Gift advice: present for high school graduate interested in math I am a PhD student in mathematics who recently found out that I will be attending my girlfriend's cousin's high school graduation party. I have never met the cousin, but hear that he is very interested in mathematics and is hoping to major in mathematics in college. He is taking Calculus BC (the equivalent of Calculus 2 at most colleges) now and is apparently doing quite well.
I am considering giving him a math book as a graduation present. The following texts immediately came to mind as decent candidates:


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*Elementary Number Theory by Underwood Dudley

*Calculus by Michael Spivak

*How to Prove it by Daniel Velleman


I think that they are all at about the right level. Further, these texts were instrumental in my early mathematical development. They provide for entertaining reads while still being substantive.
However, I would like some advice on the following matters,


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*Is a math text an appropriate graduation present?

*What other math texts might I consider?

*Would it be better to give a popular text such as Derbyshire's Prime Obsession?
 A: What about a Raspberry Pi/Arduino?  When you're getting cool gifts from everyone else, a math book might be slightly disappointing.  Not to mention anyone heading off to college is going to be downsizing their physical possessions, which might even include your book.  Hobbyist hacking on the other hand is nerdy, cool, and extremely useful.  Do you want to give him something that he could download as a PDF, or something that he could use to setup his own webserver and host that PDF?
A: Have you considered a slide rule? Every maths or engineering major should have one, even if they are hardly used these days. 
A: Yes, it is an appropriate graduation present. I would also consider a book on the history of mathematics, such as "Men of Mathematics" by E.T. Bell. That would give a broad overview of the subject and the motivations that lead to discoveries.
A: I second Kaj Hansen's suggestion of "what is mathematics" and I'd suggest also "Gödel, Escher, Bach, en eternal golden braid", it deals with very interesting topics (Gödel's incompleteness theorem, formal systems and similar things) in a very accessible and entertaining way.
I read both of them 2 years ago, when I was 17 and they made me love maths!
A: I'd go for this : Engel's Problem-Solving Strategies. 
You may argue it's a tough book,  but it's one of the only that develops any student's own mathematical thinking. It teaches willing students techniques and always challenges them with reasonings they will inevitably encounter later in their academic career. 
A: Here's a non-book suggestion:
http://www.kleinbottle.com
This site sells glassware with the topological type of a klein bottle. It even has a klein-bottle mug! (http://www.kleinbottle.com/drinking_mug_klein_bottle.htm)
It's a pretty sweet thing to have if the guy likes topology.
A: I would argue that a math text is usually not an appropriate graduation present, even for somebody very interested in mathematics.
Just because he likes math, it doesn't mean that he likes reading books on it in his free time
I am an avid book reader myself, and have shifted heavily to non-fiction in later years. I have discussed books with friends frequently, and find out that they are much less inclined to read than I expect. And when they do read, they do it for entertainment, and if they start a non-fiction book, they seldom have the motivation to finish it. These people are mostly STEM graduates, quite a few of them have a Ph.D. too. Even the ones who do read non-fiction now (in their 30s and 40s) say that they developed the taste late.
As you can see here, many people do like reading this kind of book. But as far as I can tell, many more don't like it. And don't forget that math.stackexchange is not a place representative of all people who like maths, it is a place representative of people who like spending their free time on a web site about maths, which is probably a subset of the first ones. And I argue that it is a smallish subset.
Even if he likes math and likes reading books on it, it doesn't mean that he wants to get one from you.
First, book choice can be a rather personal matter. It is very hard to meet someone's taste, even if you know the general direction of what they like.
But there is this small problem with learning: All people love to learn, nobody likes being taught. (I don't remember whose aforism is this, maybe Churchill?). If you see curiosity and learning interest in a younger person, it is good to encourage them, help them discover the information they already thirst for, etc. But by just giving them a book because you think the world would be a better place if everybody has read it, you come across as condescending, the kind of 19th century educator who thinks it is his high moral duty to bring children on the path of rightness.
From your question, it seems that you fall at neither of these extremes, and are probably closer to the first case. But it still isn't close enough. You don't know him, don't know exactly what he is interested in. Maybe he isn't taking calculus because he loves calculus, but because he feels a high pressure to get into a good college. And he doesn't know you, and doesn't know your motives; it is easy for him to assume that you are the condescending kind even if you aren't. Even if he doesn't, choosing a book he might be interested in has a high chance to fail.
Even if he likes math and would read books on it, this is not necessarily the right occasion.
He just made it through school and wants to feel that he achieved something after making it through a pile of textbooks. He needs a feeling of closure, even as an intermediate step. Yes, there is college somewhere in his future, but he probably wants some time to stop, look back at what he did, and smile, before getting back on the treadmill. Having finished one level of something only to be presented with the next, higher level immediately is a sure way to demotivate somebody from ever achieving something. No, I have no guarantee that he feels this way, but you have no guarantee that he doesn't feel this way. And I have seen many people, including bright ones open for education, who are this way.
Besides, graduation is a social occasion. No matter how much of a nerd he might be, even if he is internally proud of being a nerd, he probably won't want his classmates to remember him as "the nerd who got textbooks for graduation day". Teenagers are very sensitive about their image in their class.
Conclusion
When you start with the fact that math belongs to his favorite school subjects and conclude that he will like receiving a math text as a graduation present, you are making a lot of assumptions, and there is a high risk that at least one isn't true. He might be one of the people who will truly enjoy a math book for graduation day, but the chance that he won't is quite high.
For your decision, you can try getting better intelligence. Let your girlfriend find out through the family grapewine how likely he is to appreciate this kind of gift. If you hear that he has earlier purchased math books for himself (which were not required for a course), then this is a very good indicator that your gift will be well received. Mentioning that you knew that he reads math books for fun will also clearly remove you from the "condescending old relative" zone.
If he hasn't done this, or you can't find conclusive information, consider some other, more accessible, gift. When you meet him, you can discretely evaluate his reading interest yourself, and if the result is positive, you can still get these books for him for Christmas.
If you don't go for a math book now, here are some other options which are math-themed:

