Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof or leave it as an exercise. The type of book that I am imagining would have written "Problem: How many unit elements can a group have?" and similarly for all other theorems.

6$\begingroup$ This should be community wiki! $\endgroup$ – GMRA Jan 23 '10 at 1:43

15$\begingroup$ Why in the world would you look for such a book? Are you going to use it as a torture device on unwitting undergrads? $\endgroup$ – user577 Jan 23 '10 at 2:52

8$\begingroup$ Unit is bad too. Identity or unity are fine, but unity is typically also reserved for rings. The problem is, once you go from group theory to ring theory, and you encounter the proper usage of unit, it'll mess everything up. I mean, by the definition of a unit, every element of a group is a unit. $\endgroup$ – Harry Gindi Jan 23 '10 at 18:53

6$\begingroup$ This is known in certain circles as the "Moore Method" ... famously (or infamously) practiced by R. L. Moore. LINK: mathoverflow.net/questions/12070/… $\endgroup$ – Gerald Edgar May 19 '10 at 14:35

5$\begingroup$ (cont.): I'm not so sure that the weeding argument above is on the mark... $\endgroup$ – Jon Bannon May 14 '12 at 23:10
Algebraic Geometry by Robin Hartshorne. An algebra professor once told me that almost every exercise is a lemma or theorem from SGA.
A friendlier and more accessible book for undergraduates is "Linear Algebra Problem Book" by Paul R. Halmos. Halmos is an awesome expositor and this one is no exception.

12

$\begingroup$ I knew it was one of the two but I couldn't remember which. $\endgroup$ – user577 Jan 23 '10 at 6:44

15$\begingroup$ It is hard to imagine this book could have anymore major things (including definitions of things) as exercises and still consider it a textbook. It is basically a dictionary with exercises. $\endgroup$ – Matt May 19 '10 at 5:55

8$\begingroup$ A great alternative to Hartshorne with the same approach are Ravi Vakil's "Foundations of Algebraic Geometry" notes: math.stanford.edu/~vakil/216blog $\endgroup$ – Christopher Perez May 14 '12 at 21:41

$\begingroup$ This explains the surprising slimness of Hartshorne given the range of material covered (any lemma or theorem which involves a lot of work is pushed into the exercises and ''left to the reader''). $\endgroup$ – Hollis Williams May 23 at 16:51
Some classical books that would probably fit the bill:
Problems and Theorems in Analysis by Polya and Szego
A Hilbert Space Problem Book by Halmos
Fifty challenging problems in probability with solutions by Frederick Mosteller. It deserves to be better known than it is. Some things I like about it:
 It is elementary enough to be readable by high school students, but it introduces some serious ideas of probability.
 It is entertaining!
 It sells for $6.95.
Lovasz's "Combinatorial Problems and Exercises" is a really good example.

$\begingroup$ For a simpler text on Combinatorics with a problemoriented approach see my answer mathoverflow.net/questions/12709/… $\endgroup$ – Julien__ Jul 5 '14 at 13:45
My favorite such book is Problems in Analytic Number Theory by Ram Murty. There could not be enough good things said about it.
Onishchik and Vinberg's "Lie Groups and Algebraic Groups" (the translation, which is what I read, appeared in Springer's "Series in Soviet Mathematics") is ALL problems, and is very nice. Sadly, it is also out of print.
Problems in Group Theory, by John Dixon. I worked through a good deal of this as an undergrad, and learned a lot from it.

$\begingroup$ Great book! It even talks about linear groups and representations and characters (last two chapters), with theorems as problems. $\endgroup$ – Yannic Jan 27 '13 at 4:10
A number of books by Russian authors (cf. the one by Viro et al. from the Anton's answer) also come close to what you ask for. The two that came to my mind first are:
Theorems and problems in functional analysis by Kirillov and Gvishiani
Abel's theorem in problems and solutions by Alekseev (based on Arnol'd's lectures)

$\begingroup$ Elements of the theory of representations (By Kirillov) is another one. $\endgroup$ – Gian Maria Dall'Ara Jan 23 '10 at 9:24
Classical example
Elementary Topology. Textbook in Problems by O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov, N.Y.Netsvetaev

1$\begingroup$ This is a really good book. Students in the class don't tend to like it at the time since most of the theorems really are given as problem. But I think they tend to learn the material much better than more "standard" texts like Munkres. $\endgroup$ – Josh Jan 23 '10 at 2:21

1$\begingroup$ I used this book as a supplement for my intro topology course. I reall, really love it. $\endgroup$ – 5space Jun 18 '14 at 17:40
Here is my favorite: Victor Prasolov, Problems and Theorems in Linear Algebra. Also avaliable in PDF (mirror). However, I wouldn't recommend this to any undergrad without olympiad background.
Generally, writing textbooks in form of problem compendiums is distinctive for Soviet mathematics. I could name some more books of this kind (such as one on Lie algebras), but unfortunately they are all in Russian and most have never been translated.

