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I've been using the Art of Problem Solving series as supplemental readings throughout my A-levels(which is the equivalent of high school), and found them to be extremely helpful and informative, both in terms of the difficulty of their exercises(which is just right for me) and their styles/expositions.

Now I'm at the 2nd year of my degree and I do miss those AoPS books a lot, I wonder is there any undergraduate equivalent to them that presents materials with a series of problems and leads readers to prove the theorems themselves step by step?

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  • $\begingroup$ You might also look at books that are being produced by the discovery learning or inquiry based learning movement. For example, Michael Starbird has written books such as Heart of Mathematics:An Invitation to Effective Thinking (with Edward Burger) and also Distilling Ideas: An Introduction to Mathematical Thinking (with Brian Katz). Another example is the book Number Theory through Inquiry by Marshall. $\endgroup$ – littleO Nov 24 '15 at 21:26
  • $\begingroup$ This is not a math book, but if you're interested in learning how computers work through guided problem solving, I highly recommend the book The Elements of Computing Systems by Nisan and Schocken. It's my favorite textbook. $\endgroup$ – littleO Nov 24 '15 at 21:27
  • $\begingroup$ The 2 books you mentioned earlier seems really fascinating for me. In fact, I haven't been able to see many books like this for quite some time. Thank you for your recommendations. $\endgroup$ – wanwuwi Nov 24 '15 at 22:47
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You could check out the Linear Algebra Problem Book by Halmos. From the description:

Can one learn linear algebra solely by solving problems? Paul Halmos thinks so, and you will too once you read this book. The Linear Algebra Problem Book is an ideal text for a course in linear algebra. It takes the student step by step from the basic axioms of a field through the notion of vector spaces, on to advanced concepts such as inner product spaces and normality. All of this occurs by way of a series of 164 problems, each with hints and, at the back of the book, full solutions. This book is a marvelous example of how to teach and learn mathematics by 'doing' mathematics.

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