There is an introductory session for $n$ people. And to allow some conversation, we split the group into $k$ groups of similar sizes $s$. This is repeated until each person shared a group with everybody else. However, it can be boring to talk to someone twice.
How to split the groups such that each person shares a group with every other participant exactly once while maintaining a group size of roughly $s$?
I'm most interested in $s = 3,4,5$ as they are socially most beneficial. I'm also interested in a computational ways to solve for $10 < n < 100$.