I understood very little of anything that happened in class. When I first got to the Mathematical Olympiad Program (MOP) my sophomore year, I was in way over my head. You’re so used to getting everything right, to being the one everyone else asks, that it’s hard to admit you need help. This is hard for many outstanding students. Instead of starting from what you know and working towards what you want, start from what you want, and ask yourself what you need to get there.Īsk for help. This is particularly useful when trying to discover proofs. Look back at the problem, and the discoveries you have made so far and ask yourself: “What haven’t I used yet in any constructive way?” The answer to that question is often the key to your next step. Many problems (particularly geometry problems) have a lot of moving parts. (I’m guessing some of our more mathematically advanced readers have so internalized the solution process for this type of Diophantine equation that you don’t have to travel with Pythagoras to get there!)įocus on what you haven’t used yet. Then, I thought about how to generate that method, and the path to the solution became clear. Number theory is not my strength, but my path to the solution was to recall first the method for generating Pythagorean triples. Think about the strategies you used to solve those problems, and you might just stumble on the solution.Ī few months ago, I was playing around with some Project Euler problems, and I came upon a problem that (eventually) boiled down to generating integer solutions to c² = a² + b² + ab in an efficient manner. Some of them were even hard problems! How did you do it? Start with problems that are similar to the one you face, but also think about others that have nothing to do with your current problem. Set your sights a little lower, then raise them once you tackle the simpler problem. Perhaps more importantly, it prepared us so that when we finally hit upon the Beast Academy idea, we were confident enough to pursue it. Not a one of those pages will be in the final work, but they spurred a great many ideas for content we will use. Our lead curriculum developer wrote 100–200 pages of content, dreaming up lots of different styles and approaches we might use. We started developing an elementary school curriculum months and months before we had the idea that became Beast Academy. But there’s a chance that one of your stabs will hit something, and even if it doesn’t, the effort may prepare your mind for the winning idea when the time comes. Accept that a lot of your effort will appear to have been wasted. At some point you have to stop staring and start trying stuff. Yeah, you have no idea what to do to solve it. Here are a few strategies for dealing with hard problems, and the frustration that comes with them:ĭo something. Strategies for Difficult Math Problems - and Beyond But without that frustration, those brilliant ideas never arise. Brilliant “Aha!” moments almost always spring from minds cultivated by long periods of frustration. They will teach you a lot more than a worksheet full of easy problems. The first step in dealing with difficult problems is to accept and understand their importance. I believe we’re teaching students how to think, how to approach difficult problems, and that math happens to be the best way to do so for many people. The same sort of strategies that go into solving very difficult math problems can be used to tackle a great many problems. I think a lot of what we do at AoPS is preparing students for challenges well outside mathematics. The problem with not being challenged sufficiently goes well beyond not learning math (or whatever) as quickly as you can. If you are consistently getting every problem in a class correct, you shouldn’t be too happy - it means you aren’t learning efficiently enough. You can’t learn how to do that without fighting with problems you can’t solve. They’re training future researchers, and the whole point of research is to find and answer questions that have never been solved. This is why college classes at top-tier universities have tests on which nearly no one clears 70%, much less gets a perfect score. If they were easy, someone else would have solved them before you got to them. We ask hard questions because so many of the problems worth solving in life are hard.
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