I have been introduced to this problem recently which to me resembles of round robin, but with an additional rule which might make no solution plausible.
The idea is to find combinations of groups of people, where each person interacts with at least each other person by the end of the rounds, and each person only has to "pay" once.
So consider a group of 9 people that are trying to organize a pizza paying system. Is it possible to divide the 9 people into subgroups of 3 people, where 1 member of each group has volunteered to pay for the pizza, and never have to pay for pizza for any subsequent rounds, in addition to eating with each other person?
Is such a solution possible?
Consider for example the group
1, 2, 3, 4
For the first round we will have
paid unpaid 1 , 2 3 , 4
But then for the second round, because 1 and 3 have already paid, we need 2 and 4 to be both paying and be in a unique group
So the only solution is
2 , 3 4 , 1
But then we cannot have a situation like
1 , 3 2 , 4
so clearly not every person has interacted with each other person. Therefore a solution does not exist.
Are there any circumstances which a solution exists? If so would it have to be a square number or a special number? I was beginning to try to code a solution for this and I quickly realized that it is far more comlpicated than I initially gave credit for.