Math Problem Books

Math Problem Books

(Also read – Physics Books for self learning)  How often have you stared at a math (or physics) problem, unsure of where to begin?  Most math books are unusually pedantic and dull – all in an attempt to be more ‘rigorous’.  Screw rigor, I say.  Students are looking for a deeper understanding – lemmas, theorems are certainly not the way to provide that deep understanding.

These books fill this important gap. By providing the first step (where do I start), these problems bring those abstract theorems to life.

Here is a list of my favorite pure math problem books (Algebra, Differential Calculus, Real and Complex Analysis, Linear Algebra and more….) , theoretical (Ph.D. level physics) and mathematical physics problems.  Some of the Web Resources also have solutions to common graduate textbook problems ( Goldstein , Jackson , Griffiths , Peskin Schroder ,  Shankar  , Klauber , Merzbacher).

Berkeley Problems in Mathematics

The book is a compilation of over 1,250 problems which have appeared on the preliminary exams in Berkeley over the last twenty-five years. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph.D. program. Students who work through this book will develop problem-solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. The problems are organized by subject and ordered in an increasing level of difficulty. Tags with the exact exam year provide the opportunity to rehearse complete examinations.

The Math Problems Notebook – Boju and Funar 

A collection of non-trivial, unconventional problems that require deep insight and imagination to solve. They cover many topics, including number theory, algebra, combinatorics, geometry and analysis. The problems start as simple exercises and become more difficult as the reader progresses through the book to become challenging enough even for the experienced problem solver. The introductory problems focus on the basic methods and tools while the advanced problems aim to develop problem solving techniques and intuition as well as promote further research in the area. Solutions are included for each problem.

The Green Book of Math Problems – Kenneth Hardy

A popular selection of 100 practice problems — with hints and solutions — for students preparing for undergraduate-level math competitions. The challenging brainteasers will also appeal to anyone interested in problems concerning real numbers, differential equations, integrals, polynomials, sets, and other mathematical topics.
The hints are very helpful and the solutions are easy to follow. Questions drawn from geometry, group theory and linear algebra involve subjects ranging from multivariate integration to finite series to infinite sums and classical analysis. The only prerequisite is a high school-level background in mathematics.

The Red Book of Math Problems – Kenneth Hardy (Author)

In North America, the most prestigious competition in mathematics at the undergraduate level is the William Lowell Putnam Mathematical Competition. This volume is a handy compilation of 100 practice problems, hints, and solutions indispensable for students preparing for the Putnam and other undergraduate mathematical competitions. Indeed, it will be of use to anyone engaged in the posing and solving of mathematical problems.

Fifty Challenging Problems in Probability – by Frederick Mosteller (Author)

Marvin’s adventures in probability are one of the fifty intriguing puzzles that illustrate both elementary ad advanced aspects of probability, each problem designed to challenge the mathematically inclined. From “The Flippant Juror” and “The Prisoner’s Dilemma” to “The Cliffhanger” and “The Clumsy Chemist,” they provide an ideal supplement for all who enjoy the stimulating fun of mathematics.Professor Frederick Mosteller, who teaches statistics at Harvard University, has chosen the problems for originality, general interest, or because they demonstrate valuable techniques.

100 Great Problems of Elementary Mathematics  by Heinrich Dorrie (Author)

This interesting volume covers 100 of the most famous historical problems of elementary mathematics. Not only does the book bear witness to the extraordinary ingenuity of some of the greatest mathematical minds of history — Archimedes, Isaac Newton, Leonhard Euler, Augustin Cauchy, Pierre Fermat, Carl Friedrich Gauss, Gaspard Monge, Jakob Steiner, and many others — but it provides rare insight and inspiration to any reader, from high school math student to professional mathematician. This is indeed an unusual and uniquely valuable book.
The one hundred problems are presented in six categories: 26 arithmetical problems, 15 planimetric problems, 25 classic problems concerning conic sections and cycloids, 10 stereometric problems, 12 nautical and astronomical problems, and 12 maxima and minima problems.

A collection of problems in complex analysis  – L. I. Volkovyskii (Author), G. L. Lunts (Author), I. G. Aaramanovich (Author) 

Over 1500 problems on theory of functions of the complex variable; coverage of nearly every branch of classical function theory. Topics include conformal mappings, integrals and power series, Laurent series, parametric integrals, integrals of the Cauchy type, analytic continuation, Riemann surfaces, much more. Answers and solutions at end of text. Bibliographical references.

Problems in Group Theory –  John Dixon

From an amazon customer – ‘This is a good compilation of problems in Group Theory. Most of the problems are non-trivial and come from a variety of published research articles. The problems cover all aspects of the elementary theory, starting from subgroups, commutators up to representations and linear groups. While I think that the book is useful, it is by no means an exhaustive collection of all possible problems that one may ever encounter. By systematically working this book, one will however, definitely be able to grasp and appreciate the nuances of how to go about working with finite groups.’

The Moscow Puzzles

From an amazon reader –

This deserves a review since Shakuntala Devi and Martin Garderner have lifted problems & exercises from this book. In USSR this used to be a staple/ a must in High school Soviet curriculum. I think this statement alone is a testament to USSR’S and author’s supremacy and primacy in Math culture. This is very essential to a thorough and no-nonsense math foundation for students. If you are a puzzle freak you will recognize many problems, if not most.

Counterexamples

These are a great way to learn math. If something doesn’t fit the mould – gives you an idea of what exactly does fit the mould.  For instance – is the set of all rational numbers a group?

Counterexamples in Analysis

Summary

This list was not created overnight. Over a period of two decades, these books were part of the additions to my personal library.

Please comment if I am missing your favorite math problem book (or mathematical physics problem book) ! Also see Rare Finds in Relativity

Anuj holds professional certifications in Google Cloud, AWS as well as certifications in Docker and App Performance Tools such as New Relic. He specializes in Cloud Security, Data Encryption and Container Technologies.

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