Leo,

I think it isn’t possible, but can’t prove that conclusively. I’ve used software to examine about 2 billion possible schedules and didn’t find one. Using this search method there are about 10 times that number of schedules to consider, so the computation would take about 15 days to complete.

The best schedule that was found is below. It does not do what you want. It also has the flaw that player 1 is always in a foursome. The way to read it is that the 11 positions in each sequence are the 11 players, and the numbers tell you which group those players are in.

Day 0 : (0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2)
Day 1 : (0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2)
Day 2 : (0, 1, 0, 2, 0, 1, 2, 1, 2, 1, 0)
Day 3 : (0, 1, 1, 2, 2, 0, 1, 1, 0, 2, 0)
Day 4 : (0, 1, 2, 0, 1, 1, 0, 2, 2, 0, 1)

As an alternative, you might decide to use the schedule for 12 players below, but delete player 1. The schedule you’d obtain that way does not have the issue that one player is always in the foursome. It may not be as balanced (overall) as the 11 player schedule above though.

Day 0 : (0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2)
Day 1 : (0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 2)
Day 2 : (0, 1, 0, 2, 0, 1, 2, 1, 2, 1, 0, 2)
Day 3 : (0, 1, 1, 2, 1, 2, 0, 0, 2, 0, 2, 1)
Day 4 : (0, 0, 1, 2, 2, 0, 1, 1, 1, 2, 2, 0)

—Victoria