Gregg,

I wrote a program that looks at all of the possible schedules and comes up with one that's "most balanced". Here it is what it found. The positions are the players and the numbers are the group they are in.

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

You'll notce that, for example, players 1 and 6 never play together. There isn't a schedule where every pair of players plays together at least once, otherwise the program would have found it. It looks like the best that can be done is that some pairs are together twice, and some not at all. That's too bad. It seems the arrangement you want just isn't possible.

--Victoria