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*a popular book, as you mentioned. There are a few good suggestions in this thread already. Beside things like Goedel Escher Bach, you can also get a puzzles book like one of Smullyan's. (Yes, it is still a book. But it is entertainment, and read in a very different context).

*a puzzle, or set of puzzles. Most mathematicians I know love physical puzzles. Not the jigsaw ones, but the kind where you have to find out how to take a box apart, etc. Or just a beautiful chess set.

*an entry for some event. This works a bit better with people interested in engineering, as there are more robots exhibitions than maths exhibitions :) but depending on your budget, you could find something.

*and finally, you can of course make a gift which has to do with another hobby of his. One which is more of a hobby (= he has been seen spending free time on it without any benefit beside the joy of it) than an interest (= he perks up when he hears about the matter, and maybe prefers it in school, but does not necessarily allocate free time to it).

A: In addition to those recommended in the comments, I would highly recommend What is Mathematics? by Richard Courant.  On this book, Einstein wrote "A lucid representation of the fundamental concepts and methods of the whole field of mathematics...Easily understandable."
Fear not!  This book is certainly accessible to a high school senior with a passion for math.
A: I'd actually go off script here.

Richard Feynman - Surely You're Joking, Mr. Feynman

(The other autobiographical books are great too.)
This is to remind people that you can have a lot of fun and crazy adventures while being smart and a good student. And there's a lesson in those books about academic and scientific honesty which is applicable in mathematics just as well.
A: As a high school sophomore, I agree that the books suggested would all be appropriate level for the student, however there are some other considerations you brought up: 
Is a math text an appropriate graduation present?
I would answer a strong yes! As a student interested in both mathematics and physics, and as someone with a lot of friends interested in math, biology, and engineering, I can say that a mathematics book is very good graduation present, particularly because the occasion is particularly about education. The main question to consider is:
What type of book should I gift?
While he might love a calculus text (I wouldn't mind Apostol's Calculus), a general expository text in mathematics might be more appealing, simply because it is more accessible and takes less motivation to read (which, unless they are self studying or something of the like, is a factor). However, I wouldn't recommend the number theory text as much (although, I haven't read that specific one), simply because it is a specific area of math that may be unappealing to him.
Should you choose to give him an actual math text, though, I asked this question, which brought a good deal of answers for math texts at approximately the level you were asking. 
Now there is one question left:
What book should I get him?
I personally would recommend "How to prove it" because it is very accessible (and reads like an expository text) and really allows the student to do two things:


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*Transition from "calculate this" type math problems to "prove this" type math, a transition that is usually hard, and really interesting. The "prove this" type math is why I love math, and really allows mathematical reasoning beyond anything given in high school math.

*Allows the student the language to express mathematical statements. The usage of quantifiers and other logical symbols allows the student to express their thoughts as something other that "fuzzy" statements in natural language.


