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As university is a bit slow, we organize a group activity of doing math exercises (after learning the basics of the subject separately). Thus I am looking for a book with many exercises that are exciting and challenging enough to do in a group.

As an example, last year we did a section on topology from "Real mathematical analysis" by Charles C. Pugh. It had 145 problems ranging from basic to ones he can't solve himself.

The level is mostly undergraduate. It would be best if the books didn't assume much prior knowledge and started from the basics as we have people of very different backgrounds. The books are expected to reach something interesting though.

We are not picky about the subject. Anything could work, if the right book is found. We are slightly biased towards complex calculus, although I fear most of the exercises in that are mechanical.

In short, share books with fun exercises.

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2 Answers

up vote 3 down vote accepted

I like the freely available Combinatorics Through Guided Discovery.

Another suggestion that may or may not be along the lines of what you are looking for are the four wonderful volumes of Winning Ways for Your Mathematical Plays.

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Problem solving through problems spans many disciplines without assuming much previous knowledge of any of them. The problems, for me, were just the right level of difficulty - challenging, but not impossible. I've heard people recommend using it to study for the Putnam. Overall, seems about like what you're looking for.

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This is definitely an option. Although I am looking for something more coherent... –  Karolis Juodelė Sep 22 '12 at 20:15
What do you mean? I thought the exposition was fine. Are you saying you want something more advanced? –  Alexander Gruber Sep 25 '12 at 21:59
It's not so much a problem of difficulty. I would prefer a book that focuses on one branch of mathematics as it seems to be the only way to reach interesting results, be it Abel theorem or at least properties of Cantor set. –  Karolis Juodelė Sep 26 '12 at 7:48
Well, I don't have anything good for calculus, but if you guys like algebra I really enjoyed all the exercises in Isaacs Finite Group Theory. –  Alexander Gruber Oct 2 '12 at 4:49
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