I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but also readable.

In particular, I am looking for readable books on more obscure topics not covered in a standard undergraduate curriculum which students may not have previously heard of or thought to study.

Some examples of suggestions I've liked so far:

• On Numbers and Games, by John Conway.
• Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups, by John Meier.
• Ramsey Theory on the Integers, by Bruce Landman.

I am not looking for pop math books, Gödel, Escher, Bach, or anything of that nature.

I am also not looking for books on 'core' subjects unless the content is restricted to a subdiscipline which is not commonly studied by undergrads. (E.g. Finite Group Theory by Isaacs would be good, but Abstract Algebra by Dummit and Foote would not.)

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@ZevChonoles I think this should be made CW. –  Alex Becker Dec 24 '12 at 3:12
How about Fourier Analysis, T.W. Korner, Cambridge University Press, 1988? (the "o" in "Korner" needs an umlaut, but I can't seem to get one there...) –  David Mitra Dec 24 '12 at 3:37
@Zev I've asked some time ago on meta: Should questions about book recommendations be CW? –  Martin Sleziak Dec 24 '12 at 8:27
What about the Princeton Companion? press.princeton.edu/titles/8350.html It would be excellent for someone at that stage of mathematical life –  mt_ Dec 24 '12 at 14:41

As leery as I am of adding yet another book (co)authored by John Conway to the list, I have to plug the amazing The Symmetries Of Things, a tremendous introduction to the basics of symmetry groups and tilings. There are things it could be better at (the fact that there's no connection with root lattices and Dynkin diagrams is a little odd to me), but it's still a fine introduction that does take some deep dives into questions of group theory.

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I'd recommend the new Dover edition of Michael Barnsley's Fractals Everywhere. Barnsley is an expert on iterated function systems (IFS) and shows how fractal geometry can be used to model real world objects. Subjects include metric spaces, dynamical systems, fractal dimension, fractal interpolation, the Julia and Mandelbrot sets, and measures on fractals. The style is engaging and the book is well illustrated.

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I recommend Paul Halmos' Automathography for an account of an extremely interesting life in mathematics. It's a joy to read!

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I propose Counterexamples in Topology and Visual Group Theory.

Les nombres remarquables by François le Lionnais is also interesting, but it is written in French; I do not know if there is an equivalent in English, maybe The Penguin Dictionary of Curious and Interesting Numbers.

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Two of my favorites are written by John Derbyshire. They both combine a historical narrative with mathematical discourse with plenty of charts and illustrations to help visualize concepts:

• Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (Plume Books, 2003) ISBN 0-452-28525-9
• Unknown Quantity: A Real And Imaginary History of Algebra (Joseph Henry Press, 2006) ISBN 0-309-09657-X
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Imre Lakatos, Proofs and refutations: The logic of mathematical discovery

It's a joy to read and everybody will learn something new from it, even your math professor (if he didn't already read it).

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The Cartoon Guide to Calculus is an amazing book for calculus beginners and does satisfy all of the OP's needs.

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I just got a hold on the book Primes of the Form $p=x^2+ny^2$ by David A. Cox and I think it has the required features.

In this book the author manages to present advanced algebraic number theory via historical point of view. He starts with the works of Fermat, Euler, and Gauss, and finishes with class field theory and complex multiplication.

I would certainly recommend this book as a fun and interesting book for an advance undergrad or beginning grad student (actually to anyone who likes number theory).

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Here are several books that I have looked at frequently.

Proofs and Confirmations, David Bressoud

Winning Ways For your Mathematical Plays Vols. 1 to 4, Berlekamp, Conway, Guy

Integer Partitions, Andrews and Eriksson

Number Theory in Science and Communication, Schroeder

Fractals, Chaos, and Power Laws, Schroeder

The first part of The Road to Reality, Penrose contains a primer on the math required in modern physics.

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The Book of Numbers by John H. Conway & Richard Guy. Although this does fall into the "popular mathematics" arena, it contains breadth and depth that will be of interest. Whilst I read many formal textbooks on mathematics, The Book of Numbers is a book I often return to when working on a problem. It contains many gems.

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Off the top, no particular order:

• Conceptual Mathematics - Lawvere and Schanuel
• Sets for Mathematics - Lawvere and Rosebrugh
• A Walk Through Combinatorics - Bona
• Combinatorial Species and Tree-Like Structures - Bergeron, Labelle & Leroux
• Ordinary Differential Equations - Arnold
• What Are and What's the Purpose of Numbers - Dedekind
• Collected Works of Karl Menger - Menger
• Algebraic Number Theory and Fermat's Last Theorem - Stewart

Just a couple if you're interested in applied areas:

• Theory of Gambling and Statistical Logic - Epstein
• Theoretical Introduction to Programming - Mills
• Elements of Statistical Learning - Hastie, Tibshirani & Friedman
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I enjoyed this one:

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A Gentle Introduction to Art of Mathematics by Joseph Fields is really nice.

