone how to solve one particular problem, maybe even a class of problems, and teach them how to solve another problem. But how do you teach someone how to approach a problem that they've never seen before and still make progress on it? Well, I honestly don't know how. But what I want to do for this lecture is to talk through a couple different problems solving principles.

I've enumerated nine of them in total, and each one we're going to walk through in the context of a specific example. And I think each one is kind of simple. You'll look at it and you'll nod along thinking, "Yeah, yeah, of course that's a thing that you should do." But I would argue that each one is deceptively simple. that you would be shocked at how often you can make very meaningful progress in very hard problems just by keeping some of these tips in the back of your mind.

And to give you a little flavor for where we're going to be going today, I want to ask uh not as a quiz that I expect you to necessarily be able to solve here on the spot. Um I want to ask you a question that we will be solving later on in the lecture just so you can have it in the back of your mind, a little thing to mle over um and a a hard problem that will be fun to tackle when the time comes. So the question asks suppose that two numbers are chosen at random from the range 0 through 1 and it's done according to a uniform distribution. So maybe you pick like 385 and 58962 or something like that.

Each one is chosen at random. Suppose P is the probability that the ratio of the first number to the second rounds down to an even number. So you know maybe it rounds down to zero or ...