Polya Recommends Solving a Problem in Multiple Ways For True Understanding

"It is better to solve one problem five different ways, than to solve five problems one way." — George Pólya

In this episode, I look at a line from Hungarian-American mathematician George Pólya, author of the influential 1945 book "How to Solve It," and ask what it actually means to solve a problem versus just finding an answer to one. The discussion covers Pólya's often-skipped "look back" step, why repeating the same problem with a new approach builds transferable understanding in a way that solving new problems rarely does, and how this idea shows up well outside of math — in arguments, workflows, and decisions we make once and never revisit. I also give an honest counterpoint: depth has real limits, and sometimes breadth is the smarter move. A conversation about the difference between getting through a problem and actually learning from it.

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Welcome to the Quote of Corner. Today's quote comes from a mathematician who spent most of his life trying to figure out why some people are naturally good at solving problems, and eventually concluded that natural almost never has anything to do with it. It is better to solve one problem five different ways than to solve five problems one way. That's George Polia, a Hungarian American mathematician born in Budapest in 1887, who later taught at Stanford University for decades. If you've ever taken a math class where a teacher walked you through understand the problem, make a plan, carry out the plan, look back. You were taught Polia's method whether your teacher knew where it came from or not. His 1945 book, How to Solve It, became one of the best-selling math books ever published, not because it taught formulas, but because it taught a way of thinking. Polia wasn't interested in whether a student got the right answer. He was interested in whether they understood why the method worked, well enough to use it again somewhere else. Here's what I think this quote is actually getting at, and it's a little different than it sounds on first listen. It's not really about mathematics. It's about the difference between finding an answer and understanding a method. Most of us, when we're faced with a problem, a math problem, a legal argument, a broken process at work, we're trying to get to the other side of it as fast as possible. We find a path that works, we take it, and we move on. That's efficient. It's also, according to Paulia, the least useful thing you can do if you actually want to get better at solving problems in general. Because when you solve a problem exactly once and walk away, you've learned that one path. You haven't learned the problem. You don't know which part of your solution was essential and which part was just the first thing that happened to work. You don't know if your method would survive a slightly different version of the same problem showing up tomorrow. Solve it a second way though, and something interesting happens. You start to see the skeleton underneath the problem, the actual structure, separate from the specific numbers or facts you were handed. Solve it a third, fourth, fifth way, and you're not just getting more confident in the answer. You're building something closer to real understanding. The ability to recognize this problem's shape the next time it shows up wearing a different outfit. This connects to a step in Polia's own four-part method that almost everybody skips. Look back. Most people, the second they land on a correct answer, stop thinking about the problem entirely. Polia thought that was the single most wasted moment in the entire process because looking back, asking, could I have gotten here differently, and what does that tell me is where the actual transferable skill lives. Everything before that is just getting through today's problem. Looking back is what prepares you for tomorrow's. And I'd argue this scales up well past math. Anytime you find one approach that works, one way to structure an argument, one way to handle a difficult conversation, one workflow that gets a task done. There's a real temptation to lock it in and never look at it again. It worked, so why mess with it? But a method you've only tested once is a method you don't actually understand yet. You just know it worked, one time, under one set of conditions. We talked about something adjacent to this when we covered Chuck Close's line about inspiration being for amateurs. The idea that real competence tends to come from repetition and returning to the same material, not from waiting for a single flash of insight to strike. Polia is making almost the same case, just point it in a different direction. Instead of returning to the work in general, return to the exact same problem and make it teach you something new each time. Now, the honest counterpoint because this idea has real limits and I don't want to oversell it. Solving one problem five different ways takes time. A lot of it. And there are plenty of situations, a filing deadline, a client who needs an answer today, an exam with a clock running, where the right move isn't depth, it's getting to a correct answer efficiently and moving to the next thing. Polilla himself was talking to advanced math students who already had a baseline of fluency. Revisiting one problem repeatedly is a luxury that assumes you already know enough to try five approaches in the first place. If you're a beginner who barely has one working method, forcing yourself to invent four more before you've built basic confidence can slow you down and frustrate you for no real payoff. There's also a version of this that tips into pure perfectionism. Treating every solved problem as unfinished business, refusing to move on until you've wrung five insights out of it, when sometimes the better use of your time really is just solving five different problems and building breadth instead of depth. So the honest version of this idea isn't always solve everything five ways. It's closer to pick your spots. When a problem actually matters, when it's likely to recur, when understanding it deeply will pay off later, slow down and turn it over more than once. When it doesn't, solve it, take the win, and move on. Paulia wasn't handing down a rule for every problem you'll ever face. He was describing what separates a person who's memorized a trick from a person who's actually learned something durable. That's the version of this quote worth carrying with you. Not as a mandate to overthink everything, but as a filter for knowing which problems deserve the extra look. And as always, wisdom isn't in the quote, it's in the reflection. See you in the next episode.