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Let's say, I've posted a solution to a puzzle and it's been accepted. Later, I think of an alternate solution, distinctly different than the first but (hopefully) meriting mention.

Is it better to make an addendum to the original, accepted solution or post a new answer which can live or die on its own?

The stipulation that the first answer has been accepted isn't necessary but it might influence whether or not the original should be left alone.

Of course, a third possibility is to just ignore the alternate solution and hope that someone else posts it.

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If the solutions are distinctly different, I see no problem with posting a second answer. Sometimes there are just more ways to do something.

Also, this gives users the option to vote on their preferred solution; with one answer containing both solutions, they can't state their preference by voting alone.

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It is generally better to write a single answer. In addition to explaining the two solutions, discuss when you would prefer one to the other. This has added value compared to providing separate answers without a way to select between the two solutions.

The fact that you originally posted a single solution is neither here nor there. You can always edit your answer to improve it.

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    $\begingroup$ You can indeed always edit your answer to improve it, but if you're going to be posting a completely different solution I think it's best to use another answer. One might be more popular than the other, one might be wrong and the other right... It's separate content in that case, IMO, and should be posted separately. $\endgroup$ – WendiKidd May 24 '14 at 2:43
  • $\begingroup$ @WendiKidd I don't buy it. I've yet to see a case where the resulting content was better. Separate voting is irrelevant — an answer with two solutions includes how to select between them, and the voting takes that into account. $\endgroup$ – Gilles 'SO- stop being evil' May 24 '14 at 3:31

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