John,

Here is a schedule, though it might not be what you want. You didn’t mention the group sizes, so I assumed 4,4,3,3. You also didn’t mention balancing the number of men and women in each group. That actually makes the problem a lot harder.

The schedule below was found by computer. The 14 positions are the players, and the number in that position is the group number they play in. Groups 0 and 1 are foursome; groups 2 and 3 are threesomes.

Day 1 : (0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3)
Day 1 : (0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0, 1, 3)
Day 3 : (0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 1, 2, 0)

My suggestion is to number the players so that 1, 3, 5, 7, … are women, and 2, 4, 6, 8, … are men. And it is probably wiser to ignore the group numbers and let the threesomes go out first. A weakness in this schedule is that player 1 is always in a foursome. A strength is that it is balanced as possible, meaning that the number of different pairs that get to play together is as large as possible.

—Victoria