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How does one decide if a question is good for MathOverflow or Math.StackExchange?

I've been using the criterion of being "research-level" but that can be subjective. An expert may consider a question elementary but that is the only subject this person considers. All others may consider the same question quite advanced.

If a person gets it wrong, what options do they have to move between the two math sites?

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Moderators can migrate questions.

I don't think there will ever be a hard or fast rule; it's sort of an odd twist of fate that both sites exist. I think you just have to judge for yourself what is research-level and what isn't. If your judgement doesn't match that of the community as a whole, they'll let you know, and you should recalibrate.

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