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Please see examples of such methods.

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I find the first question quite odd. Among multiple ways to do something one is always easier.

As to the second question, the answer as ambiguous, depending on the definition of "simple" and the intended audience. If it is to say "obvious and easy to apply", then yes, unless specifically (most of the times for pedagogical reasons) it's better to use the simplier way. If it's "makes the proof the shortest", then the answer really depends on the audience. As to this difference, enjoy the diversity of mathematics and try to adopt the train of thought, maybe in some obscure case the "simple" proof doesn't work, but a "difficult" one does.

Finally, about the discovery, that's, essentially, creativity, intuition and one of the hardest parts of maths. As far as I remember, there were outstanding discoveries in finding another proofs for already solved problems.

TZakrevskiy
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  • Thank you for your answer. For the first question, I was just wondering why there may be an easiest way among multiple ways. Is this true in general? When? –  Aug 11 '13 at 10:06
  • There are situations where there are seemingly unlimited other ways of doing the task. I think that to get an exhaustive answer to the original question, one would have to duplicate most of mathematics, all the way back to the ancients. Just think of all the ways of proving the Pythagorean Theorem, for instance. – Lubin May 14 '18 at 01:08