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).

I would highly suggest

Matrix Groups for Undergraduatesby Kristopher Tapp. This is extremely readable–you feel like you are doing little more than reviewing some linear algebra and analysis, but BAM you realize that you’ve just had an extremely gentle, but useful introduction to the basics of Lie groups.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.

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, Penrosecontains a primer on the math required in modern physics.Ronald Brown’s

Topology And Groupoidsgives 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.