Good math books to discover stuff by yourself

I am asking for a book/any other online which has the following quality:

The book, before introducing a particular topic (eg. Calculus/Topology) poses some questions which are answered by the topic (eg. Calculus allows you to study motions (eg. of celestial bodies) and blah blah) and encourages the reader to answer the questions by themselves, in their own unique way. The book thus, must be read in a very active way. The main attraction of the book is not the topic itself, but the deep and super exciting questions, which can spawn up new pathways, and often reaches to very deep stuff’s exploration, by the reader, without any crutches.


0. Instead of book you can also mention any other source, be online or offline.

1. The closest book which almost fits to my criteria is Paul Lockhart’s Measurement, and some questions are very deep (There was a question about proving which functions’ integration is expression-able in closed form, just after introducing calculus!), and that book is excellent. I’m just asking for more book/online sources .

2. While there’s nothing wrong with questions that require a bit of preknowledge, I prefer more deep questions that sounds elementary like: “Can you go through the seven bridges of Koingsberg and return to the starting place ?” (and then discover Graph Theory by your own !) or “Can you compute the area of the shape traced by two pencil and a string, exactly? Can you generalize it ?” (and then discover something similar to Diff Galois Theory or something new and unique by your own !) or “Can you find out how you can solve your Rubik’s cube toy in minimal number of moves ? Can you generalize to other Erno Rubik products ?”. Anyway feel free to mention books having both/any type of questions. But, The least the preknowledge required, the better.

3. I am not asking for a regular definition-problem textbook and/or a motivation book. See my and Thorsten S’s second comment in the question for clarification

4. It’s preferable that in the book the questions should be separate and/or presented in a non-spoilery manner, so that the reader can work on the questions without spoiling him with the answer. Also, it’s preferable (as mentioned in #2) the question is simple to state, but is very deep. Though I am asking for books with question as the main feature, theory building questions (examples in 2) are more preferred than general puzzles (you can generalize any good puzzle to the point of a good theory, but you should understand which type questions I want) but feel free to add books of both/any type.

As per comments spawning from answers, This may be very very very slightly related. If you have a puzzle book in mind, which is vaguely fitting these criterion’s, you can add it there.


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.

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