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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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5 Answers 5

up vote 4 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

Fun and Fundamentals of Mathematics written by Jayant V Narlikar, Mangala Narlikar. you can find more details here in the following URL : It explores mathematics from basic level and gradually takes us to a higher level of problem solving

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As you already wrote in a comment to Alexander Gruber's suggestion, you are looking for something rather coherent. Nonetheless I cannot stop myself from writing down's Bollobas "The Art of Mathematics: Coffee Time in Memphis".

Coherency is not really its cup of tea (maybe, better... coffee), but it is a lot of fun, and – in the words of the author – it is a collection of problems that most probably would have catch a glimpse of interest from Erdos or Littlewood, which does not sound bad at all.

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I found many enjoyable and appealing problems in Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries.

If you're looking for some problems with more down-to-earth and academic mathematical content, I recommend Putnam and Beyond, as well as Problems from the Book, which both have some very challenging problems.

If you want to learn new problem solving techniques and hone them with problems, there is Problem solving strategies. You'll find lot of witty exercises in there (although the content covered does reach far beyond what a high-schooler should know).

If you read French, you may want to take a look at Exercices de mathématiques. Oraux X-ENS, by Serge Francinou, Hervé Gianella, Serge Nicolas (whole lot of breathtaking problems ranging from first-year to third-year undergraduate content).

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