Good math books to discover stuff by yourself

Solution 1:

I would highly recommend the book "Concrete Mathematics" by Oren Patashnik, Donald E. Knuth and Ronald L. Graham.

I don't know that it exactly fits your criteria—or rather, I don't know that every reader would agree that it fits your criteria—but in my opinion, it does.

The first chapter, for instance, discusses three well-known puzzles of the type that will be addressed by the techniques to be taught in the book. Each puzzle is presented in its entirety before any approach to solving it is discussed.

The book requires VERY active reading, and if you just sit back passively without making your own efforts to solve each problem as you come to it rather than after reading the entire chapter, you will probably end up completely lost. ;)

What I would recommend for reading this book is that you:

1. Play with each problem as you encounter it, before reading further.
2. Once you have either solved the problem or gotten as far as you can without help, read a few more paragraphs (or even just one more paragraph).
3. Play with the new ideas and approaches presented in that paragraph. See if you make any discoveries about them on your own.
4. Repeat.

The nice thing is that the discussion of the puzzles and the exploratory discoveries from each extend far beyond just the direct solution to the puzzle itself. So exploration is very definitely encouraged.

One more note: I highly recommend you read the preface before you start in at Chapter 1. It will make certain conventions clearer; for example, it will explain why there are comments from students of the course scattered throughout the book. :)

An excerpt from the preface (my favorite part of the preface, actually):

The course title “Concrete Mathematics” was originally intended as an antidote to “Abstract Mathematics,” since concrete classical results were rapidly being swept out of the modern mathematical curriculum by a new wave of abstract ideas popularly called the “New Math.” Abstract mathematics is a wonderful subject, and there’s nothing wrong with it: It’s beautiful, general, and useful. But its adherents had become deluded that the rest of mathematics was inferior and no longer worthy of attention. The goal of generalization had become so fashionable that a generation of mathematicians had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. Abstract mathematics was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance.

And another excerpt from the preface, one which (for me at least) shows very clearly that this is "my kind of book":

Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren’t ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive.

Solution 2:

I'd highly recommend Paul Sally's Fundamentals of Mathematical Analysis. He includes several independent projects at the end of each chapter, where he guides you (loosely, all the work is up to you) through developing things such as topology, integration on $\mathbb{R}^n$, measure, the spectrum etc. on your own!

Solution 3:

My favorite one so far: Calculus by Michael Spivak (disclaimer: I'm still working on it). While the text is excellent, it is relatively concise and each chapter reads like a good lecture, one that lays down the definitions and main results and introduces the subject while keeping the reader interested. In my opinion, the text could be skimmed and used as reference for someone wanting to jump into the problems, its real treasure. Most of the problems are quite interesting and quite hard, with little to no clue on how to solve them present in the text. Many times I have struggled for hours trying to solve one of its problems, only to come back feeling very enriched from my wikipedia bingings, hour-long reflections, trial-and-errors and sweat.

Solution 4:

Check out "Combinatorial Problems and Exercises" by László Lovász. A professor of mine recommended the book, it seems like the idea of it is that you learn the subject through doing the problems, so hopefully this fits well.