When guessing is preferred over analysis

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Nontrivial mathematical equations can sometimes be more easily approached through educated guessing rather than rigorous analysis. The discussion highlights the role of iterative methods that begin with a guess, questioning whether these methods qualify as guessing. The importance of verifying solutions after making educated guesses is emphasized. The conversation also touches on the concept of what constitutes a "good enough" solution in mathematical contexts. Overall, educated guessing can be a valid strategy in problem-solving, provided it is followed by verification.
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Are there nontrivial mathematical equations more easily solved by guessing than by analysis?

In part, P/Q for polynomials comes to mind.
 
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What counts as a (good enough) solution? Do iterative methods initialized with a guess count as guessing? Do you not have numerical analysis in mind?
 
An educated guess is always a good way of solving a problem. As long, of course, as you check the answer and verifty it.
 

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