One good option is "The Magic of Numbers" by Gross and Harris (not to be confused with a book of the same title by ET Bell), which was written for the eponymous class Gross used to teach at Harvard. The problems include some stuff on, say, Catalan numbers, and some reasonably serious modular arithmetic (e.g. RSA encryption) with a minimum of baggage, which should recommend the book to non-mathematicians.
The Art of Problem Solving series (here) is also quite good. I learned a lot from some of those books when I was in high school--they have lots of exercises, ranging from very easy to problems I, at least, found quite difficult. And there is a lot of discussion of technique, which I think non-mathematicians often find lacking in other textbooks.
And Martin Gardner's entire oevre is great, and I think that recommendation probably doesn't require any explanation.