3$\begingroup$ If you mean Vinberg and Onishchik, "Семинар по группам Ли и алгебраическим группам", it has been translated by Springer as "Lie groups and algebraic groups". Kirillov's "Elements of representation theory" has also been translated. $\endgroup$ – Victor Protsak May 19 '10 at 2:26
Problems in Algebraic Number Theory, by Esmonde and Murty, is very good.
Pinter's A Book of Abstract Algebra is half problems, half text. Many important topics are covered as problems. For example, direct products of groups are introduced and their properties developed in a set of problems. Cauchy's Theorem and Sylow's Theorem are introduced as problems. I taught myself a good deal of abstract algebra from this book one summer. The high proportion of problems to exposition kept me stimulated, and his decomposition of proofs of theorems into bitesized problem chunks was a confidence booster. I remember this book with great affection.

1$\begingroup$ +1: Pinter's book really is a wonderful alternative to the standard texts used for a first (undergraduate) course in abstract algebra. $\endgroup$ – J W May 15 '12 at 5:25
"Elements of the Theory of Representations" by A. Kirillov. This is a concise introduction to the representation theory of both finite and Lie groups. It contains necessary background from other fields, e.g. analysis on manifolds. Many theorems are formulated as problems, often with hints. Originally the book was written in Russian, but there is also English translation published by SpringerVerlag in 1976.
Convex Figures I.M. Yaglom and V.G. Boltyanskii Holt, Rinehart and Winston, NY, 1961
The first half of this book has definitions and results related to convexity to be proved by the reader and the solutions to these problems (theorems) is given in the second half of the book.
The topics treated include Helly's Theorem, isoperimetric results, Minkowski addition of sets and curves of constant width.
All of this material is clearly and well handled.
I had a lot of fun skimming through Jim Henle's An Outline of Set Theory. (It's now out of print, but I suspect Springer's Problem Books in Mathematics series has a few more such titles still in print.)
Also, a nice little freebie is Stefan Bilaniuk's A Problem Course in Mathematical Logic.
In a different area, there is Number Theory Through Inquiry by David C. Marshall, Michael Starbird, Edward Odell.

3$\begingroup$ There is an absolutely scathing (probably overthetop) review of Henle's book by Craig Smorynski (Amer. Math. Monthly, Vol. 95, no. 4 [April 1988]). Here is a link, which hopefully works: thevelho88.free.fr/books/AMS19884.pdf $\endgroup$ – Todd Trimble♦ Jan 27 '13 at 4:18

1$\begingroup$ Awesome review! He makes some really good points. But you're right, asking for the author to be "raked over the coals or at least flogged" is probably a bit over the top. $\endgroup$ – Pablo Lessa Aug 16 '14 at 12:37
Ian Adamson has 2 really nice books pitched at the upper level undergraduate/graduate level: A General Topology Workbook and A Set Theory Workbook. Set Theory and point set topology can mostly be developed directly from the definitions,so these are nice subjects to present in this manner,particularly to students just learning how to do rigorous proofs. Best of all,both books come with complete solutions in the back.

$\begingroup$ Ah, beat me to it. This was definitely the first thing that came to mind. $\endgroup$ – Matt May 19 '10 at 5:56

$\begingroup$ Then again, I don't consider it a good idea to make general topology the students' first encounter with rigorous proofs . $\endgroup$ – darij grinberg May 19 '10 at 19:40

$\begingroup$ @Danj Depends on how gentle a treatment you give them,Danj. If you hand them Kelly and then tell them to have at it while you're off to a conference,then I agree........... $\endgroup$ – The Mathemagician May 20 '10 at 7:23
Joe Roberts, Elementary Number Theory, A Problem Oriented Approach. The 1st half of the book is all problems, the 2nd half is the solutions. This book is unusual for another reason; it's done entirely in calligraphy.

2$\begingroup$ Great book  I spent a fun summer working through its material on continued fractions with a fellow UIUC student. $\endgroup$ – Scott McKuen May 19 '10 at 6:04

1$\begingroup$ I'm that fellow UIUC student. This book shepherded (for me) continued fractions from the "know about" column to the "can use effectively" column. It's my alltime favorite text. $\endgroup$ – Kevin O'Bryant May 26 '10 at 2:16
Kenneth P. Bogart's "Combinatorics through Guided Discovery" is written exactly this way and all properties touched on in the book are discovered in the book through problems.
You can download it here:
http://www.math.dartmouth.edu/newsresources/electronic/kpbogart/

$\begingroup$ For a published alternative recommended by the AMS see also my answer here : mathoverflow.net/questions/12709/… $\endgroup$ – Julien__ Jul 5 '14 at 13:46
The little commutative algebra book by Atiyah and MacDonald is one such—the reason it's so little is that probably twothirds of the results in it are in the exercises. I guess you know the subject if you can skim through it looking for a fact (which is quite likely to be present, if not in the main text) all the while nodding your head as though the proofs were actually given.

