Rich,
Mathematically speaking, 16 is a much nicer number of players to work with than 20. In fact I believe that a six week schedule of foursomes where players play together at most once is impossible, so it's no wonder you are finding the problem hard. A five week schedule with the same property is possible, for example use the first five weeks in the table below.
Week 1: (A B C D) (E F G H) (I J K L) (M N O P) (Q R S T)
Week 2: (A F K P) (E J O T) (I N S D) (M R C H) (Q B G L)
Week 3: (A J S H) (E N C L) (I R G P) (M B K T) (Q F O D)
Week 4: (A N G T) (E R K D) (I B O H) (M F S L) (Q J C P)
Week 5: (A R O L) (E B S P) (I F C T) (M J G D) (Q N K H)
Week 6: (A E I M Q) (B F J N R) (C G K O S) (D H L P T)
I realise that many golf courses would not allow it, but if it were possible to play 4 fivesomes in week 6 as shown above, then everyone would have played with everyone else exactly once. If fivesomes are out of the question then let us know for how many weeks you will be playing? For foursomes the perfect number would be 19 weeks, as then it can be arranged that all players play with each other exactly 3 times.
Ian.
Rich,
From now to the end of September would be around 26 rounds of play, I have software that would generate a schedule for this, but my worry is that while the whole would be balanced, it would no longer look balanced if you split it into 4 parts of 6 or 7 weeks. For example it is quite likely that in the first 7 weeks a few pairs would play together 3 or 4 times while a few other pairs not at all; I think your colleagues might notice this. So if it's the social mix that is important, then you may well be better off making adjustments to the 5 week schedule above.
There is an important property of this 5 week schedule that I did not mention earlier. Within each week above, rotate the order of the foursomes so that they look like this:
Tee-off Order
first second third fourth fifth
Week 1: (A B C D) (E F G H) (I J K L) (M N O P) (Q R S T)
Week 2: (E J O T) (I N S D) (M R C H) (Q B G L) (A F K P)
Week 3: (I R G P) (M B K T) (Q F O D) (A J S H) (E N C L)
Week 4: (M F S L) (Q J C P) (A N G T) (E R K D) (I B O H)
Week 5: (Q N K H) (A R O L) (E B S P) (I F C T) (M J G D)
Now players get to tee-off once each at the five tee-off times.
Ian.