# Problem Solving Through Problems

The purpose of this book is to isolate and draw attention to the most important problem-solving techniques typically encountered in undergradu ate mathematics and to illustrate their use by interesting examples and problems not easily found in other sources.Each section features a single idea, the power and versatility of which is demonstrated in the examples and reinforc The purpose of this book is to isolate and draw attention to the most important problem-solving techniques typically encountered in undergradu ate mathematics and to illustrate their use by interesting examples and problems not easily found in other sources.

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I think a lot of what we do at Ao PS is preparing students for challenges well outside mathematics.

The same sort of strategies that go into solving very difficult math problems can be used to tackle a great many problems.

Our lead curriculum developer wrote 100–200 pages of content, dreaming up lots of different styles and approaches we might use. (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! Many problems (particularly geometry problems) have a lot of moving parts.

Not of those pages will be in the final work, but they spurred a great many ideas for content we will use. 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?

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.Each section features a single idea, the power and versatility of which is demonstrated in the examples and reinforced in the problems.The book serves as an introduction and guide to the problems literature (e.g., as found in the problems sections of undergraduate mathematics journals) and as an easily accessed reference of essential knowledge for students and teachers of mathematics.Perhaps more importantly, it prepared us so that when we finally hit upon the Beast Academy idea, we were confident enough to pursue it. ” The answer to that question is often the key to your next step. This is particularly useful when trying to discover proofs. You’re so used to getting everything right, to being the one everyone else asks, that it’s hard to admit you need help.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. When I first got to MOP my sophomore year, I was in way over my head.Each problem is chosen for its natural appeal and beauty, but primarily to provide the context for illustrating a given problem-solving method.The aim throughout is to show how a basic set of simple techniques can be applied in diverse ways to solve an enormous variety of problems.You can’t learn how to do that without fighting with problems you can’t solve.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. The problem with not being challenged sufficiently goes well beyond not learning math (or whatever) as quickly as you can.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.