We need to find out how many days are needed for every player to partner and oppose every other player. Take partnership first. Take one player leaving 7. We have a choice of 8 available players to start with, then 7, so there are 8*7=56 ways of picking two players out of each, but since the order is irrelevant, and there are two ways of ordering two players, there are half as many combinations, that is, 28. Four players (2 against 2) constitute a game so we would need 14 games. At two games a day, 7 days are required and we're only given 6. Therefore two games cannot be played, or we need an extension to 7 days, or 3 games need to be played on two days out of 6.
To make nomenclature easy, let's use letters A to H represent the players. The list below shows all partnerships.
AB AC AD AE AF AG AH
BC BD BE BF BG BH
CD CE CF CG CH
DE DF DG DH
EF EG EH
FG FH
GH
In the next step we need to pair partners. So we list the first 14 partners into team set 1, leaving 14 partners remaining as team set 2. Then, against each of the partners in team set 1 we attempt to continue with team set 2 partners, but where one of the team set 2 partners is the same as a team set 1 partner we skip to the next and fill in the "hole" at the earliest opportunity. In this way we should end up with 14 pairs of partners and a set of 14 tentative gamesters; for example:
ABCE ACDH ADCF AECG AFCH AGDE AHDF BCDG BDGH BEFG BFEG BGEH BHEF CDFH
Now we need to investigate pairs in opposition and swap wherever necessary to cover all opponents.
For example, we have AB as partners (ABCE), but not as opponents. So amongst the remaining partnerships beginning with A we need to find a pair of opponents beginning with B (BCDG ... BHEF). If we pick ACDH and swap out DH for a pair of B opponents, then we can choose from BDGH ... BHEF, i.e., GH, FG, EG, EH, EF. Let's pick BEFG. So ACDH becomes ACBE and BEFG becomes DHFG. We couldn't pick BCDG because replacing BC with DH would give us DHDG, and players cannot partner or oppose themselves.
It's not necessary to swap out opponents as in this example because there may already be situations where the 4-player arrangements include two players as partners and opponents. So it's a good idea to keep checking and identify which players still need opponents using a checklist. If we perform the swap-out in the example and checkout all oppositions, we discover that there are only two oppositions lacking: D against E and E against F. We now look for DE and EF as partnerships and find AGDE and BHEF. If we remove these two from the list of 14 4-player combinations we're left with sufficient games to fill the 6 days. However, we also lose B against F because there are no other games in which B and F are opponents.
The final list becomes:
Day 1: ABCE ACBE Day 2: ADCF AECG Day 3: AFCH AHDF
Day 4: BCDG BDGH Day 5: DHFG BFEG Day 6: BGEH CDFH
This arrangement removes DE and EF as partners and opponents and BF as opponents. All other players in some game play as partners or opponents.
