Fun but serious mathematics books to gift advanced undergraduates.

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).

Solution 1:

• Check into Generatingfunctionology by Herbert Wilf.
  
  From the linked (author's) site, the second edition is available for downloading as a pdf. There is also a link to the third edition, available for purchase.
  
  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.

Solution 2:

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.

Solution 3:

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.