This question is related to Counterexample Math Books, but I'm looking for books in areas which aren't covered there (for example, number theory).
In addition, books that focus on clever observations that comment on theorems, proofs, examples and counterexamples are also very welcome.
A great book about proofs is Martin Aigner, Günter Ziegler: Proofs from THE BOOK.
Three additional recommendations for your extensive book list (considering your related questions :-))
Number Theory
Books dedicated to counterexamples in number theory are not known to me, but since you pointed to number theory here are books with related information
- Unsolved Problems in Number Theory by Richard K. Guy.
This book is not dedicated for a beginner in mathematics. Although many of the questions are formulated using basic number theoretical terminology, most of them are really tough problems (namely at least partly unsolved). But it provides a good impression which problems are subject to research and valuable for the interested reader: For each problem there is an extensive list of references. Hundreds of problems are listed organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous.
- Problems in Algebraic Number Theory by Jody Esmonde and M. Ram Murty.
This is a fine book appropriate for self-study after you have mastered a course in undergraduate algebra. It provides 500 problems and presents thereby concepts and ideas from algebraic number theory. It consists of two parts. The first presents theorems and problems (exercises), the second part provides complete solutions.
Comments on theorems, proofs, etc.
- A Problem Seminar by Donald J.Newman.
This is a helpful booklet in training for the advanced undergraduate. The problems are organised in Estimation Theory, Generating Functions, Limits of Integrals, Expectations, Prime Factors, Category Arguments and Convexity. My impression is the book is in degree of difficulty and in its spirit similar to the book Counterexamples in Calculus. I appreciate both of them.
A word of warning: Please keep in mind that the Counterexamples books differ significantly in their degree of difficulty from less demanding (Counterexamples in Calculus) to highly sophisticated (e.g. Counterexamples in Topological Vector Spaces). While the first is accessible (and a good training!) after you have mastered a course in undergraduate analysis, the latter demands a profound knowledge in point-set topology and functional analysis.
- Problem-Solving Through Problems by Loren C.Larson
This is a helpful guide explaining how to solve problems. E.g. chapter 1 Heuristics provides helpful hints like Searching for a Pattern, Drawing a Figure, Formulate an Equivalent Problem, Modify the Problem, Choose Effective Notation, Exploit Symmetry, Divide into Cases, Work Backward, Argue by Contradiction, Pursue Parity, Consider Extreme Cases and Generalize. This book is accessible for the undergraduate student and it can be considered as good training on the job and prerequisite. Namely a prerequisite before looking at the more demandig problems from Problem-Solving Strategies which is an answer according to this question from you.
Clever observations that comment on examples
- Irresistible Integrals by George Boros and Victor H. Moll.
This is a great collection of beautiful integrals and techniques on how to solve them. It provides a nice survey of standard techniques which you may know after you have mastered an undergraduate course in one-variable calculus. One important fact is that knowledge in real calculus is sufficient. The techniques presented here do not require any knowledge in complex analysis.
Gary L. Wise, Eric B. Hall - Counterexamples in Probability and Real Analysis, is a very good book related to what you are looking for.
