# 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.

Notes/Disclaimers:

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.

• Here is a closely related / possibly identical question: Big list of guided discovery books – littleO Jan 3 '17 at 6:51
• @littleO It's roughly related, but I don't want to be guided. Thanks for the link anyway. – user403711 Jan 3 '17 at 6:54
• I am sorry but I don't get the pertinence of the question. Why do you want a book which motivate you ? Aren't you motivated enough if you want self-study ? How can you know if a question is deep before being familiar with this topic ? Of course it is good to have a bit of motivation, but I think looking wikipedia, talking with older students is enough. You can't fill up a book with motivation, but with theorems, exercises and example you can. – user171326 Jan 3 '17 at 9:26
• Example: Taylor series. Most of the mathematical books will start with...Taylor series...definition...polynominal...order n....error bounds....*blahblahblah*. What AK wants is a book which gives thim this question: You know what a polynominal is. If you have an arbitrary function, how would you approximate it with polynominals that the approximation gets better and better in the vicinity of a specific point? – Thorsten S. Jan 3 '17 at 13:20
• It's more than a little tricky because "guided" is a gradient scale, not an absolute. Fundamentally speaking, if you don't want to be guided at all, then you shouldn't be reading a book, since it "guides" you through the author's ideas and communication to you. (Perhaps you should be writing one.) The "simple questions that have deep answers," if written in a book such that they are known to have deep answers, have already been explored. By contrast if you explore your own questions, perhaps you will find a deep area of exploration that has not been considered before. ;) – Wildcard Jan 7 '17 at 16:26

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.

• I didn't read the book, but it looks interesting. And the penultimate paragraph is exactly what I'm looking. – user403711 Jan 3 '17 at 6:51
• (Note: as the answer is edited, the penultimate paragraph was "The nice thing is that ..."). But as of now, the last paragraph sounds cool too. – user403711 Jan 3 '17 at 7:26

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!

• Cool ! But are there any preknowledge expected or I can just jump to the end of each chapter ? – user403711 Jan 3 '17 at 7:26
• @ArbitraryKangaroo It's a pretty challenging book that covers high level topics, such as Fourier Analysis on Locally Compact Abelian Groups, but it's all self-contained I think. There are lots of exercises interspersed throughout the chapters, which are nearly independent projects on their own (you prove the Implicit Function Theorem in a series of 4 exercises). The independent projects probably require reading the chapters they're attached to, but not always. – David Bowman Jan 3 '17 at 18:59

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.

• You might consider adding a link. – Wildcard Jan 12 '17 at 1:11
• Thanks for the recommendation. Newbie mistake. – RealityRoadie Jan 13 '17 at 17:22

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.

A Hilbert Space Problem Book by Paul Halmos. It's an introductory textbook, but the style, throughout, is to introduce a def'n, give some examples, and then ask the reader to prove results (theorems) about it. (E.g. "Prove that every non-empty weakly-open set is unbounded".) The author' proofs are collected at the end of the book.

From your edits it appears you are much more interested in the questions than in any sort of textbook.

In which case, I recommend the book of old classics; "Amusements in Mathematics" by Henry Dudeney.

Henry Dudeney lived from 1857-1930. There are some real classic puzzles in there, covering many widely different zones of mathematics.

An excerpt from the heading of the section "Arithmetical and Algebraical problems" with emphasis added:

The puzzles in this department are roughly thrown together in classes for the convenience of the reader. Some are very easy, others quite difficult. But they are not arranged in any order of difficulty—and this is intentional, for it is well that the solver should not be warned that a puzzle is just what it seems to be. It may, therefore, prove to be quite as simple as it looks, or it may contain some pitfall into which, through want of care or overconfidence, we may stumble.

Also, the arithmetical and algebraical puzzles are not separated in the manner adopted by some authors, who arbitrarily require certain problems to be solved by one method or the other. The reader is left to make his own choice and determine which puzzles are capable of being solved by him on purely arithmetical lines.

You can't get much less guided in your mathematical exploration than a pure puzzle book, which doesn't even warn you about the relative difficulty levels of the problems.

I still prefer Concrete Mathematics as I am more certain that the puzzles actually go somewhere (lead to deep development of interesting ideas), but if you just want questions you could give this one a try.

• See the update to note 4. +1 for another good suggestion. (BTW, I am a bit more fan of geometry topology than algebra arithmetic, so do you know any similar book in that genre ?) – user403711 Jan 7 '17 at 5:52
• @ArbitraryKangaroo, check out the table of contents for this book on Amazon's "Look Inside." Algebra/arithmetic is just the first section; it's got all kinds of stuff, definitely including geometry. Understood on the update to note 4. :) – Wildcard Jan 7 '17 at 5:53
• (I have updated the question to add another similar question. You might want to delete this answer, and add it there, where it's more appropriate) – user403711 Jan 10 '17 at 9:42

I'm surprised that this one hasn't been put up yet. My suggestion is Flatland, the classic by Edwin Abbott. The Flatlanders (2-dimensional geometrical objects) are divided into social classes, where the more sides any shape has, the higher ranking it is. Circles are the nobles, Isosceles triangles are workers, etc. They all believe that there are no more dimensions than 2, while the Line-landers (those who live in a line outside of Flatland) believe that the highest dimension is the first, and the Point believes only in himself. When the main character gets a visit from a Sphere, he learns a lot more about geometry, topology, etc.

• Well, I must admit that I didn't read the book, but I did watch the movie. Though I see there are some good geometric questions, I don't see any good topology question which may spawn from there. – user403711 Jan 7 '17 at 5:02