Fixtures for doubles pétanque league

Question

I have a problem which I'm hoping for some help with. I'm setting up two Pétanque leagues. One league has 9 players and the other has 8.

The format is doubles, so each game will be 2 players vs 2 players. I want each player to play on the same team as each other player once, and each player to play against each other player twice.

Can someone work out whether this is possible and, if it is possible, give me a list of fixtures for each league that fulfills the criteria?

Thanks

Jack

Answer 1

These are called Whist tournaments.
In Baker, R. D. (1975). Whist tournaments. Congr. Numer, 14, 89-100, it is shown that a Whist tournament of size $n$ exists for all $n\equiv 0\pmod 4$.
In Anderson, I. (1990). Combinatorial designs: construction methods (Vol. 82). Chichester: Ellis Horwood., it is shown that a Whist tournament of size $n$ exists for all $n\equiv 1\pmod 4$.

Hence, for both your values of $n=8,9$ a tournament is possible.

According to this article on ScienceDirect, there is essentially a single solution for $n=8$, and two for $n=9$.

---

The unique schedule for $n=8$ can be found below.

      i       ii    
1  H+G:F+C  B+A:D+E 
2  G+D:H+E  C+B:F+A 
3  H+C:E+B  D+A:G+F 
4  E+G:B+F  A+C:D+H 
5  A+E:C+G  F+H:B+D 
6  C+D:G+B  E+F:H+A 
7  D+F:E+C  B+H:A+G 

---

A schedule for $9$ players can be found below. Notice that on each a round a player does not play any games ('Bye').

  Bye    i       ii    
 1 A  B+C:D+G  E+I:F+H 
 2 B  C+A:E+H  F+G:D+I 
 3 C  A+B:F+I  D+H:E+G 
 4 D  E+F:G+A  H+C:I+B 
 5 E  F+D:H+B  I+A:G+C 
 6 F  D+E:I+C  G+B:H+A 
 7 G  H+I:A+D  B+F:C+E 
 8 H  I+G:B+E  C+D:A+F 
 9 I  G+H:C+F  A+E:B+D

---

You can find other Whist tournament configurations in this handy link