Close runner ups:


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*"What is mathematics" by Courant

*(too elementary, but would be good for someone with only a high school algebra background) "The joy of X"
Edit for posterity:
"A book of Abstract Algebra" (Dover, by Pinter) is such a beautiful and approachable text that I only wish I would have discovered sooner. Although it does have as a prerequisite "How to prove it" (or knowledge of proofs), it is definitely appropriate once that knowledge is gained. 
A: I recommend Polya's "How to Solve it" and possibly "Mathematical Discovery - On Understanding, Learning and Teaching Problem Solving" (but I haven't read this second one). It gives you a real feeling for how mathematics is (and should be, at all times) done.
A: I gave a copy of "The Man Who Loved Only Numbers" about Paul Erdos to a young cousin who had made some comments about going on in math. It's really the only book I've tripped over that truly describes what it is to love mathematics and numbers. It is also a great way to expose someone to the world of mathematics beyond calculus
The book was recommended to me by one of my math teachers, a specialist in cooperative game theory and discrete math.
A: One nice introductory book into pure maths is "A Concise Introduction to Pure Mathematics" by Martin Liebeck. It is extremely accessible to high school graduates, yet it introduces more rigour than is often given in first year university, but in a fun and elegant manor.
A: An outstanding book it Aigner and Ziegler's "Proofs from THE BOOK" (3rd edition, Springer, 2003). If it has it's own Wikipedia page...
A: I enjoyed The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan.  It was given to me as gift after completing a teaching internship.
A: I am of the opinion that a text like the ones mentioned is a perfect graduation gift for a recent high school graduate interested in mathematics and college, especially considering the fact that you do not know him quite well and most of the people that know him in this way at the party will be giving him money or trite gifts.  This type of gift provides a more personal touch and shows that you have tried to find out about what he would enjoy as a gift and what he is interested in. In terms of the text that you should choose, Calculus by Michael Spivak is a great way to introduce a new mathematics student to higher mathematics.  This was my first math book going into college and I wouldn't have traded it for the world.  Real reality check as well compared with all high school texts - this book completely eliminated my giant ego coming out of high school (not just in math, gave me some perspective about university learning as a whole) and motivated me to really study the content of mathematics. Also, as a side note, this student sounds very familiar to me...a little bit of a throwback.
A: To answer your first question: If the text is really usefull it always is an appropiate present be it math or any other major. Good professional literature is often expensive, especially for a student. Getting such a book as a gift might really help. This is under the prerequisite that you are a hundred percent sure that it is needed.
Anyway I can't shake this feeling that it's a little like giving clothes to a child for christmas ;-)
To consider something less barren and more entertaining (and a lot less usefull for that matter) I recommend The Simpsons and Their Mathematical Secrets by Simon Singh.
A: If the graduate has some familiarity with computers, I would suggest a CAS such as Maple or Mathematica.
If that is more than you want to spend, I suggest these two books:
1) Mathematics by Experiment
2) Experimentation in Mathematics
A: I think a math text is an appropriate.
I'd recommend 


*

*The Princeton Companion to Mathematics: Great for just browsing, it offers a motivated, understandable introduction to every major subject in mathematics, as well as lots of commentary by mathematicians on how to "do math." I very much wished I had gotten this book at the beginning of my undergraduate career, as it would have saved me from trudging through a lot of math courses without knowing why they were useful or seeing the big picture.

*Problems from The Book: Lots of beautiful problems (patterned off Proofs from The Book). Note this may appeal more to someone who has had experience with math olympiads.

A: Well, this will be very different, and may reveal a hobby of mine, but take it for what it is- a suggestion.
Instead of a math book, get him a fountain pen.  (I'd suggest this one, for a gift, or this if you want a bit more for a bit cheaper.) Doing math in pen is a good habit. Going to college for mathematics means writing notes constantly.  Clear handwriting counts more than any other major, and good tools make the job easier and more fun.   He will use this thing all the time.  Rather than giving him math to learn, you'll be giving him an interesting thing to learn it with.  I wish I'd have discovered nice stationary and writing tools earlier in college.
A: I would think very carefully about buying someone a maths book as a gift. I work in maths but would despise receiving a textbook or general math book as a present. If I had just done exams and am graduating one of the last thing I would want is such a book. If the student is interested in pursuing maths I'm sure he'd look up the right material of his own volition or would ask someone.
A: My (somewhat biased) $0.02 goes to An Intro. to Mathematical Cryptography by Hoffstein, Pipher and Silverman. Because it introduces all the necessary tools as they are used, it's a nice intro. to concepts of "higher mathematics" with some added motivation. It is also reasonably priced.
A: My favorite general-read math book (and one that I got my dad for his birthday): The Joy of $x$ by Stephen Strogatz.
http://www.amazon.com/The-Joy-Guided-Tour-Infinity/dp/0547517653
A: I would like to recommend: 1. P R Halmos's written "Naive Set Theory" 2. C E Weatherburn's Elementary Vector Analysis, together with its Solution book by R K Sinha. These are too cheap books of utmost value and interest.
A: As a high school student in the same situation, I would recommend Excursions in Calculus by Robert M. Young. It contains almost all the fun of Spivak with none of the head bashing boringness (just my opinion). 
A: He might be a tad too old for it, but Derbyshire's Prime Obsession really kindled my interest in mathematics as a pursuit in itself, i.e. outside of its use in physics and related fields.
