*This is why college classes at top-tier universities have tests on which nearly no one clears 70%, much less gets a perfect score.*They’re training future researchers, and the whole point of research is to find and answer questions that have never been solved.Whenever possible, problems within sections are chosen to cut across expected course boundaries and to thereby strengthen the evidence that a single intuition is capable of broad application.

It took a few hundred hours (IIRC), but was time very well spent and was repaid later with interest.

That's not to say it's a silver bullet - working through the book requires that you be at the right current problem-solving level (neither too far ahead of the book's level, or behind). But for me it was immensely helpful.[*] TWorking through most of this book was one of the two most useful [*] things I ever did for learning to solve mathematical problems.

She does fantastic work in our courses, and was easily among the very top students in the class she took with me, and yet she still occasionally hits problems that she can’t solve.

Moreover, she has access to an excellent math teacher in her school who sometimes can’t help her get past these problems, either.

You should also be prepared for the question to go off in a non-intuitive direction and be ready to follow it.

A parent of one of our students wrote today about his daughter’s occasional frustration with the difficulty of some of the problems in our courses.Here are a few strategies for dealing with hard problems, and the frustration that comes with them: Do something. 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. A 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 is not my strength, but my path to the solution was to recall first the method for generating Pythagorean triples.We started developing an elementary school curriculum months and months before we had the idea that became Beast Academy. Set your sights a little lower, then raise them once you tackle the simpler problem. Then, I thought about how to generate that method, and then the path to the solution became clear.I recommend extensively trying it; among other things, getting good at being stuck is an immensely valuable part of learning to problem solve.I still generally only get half marks on BMO/IMO papers if I work to the time limit so you can't take this as gospel but if I was just starting out on this style of paper these are two bits of advice I'd want (beyond the mathematics staple of 'if in doubt, sketch it out').1.Don't be afraid to play with the numbers for a good while…simple number patterns (especially the triangular numbers) have a part in a surprising proportion of the solutions but if you haven't extrapolated numerically this can be difficult if not impossible to see.2.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 first step in dealing with difficult problems is to accept and understand their importance. They will teach you a lot more than a worksheet full of easy problems.Brilliant moments almost always spring from minds cultivated by long periods of frustration. Start with problems that are similar to the one you face, but even think about others that have nothing to do with your current problem.But without that frustration, those brilliant ideas never arise. At some you have to stop staring and start trying stuff. Accept that a lot of your effort will appear to have been wasted. Think about the strategies you used to solve those problems, and you might stumble on the solution.

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## Problem-Solving Through Problems

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Isolate and draw attention to the most important problem-solving techniques. undergraduate mathematics. interesting examples and problems not easily.…