I enjoy giving interesting problems to my peers who have not seen a lot of mathematics. It is a constant conversation I have with many friends who would interchange "mathematical thinking" with "mechanical thinking" in conversation. Consequently, I often make it my mission to elucidate why mathematics is a beautiful subject, and one that essentially deals with the ideas, and an unreserved asking of questions. I'm of the opinion that the symbolic representation of a concept is over-emphasized in the earlier years of math and pushes people away from a subject.
A few days ago, I was speaking with a good friend of mine who graduated with a degree in philosophy, but had taken courses through "vector calculus" in mathematics. I spent some time trying to explain why his vision of mathematics was incorrect-- but a problem speaks a thousand words. I gave him a problem I saw in Paul Lockhart's "A Mathematician's Lament"
Given two points who rest above a horizontal line, find the shortest distance between them so that the path touches the line only once. It can be found here on SE: An elementary (?) minimization problem
He spent a solid hour on it, intermittently asking questions throughout (kind of a game of "20 questions") and we eventually came to the correct answer.
I want him to perhaps see some more mathematics, but in a more "abstract" or "Advanced setting." He is out of college, and so, has extra time to enjoy mathematics, and I think he has a real knack for it.
In response to the close vote, I will rephrase my main question, I agree it was perhaps too broad.
My main question: I'm requesting a book suitable for someone who is intelligent, and also interested in more abstract/pure mathematics. His interests are rested in solving interesting questions, but also in more general mathematical structure. I was tempted to give something like Paul Zeitz's Art and Craft of Problem Solving but elected against it, since it didn't give any insight regarding the coherence of mathematical concepts.
Thus, I'm asking for a reference that you may have found inspiring, but also elucidated a novel mathematical concept in a natural way. Recommendations for a particular subject in math is also appreciated, I'm not sure where to start him.
I ended up using some of the recommendations:
Mathematician's Delight and Euler's Gem seemed like good starting points.
I really considered Visual Complex Analysis, since my acquaintance had some familiarity with vector calculus, and I thought it may have been an interesting place to start.
I also supplied the introduction (and 1st half of chapter 1) of Mathematics Made Difficult, because I find it to be very funny, but also I thought the Platonic Dialogue in the first chapter is quite captivating. I may also send over some of his discussion of Topology and the "open men," for similar reasons.