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As someone in your target audience, I recommend A=B, by Marko Petkovsek, Herbert Wilf and Doron Zeilberger (with a foreword by Donald E. Knuth).

It's basically a book on generating combinatorial identities programmatically, inspired by Exercise 1.2.6.63 in Knuth's Art of Computer Programming, Volume 1:

[50] Develop computer programs for simplifying sums that involve binomial coeffcients.

The mathematics is not above an undergraduate's head, and a lot of the results are intuitive and attractive.

Best of all, the book is available in its entirety from the website (although paper copies can also be purchased)

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Saunders MacLane, Mathematics Form and Function. I'm reading it right now. It gives a wonderful birds eye view of (undergraduate) mathematics. The book is mostly self-contained for undergraduates and upwards, but it certainly helps to know a lot of math already.

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• Check into Generatingfunctionology by Herbert Wilf.

It's a very helpful, useful, readable, fun, (and short!) book that a student could conceivably cover over winter break.

• Another promising book by John Conway (et. al.) is The Symmetries of Things, which may very well be of interest to students.

• One additional suggestion, as it is a classic well worth being placed on any serious student's bookshelf: How to Solve It by Georg Polya.
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+1 for Wilf. One of the great combinatorics texts of all time. –  Mathemagician1234 Dec 24 '12 at 6:04
+1 for Conway too; I picked up The Symmetries of Things as a gift for myself this year and it's an absolute delight - I came here to post it myself! –  Steven Stadnicki Dec 25 '12 at 4:32

I should like to add In Pursuit of the Traveling Salesman by William Cook.

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I personally did not read this, but a friend of mine read A History of Abstract Algebra by Israel Kleiner for a term paper he was writing. It was real good, especially the parts about Noether and Dedekind. From what I gather all the information came from this book. I plan to buy it soon myself.

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I suggest Information Theory, Inference and Learning Algorithms by David MacKay. It is a book about Information Theory and codes, topics of great practical importance which aren't well covered in undergraduate courses. It has many exercises, some of which are very challenging. Opening it at random is bound to reveal something interesting, such as how the Bletchley Park codebreakers worked, what's the difference between British and American crosswords, and why we don't reproduce asexually. It requires a minimum of prior knowledge and is also a great introduction to probability and Bayesian statistics.

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The Sensual (Quadratic) Form, by John Conway.

I confess that I've only read the first chapter, but what I've read seems to fit the bill perfectly: Easily readable essays on an interesting topic that students don't normally see. Each chapter is independent from the others, and even many number theorists I know haven't heard of "topographs," the unique approach to visualizing quadratic forms that Conway develops in the first chapter.

EDIT: I want to add Proofs that Really Count by Art Benjamin and Jenny Quinn. Again, haven't read it myself, but I've met both the authors and seen them give (excellent) talks, and I've never heard anything but praise for this book.

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For another very nice book (in progress) that also discusses topographs, see Allen Hatcher's page for his "Topology of numbers", math.cornell.edu/~hatcher/TN/TNpage.html –  Andrés Caicedo Dec 24 '12 at 7:31

Ronald Brown's Topology And Groupoids gives a highly original and unusual first course in topology through basic category theory and the fundamental groupoid instead of the fundamental group. This allows Brown to present homotopy constructions in a very geometric way and to exclude homology altogether.

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This is a great book on the art of inequalities. Lots of problems inside as well: Cauchy-Schwarz Inequality

Also available online in PDF form here:

Cauchy-Schwarz PDF

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Look at Numbers, by Ebbinghaus and 7 co-authors. It has nice discussions about the real and complex numbers (aimed at mathematicians, not neophytes), and also the quaternions, octonions, p-adic numbers, and infinitesimals.

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For a fun read, which has the additional advantage of dividing into independent chapters which can be consumed in bite-sized chunks over the holiday season, how about

Proofs from The Book, by Martin Aigner and Günter Ziegler

And +1 to the OP's initial suggestion of Conway's On Numbers and Games.

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Fifty challenging problems in probability. Here is a larger list. Hope it helps

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Euler's Gem is a great book, you should check it out!

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Modern Graph theory by Bela Bollobas counts as fun if they're interested in doing exercises which can be approached by clever intuitive arguments; it's packed full of them.

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I would gift "Visual Complex Analysis" by Needham. It is a very beautiful book with a deep geometric intuition about complex numbers that is not typically covered in an undergraduate complex analysis course. I would also gift Stillwell's "Roads to Infinity: The Mathematics of Truth and Proof", one of the most beautiful treatment of the infinity concept that I encountered and that does not stay at the undergraduate level.

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+1: I was also thinking of Visual Complex Analysis. –  Jair Taylor Dec 24 '12 at 3:32
+1 for a terrific and unorthodox book. –  Mathemagician1234 Dec 24 '12 at 6:01
How is it that one can find something so beautiful in visual complex analysis but desires to use a website that burns ones eyes. –  John Riselvato Dec 24 '12 at 14:26

Surreal Numbers by Knuth. A Novel which turns into pure mathematics.

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