# Adventures in Problem Solving : Mathematical Marvels by Shailesh Shirali PDF

What is “problem solving”? The great mathematician David Hilbert said that, in essence, problems are the life blood of mathematics, and Paul Halmos added a rider: that the complementary activity – theory building – provides the soul of the subject.

Here is Halmos’ view:
Mathematicians sometimes classify themselves as either problem solvers or theory-creators. The problem-solvers answer yes-or.no questions and discuss vital special cases and concrete examples that are the flesh and blood of all mathematics; the theory creators fit the results into a framework, illuminate it all, and point it in a definite direction—they provide the skeleton and the soul of mathematics. One can be both a problem-solver and a theory-creator, but is usually one or the other. The problem solvers make geometric constructions, the theory-creators discuss the foundations of Euclidean geometry; the problem-solvers find out what makes switching diagrams tick, the theory. creators prove representations for Boolean algebras.

## Adventures in Problem Solving pdf

At the research level, good problems often play a vital role by directing the future course of a subject. When Hilbert offered his famous list of twenty-three problems at the International Congress of Mathematicians in 1900, he must surely have had this in mind; but perhaps even he would have been surprised at the extent to which his list shaped the course of research in mathematics in the 20th century. It is not hard to see why this would happen; a concerted attack on a difficult problem inevitably throws up a large number of challenges-new notation has to be devised, new concepts introduced, new connections found; and often this work provides the foundation of a new branch of mathematics.

This phenomenon can be seen many times in the history of the subject. Consider, for instance, the role played by the Königsberg bridges problem (posed and solved by Euler) in the origins of graph theory, or the growth of this subject achieved through work on the four-colour problem; or the progress achieved in number theory as a result of the work on Fermat’s ‘last theorem’; or the discovery of non-Euclidean geometries by Lobachevsky and Bolyai (independently) as a result of their efforts to prove the fifth postulate of Euclid (the “parallel postulate”); or the growth of recursive set theory through the efforts to solve Hilbert’s tenth problem (on algorithms to solve Diophantine equations). More such instances can readily be quoted.

As for pedagogy, the benefits of problem solving will be attested to by practically every teacher of mathematics. In some countries, notably Hungary, Russia and Romania, problem solving and its dual activity, that of problem posing, have been developed into a fine craft and have entered the teaching of mathematics in a big way.

### Adventures in Problem Solving by Shailesh Shirali PDF

The sizeable growth of interest in the Mathematics Olympiads over the last two decades attests to the increasing awareness of the importance of problem solving; problem solving is being increasingly talked about in educational circles these days. This book is written for problem buffs, but it is not a book about strategies or heuristics. There are many excellent books on the art of problem solving already written; the best amongst them (Polya’s two books, How To Solve It and Mathematical Discovery) would be difficult to better. The present work is more in the spirit of Newman’s A Problem Seminar and Polya and Szego’s Problems and Theorems in Analysis, both of which offer problem sequences rather than isolated problems. When I first started writing this book, the title I had in mind was The Art of Problem Solving; but once the book got under way, it became clear that an alternative title was needed. I briefly considered Episodes in Problem Solving before finally opting for the present title.