$\begingroup$ The point is that "understanding" this text is a diagnostic, an exam, a test, in itself. The book is not a learning device, in many regards, but a test, in itself, ... as are (too...) many. $\endgroup$ – paul garrett Nov 16 '14 at 0:58

$\begingroup$ @paulgarrett If I understand you correctly: I've never understood why people like this book so much either. $\endgroup$ – Ryan Reich Nov 16 '14 at 1:26

$\begingroup$ @RyanRiech, yes, I'm afraid that many people, in good faith, somehow misinterpret, or overinterpret, suffering as genuinely virtuous activity. In fact, I don't think there's much moral virtue, and certainly not professional virtue, in reinventing reallycrappy wheels... given that one's time and energy are finite, especially. But various sadomasochistic relationship themes have an enduring popularity in the species... so thereweare. $\endgroup$ – paul garrett Nov 16 '14 at 1:29
An introduction to the theory of groups by Joseph J Rotman makes for a good DIY second course in group theory.

$\begingroup$ It makes a good FIRST course for those with a good algebra background,Sonia. Like say,from Rotman's ADVANCED MODERN ALGEBRA. $\endgroup$ – The Mathemagician Apr 2 '10 at 6:51
Learning mathematics by solving problems is part of the french tradition. You will find many problems in Bourbaki or Dieudonne's Elements d'Analyse. At a more elementary level there are several problems covering a large amount of material at the end of Colmez's Elements d'analyse et d'algebre (et de theorie des nombres).
There are several introductory problem texts by R.P. Burns:
A nice introduction (highly recommended!) to abstract algebra which is almost entirely presented in problem format is the following by Clark:
I recently found the following books (which I have no experience with):
"Selected Problems in Real Analysis" by M. G. Goluzina, A. A. Lodkin, B. M. Makarov, A. N. Podkorytov. (The authors are modest with the title, I would rather say that this book contains all problems in real analysis.) This is a perfect complement to Polya and Szego, whose book is rather on complex analysis.
There are number of books in the Schaum's Outline series that I would recommend to anyone beginning in the subject of choice, Group Theory, Linear Algebra, General Topology to name a few. They are good in making the initial learning curve less steep, and help to make many of the other books mentioned more accessible to someone new to the subject.
Gerhard "Ask Me About System Design" Paseman, 2012.05.13
FiniteDimensional Linear Analysis: A Systematic Presentation in Problem Form (Dover Books on Mathematics) by GlazmanLjubic A (difficult)introduction to finite analysis(no solutions)
Theorie des groupes Jean Delcourt(in french) (has solutions)
I would like to mention about Serge Lang's Algebra. Many of the "standard" results/theorems appear as exercises: one particular example is the construction and properties of Witt ring as exercises in chapter XV; of course, there is "that famous" homological algebra problem too.

$\begingroup$ famous" homological algebra problem too is now gone is newer editions $\endgroup$ – Koushik May 23 '14 at 7:03

$\begingroup$ just out of curiosity, can you give a reference to or restate here "that famous homological algebra problem"? $\endgroup$ – AnatolyVorobey Jun 1 '18 at 10:03
See these two good books :
Combinatorics : a problem oriented approach  Daniel A. Marcus
Graph Theory : a problem oriented approach  Daniel A. Marcus
The first one was recommended for selfstudy by an AMS article :
Marcus’s elementary Combinatorics: A Problem Oriented Approach is appropriate for self study; more advanced is Lovász’s Combinatorial Problems and Exercises.
See also :
Problems and Theorems in Classical Set Theory by Péter Komjáth and Vilmos Totik.
Some answers mention problem books (quite different from standardformat textbooks in that they consist almost entirely of problems and their solutions). Such books have been widely used in Eastern Europe at every level of education (at least when I was getting it). Let me add another one to the list:
MR0447533 Krzyż, Jan G. Problems in complex variable theory. Translation of the 1962 Polish original. Modern Analytic and Computational Methods in Science and Mathematics, No. 36. American Elsevier Publishing Co., Inc., New York; PWNPolish Scientific Publishers, Warsaw, 1971. xvii+283 pp.
In his foreword, the author states: ``Most exercises are just examples illustrating basic concepts and theorems, some are standard theorems contained in most textbooks. However, the author does believe that the reconstruction of certain proofs could be instructive and is possible for an average mathematics student."
Besides standard material, there is a collection of quirky little facts in e.g. nonEuclidean geometry in the disk or logarithmic potential theory (and much more). All stated as problems for the reader to solve. However, many solutions are included.
There were subsequent editions in Polish. I used one as an undergraduate student and still have a copy.