John Stallings's article "How Not To Prove The Poincare Conjecture", available here http://math.berkeley.edu/~stall/ 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.