Tenpin bowling league schedule

[C_Ball]

18 people playing in triples playing every other player twice (across 17 weeks).  Ideally no triple is repeated and no one plays the same person two weeks in a row but not essential.  

Any help appreciated.

[Ian Wakeling]

I think the following schedule meets all your requirements.

(18 14 9) (11 2 6) (13 7 5) (8 15 12) (1 3 4) (16 10 17)
(18 15 10) (12 3 7) (14 8 6) (9 16 13) (2 4 5) (17 11 1)
(18 16 11) (13 4 8) (15 9 7) (10 17 14) (3 5 6) (1 12 2)
(18 17 12) (14 5 9) (16 10 8) (11 1 15) (4 6 7) (2 13 3)
(18 1 13) (15 6 10) (17 11 9) (12 2 16) (5 7 8) (3 14 4)
(18 2 14) (16 7 11) (1 12 10) (13 3 17) (6 8 9) (4 15 5)
(18 3 15) (17 8 12) (2 13 11) (14 4 1) (7 9 10) (5 16 6)
(18 4 16) (1 9 13) (3 14 12) (15 5 2) (8 10 11) (6 17 7)
(18 5 17) (2 10 14) (4 15 13) (16 6 3) (9 11 12) (7 1 8)
(18 6 1) (3 11 15) (5 16 14) (17 7 4) (10 12 13) (8 2 9)
(18 7 2) (4 12 16) (6 17 15) (1 8 5) (11 13 14) (9 3 10)
(18 8 3) (5 13 17) (7 1 16) (2 9 6) (12 14 15) (10 4 11)
(18 9 4) (6 14 1) (8 2 17) (3 10 7) (13 15 16) (11 5 12)
(18 10 5) (7 15 2) (9 3 1) (4 11 8) (14 16 17) (12 6 13)
(18 11 6) (8 16 3) (10 4 2) (5 12 9) (15 17 1) (13 7 14)
(18 12 7) (9 17 4) (11 5 3) (6 13 10) (16 1 2) (14 8 15)
(18 13 8) (10 1 5) (12 6 4) (7 14 11) (17 2 3) (15 9 16)

You probably want to randomly allocate triples to lanes each week to avoid player 18 staying in the same place.

[C_Ball]

Thank you Ian, the schedule does meet all of the requirements although it seems to have taken me longer to check than it did for you to put the schedule together.

Looking at the schedule it seems like an extension of a circle method round robin, with sequential numbers going down columns, fixed team and court balancing aside from one team.  If so, does this mean the circle method generalises to other total number of players in a triples format such as 9,12,15,21,... or even other groupings such as 4,5,6,... players per match?

The other question that came to mind was about producing a court balanced schedule for this type of problem.  Can something similar to the 2 players per match approach work?

[Ian Wakeling]

I was lucky that you happened to pick 18 players, there are 17 teams around the circle and it makes it a lot easier that 17 is a prime number.  So in general the circle method will not always work here.  But there are alternatives, for 9,15,21,... players where it is possible to play each other once (see here).  Or if you want to play twice, then all you need to do is find a randomisation of the player numbers that does not repeat any triples from the original.  As the group size gets bigger, constructions will become harder and harder - although you could easily do 16 players in groups of 4.

Unfortunately, I am not aware of any court balancing method, similar to the method for pairs, that extends to triples.