Ken,

I was able to search all possibilities using a computer. The most balanced schedule it found is below. In this schedule, player 1 is always in a threesome, so that “player” represents the two who are sharing the cart.

The positions in each sequence are the player numbers, and the entries are the groups (0, 1, 2). The groups on Day 0 are 1, 2, 3; 4, 5, 6, 7; 8, 9, 10, 11. The numbering of the groups is immaterial. Player 1’s group does not have to always go off first.

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

Have fun!
—Victoria