I work in an office of about 600 people, 8 of us have made it into the office. So I've decided I'm going to spend my day trying an attempt this problem. I can't do it mathematically. So I'm using Excel.
I've created a work book which has each play date along the top and each child along the side. There are then 3 validation tests to ensure they meet your criteria.
I've created 5 groups called a,b,c,d,e so each child needs to appear in each group 4 times. And 4 groups within each play date.
There's another validation where you add a childs name and it counts the number of times those children appear in the same group at the same time.
I'm just populating the schedule with different patterns, I've tried a few so far and none work. I will continue to try but I thought you might like to try my method.