Dean,

Here is one that works for 4 days. Every team sits out once. Every two teams are “together” exactly once in the sense of being on the ice at the same time, or having a bye at the same time. That means pairs of teams that are not on the ice together are exactly the pairs who have the byes. (As an aside, this is golf schedule for 9 players and 4 rounds, with one of the players deleted.)

The way to read the schedule is that the 8 positions in each sequence are the teams, and the numbers indicate the time slots. The zeros indicate the teams with a bye. You'll notice that team 1 is always in slot 1 and team 2 is always in slot 2. Something like this is bound to happen, but to assure fairness you night randomly assign the time slots once the groupings are known.

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

You can use the schedule over and over, just renumber the teams each time so that no two teams who had the same bye before have a bye together again.

—Victoria