Sheri,

Ideally each player misses three weeks. And that’s possible. List the players 1 through 9 in random order, then list them again as 10 through 18 (in a possibly different order — say put the names in a hat and draw them without replacement until all 9 names are listed). And then do it again to list the players in positions 19 through 27. The first 3 players listed miss game 1, the next three miss game 2, and so on.

—Victoria