Playing outwith your group

[David_Shields]

Hi,

I am trying to run a round robin bowls competition with a twist and looking for any help possible! There are normally between 10-12 teams, however there only 5 rinks to play on. creating 2 groups  of 5 or 6 teams, and playing "outwith" your group to have 4 or 5 games in my mind maximises the games played for everyone. However I cannot seem to get it to work with the restriction of 5 rinks - with everyone playing on different rinks if possible.

Any help would be greatly appreciated!!

[Ian Wakeling]

If there is one group of 6 teams, then it is mathematically impossible to schedule the 15 games (with 5 rounds and 5 rinks) in such a way that each team plays exactly once on each rink.  I believe it is also impossible to schedule a group of 5 teams on a 5x5 grid in such a way that teams play at most once on each rink.  To get the rink balance to work, then more than 5 rounds are necessary and consequently there would be a number of unused slots in the grid.

If there are 10 teams it may be best not to divide into 2 groups, as it is possible to give everyone 5 different opponents, while playing once on each rink, like this:

(B F) (I G) (J A) (H C) (D E)
(I E) (D A) (H F) (B G) (J C)
(H A) (B C) (D G) (J E) (I F)
(D C) (J F) (B E) (I A) (H G)
(J G) (H E) (I C) (D F) (B A)

[David_Shields]

Thanks for getting back to me so quickly!

when you say that to get the rink balance to work, then more than 5 rounds are necessary and consequently there would be a number of unused slots in the grid - do you mean that there would be "bye" sessions?

Are you able to provide a similar grid for 8 teams, 5 different opponents, while playing once on each rink?

My thanks again for any help you can provide!

[Ian Wakeling]

8 teams on 5 rinks is also impossible, however 8 teams on 4 rinks and 4 opponents does work. For example:

(A B) (H D) (C F) (G E)
(C E) (G F) (A D) (H B)
(G D) (C B) (H E) (A F)
(H F) (A E) (G B) (C D)

With regard to the unused slots, I mean that I can balance the rinks if allowed extra rounds.  This leads both to empty slots and byes.

For example, 6 teams and 8 rounds.

(---) (A F) (---) (B D) (---)
(---) (---) (B F) (---) (D E)
(A E) (---) (---) (---) (---)
(---) (B E) (---) (---) (C F)
(B C) (---) (A D) (---) (---)
(---) (C D) (---) (E F) (---)
(---) (---) (C E) (---) (A B)
(D F) (---) (---) (A C) (---)

It is possible to schedule another group of 6 teams (G to L) in the empty slots to the right of the games above (or in the case of the 5th rink, the empty slot in the 1st rink).  So the closest I can come to your original 12 team scenario is:

(---) (A F) (G L) (B D) (H J)
(J K) (---) (B F) (H L) (D E)
(A E) (G K) (---) (---) (---)
(I L) (B E) (H K) (---) (C F)
(B C) (H I) (A D) (G J) (---)
(---) (C D) (I J) (E F) (K L)
(G H) (---) (C E) (I K) (A B)
(D F) (J L) (---) (A C) (G I)