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I refer to this question.

Basically it asks for a proof of a known result, without using a very general & basic formula. The question has received a decent amount of views and there seemed to be a common approval from all comments that the answer should probably be "No".

The question is only a week old, so someone might still find a positive answer, but this is starting to feel quite unlikely.

How should such a question be dealt with ? Should it stay open as long as no positive answer is found? or should someone just basically answer "Very unlikely - not possible as far as I (or MSE community) knows"

Edit - 06/11 This post has now an accepted answer. The piece of mathematics used is very interesting and well done. I'm still unsure if this answers perfectly the question, because (as mentioned in the answer) it is very close to reproving Euler's formula. But that's only my opinion. The general question on how we should treat these cases is still interesting I think.

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    $\begingroup$ For this specific question: they want to use the planar property, but not the Euler formula. It seems to be an interesting question to me, at least I would wonder if the validation of Euler formula is equivalent to being planar. It don't think this question is really unanswerable (of course I might be wrong). $\endgroup$ – Arctic Char Oct 28 at 19:08
  • $\begingroup$ What is your opinion of the cases where a proof is possible without the classic and obvious method, but the alternative proof is so complicated that it validates the classic "boring" solution? $\endgroup$ – Robert Soupe Nov 4 at 16:28
  • $\begingroup$ @RobertSoupe Are you referring to the specific case of David E Speyer comment? I really don't know. He's using the circle packing theorem, which is kind of a much more difficult generalisation of the Euler's formula (the classic solution)... I'm really open for opinion here. I think in the spirit of the question, an acceptable proof should not directly imply the simple formula. The question is trying to see if the result is in some sense dependent or not from Euler's formula. $\endgroup$ – Thomas Lesgourgues Nov 4 at 16:48
  • $\begingroup$ I was thinking more of elementary number theory questions that are like "prove it without induction" or "prove it, but not by contradiction." $\endgroup$ – Robert Soupe Nov 4 at 19:51
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    $\begingroup$ I think there are interesting exercises, and I'm always in favour of new proofs of existing theorem. They help understand new techniques and sometimes the initial proof. I remember some years ago looking at dozen of proofs for the quadratic reciprocity, some were very interesting. More recently on the Carne bound, I knew a probabilistic proof, but discovered the "standard" proof involving Chebishev Polynomial, pure magic :) I guess, that kinda answer my question, leave it open, who knows someone someday might find something interesting. $\endgroup$ – Thomas Lesgourgues Nov 5 at 7:16

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