I was presented with this problem at work. Say you have $24$ people and $8$ tables in a room. You want to set people at these tables in groups of three such that during each new round (where people switch tables and partners) no one meets someone twice and by the end everyone has met everyone.
At first this problem seemed very simple because it's possible to find many different groupings where no one meets twice, however putting the groupings of three into groupings of $8$ ($8$ groups of $3$) so everyone is active once per round is much harder.
Is this problem possible to solve? I think it might not be.