Suppose 20 people attend an event where there is 4 different activities to do. Suppose we want to split the group in subgroups, each subgroup attending one session of an activity, then moving on the other activities...
Is it possible to arrange things so that at the end of the day everybody has attended every activity exactly once and so that everybody has meet all 19 other people exactly once (by participating to an activity with them) ?
i.e. if A and B attend the first activity together, they wont meet again later that day.
For what size of crowd is this possible (if my random choice of considering 20 people doesnt give a solution) ?