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Let us suppose that I have $n$ students in my class, and I break them up into $k$ groups per week. Let's also suppose that I want to repeat this each week, except that I don't want any student to work with any other student more than once.

For how many weeks can this continue?

This is an extension of some of the classic tournament-scheduling questions, and it wouldn't surprise me if there were an equally elementary approach. But I haven't found it yet. And since I haven't found it yet, I haven't come across the divisibility relationships that would facilitate a 'nice' answer, other than $k \mid n$.

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This isn't a hint since I don't know the answer but perhaps you might find it useful. For a given person, they will meet $\frac{n}{k}-1$ new people each week. So if $m$ is the number of weeks this continues, then certainly we must have $m(\frac{n}{k}-1) < n$. Could it ever be this large? – nullUser Sep 23 '12 at 3:51
Is it assumed that the groups must be equal-sized? – Erick Wong Sep 23 '12 at 4:27

This is the Social Golfer Problem; no general solution is known. Some data can be found here, and there is a Wolfram Social Golfer Problem Demonstration where you can find a useful links and information. Kirkman’s Schoolgirl Problem is a special case that is quite well known.

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