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Tips to be a better problem solver [last live lecture]

Grant Sanderson · · ·69 min read
Commentary by Hex Index staff

Grant Sanderson's lecture on mathematical problem-solving offers a refreshing approach: acknowledging that teaching someone to solve novel problems is genuinely difficult. "I honestly don't know how" to teach this skill, he admits — a rare admission from an educator that most students would never hear in a formal classroom setting.

The Deceptively Simple Principles

Sanderson frames his nine tips as deceptively simple — principles that seem obvious once explained but prove surprisingly powerful. He argues 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." This framing is effective because it reframes problem-solving from an unattainable talent into a learnable habit.

Tips to be a better problem solver [last live lecture]

The core insight he offers centers on definition and naming. His first tip — "Make sure you're always using the defining features of whatever your setup is" — is presented as foundational: "I swear a good 70% of the problem sets that I did as an undergraduate math major essentially came down to looking at what was given, asking very critically what is the definition of each term involved." This is practical advice from someone who has spent years teaching mathematics.

When you encounter a geometry problem you've never seen, start by unpacking the definitions — that's where most solutions hide.

Geometry and Symmetry

The inscribed angle theorem serves as his primary example. He walks through adding elements to diagrams — "quite often when you see people solve hard geometry problems, it comes down to adding something to the picture" — and finding symmetry. The technique of naming angles ("I'll call this little angle that we've formed with our radius alpha and this little angle beta") becomes a tool for reasoning: "when you give things meaningful names, that actually helps you move forward in your problem."

Sanderson also demonstrates how to leverage symmetry in geometric figures — recognizing isosceles triangles and using their properties to draw connections between angles. This approach transforms what appears to be an arbitrary diagram into a structured system of relationships.

The Probability Puzzle

The quiz question about random numbers — "suppose that two numbers are chosen at random from the range 0 through 1" and finding the probability that their ratio rounds down to an even number — serves as both an engaging problem and a demonstration of his teaching method. He encourages participants to develop intuition first ("what do you think that probability is going to be?") before solving rigorously.

Critics might note that the lecture contains significant platform self-referencing — mentions of Khan Academy, item pool features, live quiz software, and links in descriptions — which can distract from the core mathematical content. However, these references also contextualize the lecture within his broader educational ecosystem.

Bottom Line

Sanderson's strongest contribution is his honest admission that problem-solving itself cannot be easily taught — a counterintuitive claim backed by nine concrete techniques. His vulnerability lies in the gap between explaining principles and actually solving novel problems: knowing these tips doesn't guarantee creative breakthrough. The lecture succeeds because it lowers the barrier between mathematical thinking and everyday reasoning, showing that geometry and probability are approached with the same systematic tools.

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Tips to be a better problem solver [last live lecture]

by Grant Sanderson · · Watch video

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.

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 not as a quiz that I expect you to necessarily be able to solve here on the spot. 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 and 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 maybe it rounds down to zero or it rounds down to two or to four. ...