van der Waerden's discussion of his (and two colleague's) proof of Baudet's Conjecture describes how the proof was found after a sequence of about half a dozen episodes of trying an approach, seeing that it failed, analysing the failure, and using this analysis to discover a fresh approach that patches the failures in the earlier one. Typical patches are: try a more complex induction rule, generalise the original conjecture, invent and prove an intermediate lemma. If the final proof is presented without the story behind it's invention, it appears to be magic, but actually it's the result of a prolonged trial and error.
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