The solution to Putnam 1985/B-4 in another post on this blog illustrates something about real problem solving rather answers in exams/competitions. That is that for a real life problem a lot of exploration/experimentation is undertaken before a formal solution/proof is developed. In fact for the problem in that other blog post there was significantly more exploration involved in developing the ideas for a solution than is shown there. Mainly in the form of sketches of particular cases.
When results are published usually none of the exploratory work is mentioned. The solution/proof leaps fully formed like Athena from the head of Zeus. Probably the best known practitioner of extensive exploratory work hidden from the final publish result is Gauss.
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