Some of them will magically disappear every hour until only one remains, but imagine the firepower!
They wouldn't disappear. They would, after a period, go back in time in order to become one of the other people in the battle.
Using a time turner to make clones in battle is a very, very dangerous idea. Harry has been warned, strenuously, by Professor McGonnagal that he should not directly interact with himself, and we have an anecdote about an auror/criminal pair that went insane because they abused time turners. I imagine that one of the more stable time loops would involve the original Dumbledore/Harry getting disabled before going back in time for the first time.
But yeah, the cloning objects thing is a reasonable use of a Time Turner.
Edit to add: If by collaborating on tricky problems, you are referring to e.g. academic problems, rather than problems of strength, this amounts to a rather absurd charade. If you use a Time Turner to put 6 copies of yourself in a room, and in an hour they succeed in answering the problem, that means that at the beginning of the hour, 5 of them already knew the answer.
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EDIT: New discussion thread here.
This is a new thread to discuss Eliezer Yudkowsky's Harry Potter and the Methods of Rationality and anything related to it. With two chapters recently the previous thread has very quickly reached 500 comments. The latest chapter as of 17th March 2012 is Ch. 79.
There is now a site dedicated to the story at hpmor.com, which is now the place to go to find the authors notes and all sorts of other goodies. AdeleneDawner has kept an archive of Author's Notes. (This goes up to the notes for chapter 76, and is now not updating. The authors notes from chapter 77 onwards are on hpmor.com.)
The first 5 discussion threads are on the main page under the harry_potter tag. Threads 6 and on (including this one) are in the discussion section using its separate tag system. Also: one, two, three, four, five, six, seven, eight, nine, ten.
As a reminder, it's often useful to start your comment by indicating which chapter you are commenting on.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: