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(this is basically the same question, only in math, as Alessandro Cossentino asked about theoretical computer science at TCS.SE:; if a question similar to this one already exists at MO, feel free to close this)

Recently Ryan Williams published (see a more "pedagogical" version of his proof in complexity theory concerning NEXP and ACC - in his own words, "the proof will be described from the perspective of someone trying to discover it". The paper discusses more intuition, failed attempts at solving the problem etc. much more extensively than a typical journal paper.

Personally I find such efforts extremely valuable, because they give you an opportunity to learn how somebody thinks about mathematics and not only verify the formal correctness of some abstract reasoning.

What other examples of such approach are you aware of?

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This sounds like what I call a "motivated proof". – zeb Nov 9 '11 at 4:38
I think it's common to make these big-list soft-questions community wiki (so nobody gains/loses reputation for voting answers up or down), and ask for one paper per answer (so that votes up or down reflect exactly the content, and not some proper subset). – Zack Wolske Nov 9 '11 at 6:41
Timothy Gowers's papers are a very good example. You should also take a look at his blog; for example… – Andrés E. Caicedo Nov 9 '11 at 7:23

John Stallings's article "How Not To Prove The Poincare Conjecture", available here walks us through 4 conjectures - with implications between each - which would prove the famous question, if they were actually true. In the end the ideas don't work, but the reader does get a nice glimpse of standard tactics for classifying manifolds, and how one might leap between algebraic and geometric arguments to make the pieces fall into place.

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I like the excuse at the beginning of the article: "I have committed the sin of falsely proving Poincaré's Conjecture. But that was in another country..." – Georges Elencwajg Nov 9 '11 at 10:45
Thanks to more recent developments by Perelman et al, the rest of that quote from Marlowe is now also apropos. Perhaps inappropriate in other ways, though, given its sexist and anti-Semitic context. – Nate Eldredge Nov 9 '11 at 15:28

Timothy Chow's article on forcing (called A Beginner's Guide to Forcing) is one of the best of this general type.

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Polya's "Mathematics and Plausible Reasoning" is a good source. In chapter XVI he compares how many textbooks present proofs (which he calls a "deus ex machina" approach) and how it could be done showing the discovery process. The example he uses to illustrate is: If the terms of the sequence a1, a2, a3... are non-negative real numbers not all equal 0, then sum(1, infinity) (a_1, a_2, ...a_n)^1/2 < e sum(1, infinity)a_n.

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Definitely not at the level of an article, but following this idea of explaining the reader what comes to his mind, the book by T. Tao, "Solving mathematical problems, a personal perspective", is a masterpiece in my opinion; even tough it is aimed at solving IMO's level (or a bit lower) problems, it gives good tricks to transpose to higher level problem (where the difference is essentially only knowledge and time to master the theory).

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