Harold,

Here is a schedule for 2 teams per week for 10 weeks. Just separate the teams and you have 1 team per week for 20 weeks.

I was able to check all of the possible schedules by computer — there are well over 200 million of them — and this one is the most balanced in terms of how many times each two players are together.

The positions in each list are the players, and the numbers designate the foursomes. Day 0 gives you the foursomes 1, 2, 3, 4 and 5, 6, 7, 8. Day 2 gives 1, 2, 5, 6 and 3, 4, 7, 8. And so on. In all there are 20 foursomes, with each golfer in 10 of them and the pairs as balanced as possible. If you take the foursomes in order, that is Day 0 gives the first two Men’s nights, Day 1 gives the next 2, etc, then no player misses more than 2 nights in a row. It is not possible to do better than that.

Day 0 : (0, 0, 0, 0, 1, 1, 1, 1)
Day 1 : (0, 0, 1, 1, 0, 0, 1, 1)
Day 2 : (0, 1, 0, 1, 0, 1, 0, 1)
Day 3 : (0, 1, 1, 0, 1, 0, 0, 1)
Day 4 : (0, 0, 1, 1, 1, 1, 0, 0)
Day 5 : (0, 1, 0, 1, 1, 0, 1, 0)
Day 6 : (0, 1, 1, 0, 0, 1, 1, 0)
Day 7 : (0, 0, 0, 0, 1, 1, 1, 1)
Day 8 : (0, 0, 1, 1, 0, 0, 1, 1)
Day 9 : (0, 1, 0, 1, 0, 1, 0, 1)

Hope this works for you.
—Victoria