Getting around ghost entries in odd-numbered bridge games!

[McBruceWX5]

At the beginning of the pandemic I decided to try a round-robin problem that had eluded me since it came up in the 1990s: a bridge league expecting 12 teams that gets a 13th.  The interesting catch is that in tournament bridge we have a way to make three teams play a round-robin amongst themselves over two rounds, avoiding byes.  So the question became: can we make a schedule of six two-round gamedates where three of the 13 teams play a round-robin (AvB BvC CvA) and the other ten play two matches against different opponents, and have it all work out so all 13 teams play each other once after six matchdays and twelve rounds?

I've posted at

http://mcbruce.ca/13teams/

a statement of the problem in pdf, along with a LibreOffice spreadsheet you can use to try fitting the 13 teams into the 12 rounds, and a second LibreOffice spreadsheet with the solution that I stumbled upon after a few days tinkering.  The spreadsheets check as you tinker with the schedule and turn cells red when a problem is discovered.

What I'm really after is some generic rule that dictates what to do with ANY odd number of teams, 2t + 1, playing t rounds of two matches, with three teams each week in a round-robin to make a complete schedule.  The solution I found is random and was arrived at by tinkering.  Most bridge schedules have simple rules that allow Directors like me to improvise solutions when an unexpected number of people arrive.  When this problem came up in the 1990s, we were stumped and ended up convincing two teams to merge and get us back down to 12!

[Ian Wakeling]

There is a way to do what you want, however it is not going to be possible to express it in a simple set of a rules that can be used to build the schedule from scratch.  Of course that is not a big problem,  all it means is that you will need to prepare a collection of schedules for 2t + 1 teams, and have them to hand when the tournament starts.

The key to a solution is knowing about Partitioned Balanced Tournament Designs - which I discuss here and give examples in replies #1 and #5.

Given a PBTD for 2t teams, then it is possible to construct your bridge schedule for 2t+1 teams. As an example I will use the 10 team PBTD to make an 11 team bridge schedule. Here is the PBTD

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

The first two sessions of the bridge schedule are made from the 1st column above as follows.  Add the 11th team K to the middle pair from round 5, to make the sub-group of 3 teams. The ordinary pairings for the two sessions are then the games taken from above and below the middle pair respectively.

Session 1: (A B K) (I E) (J F) (H C) (G D)
Session 2:  -----  (F H) (E G) (D J) (C I)

continue in the same way for all 5 of the columns, so sessions 3 and 4 are:

Session 3: (C D K) (H J) (G I) (E B) (F A)
Session 4:  -----  (J E) (I F) (A G) (B H)

etc...

A PBTD for 18 teams was discovered relatively recently and can be found here on arXiv.org.  In summary using PBTDs will give you a way to make the 2t+1 bridge schedules for  t >= 5, excepting t=11 which is still unknown.

Hope that helps.

[McBruceWX5]

Wow, very interesting solution.  I especially admire the way you have taken your familiarity with a schedule type where each round is run in rows and realized that running it in columns would work for my problem!  Bravo!

The original scenario was for a summer league which was going to go for six consecutive weeks, but I am already seeing possible applications for two or three session team events where a complete round robin is better than a swiss when there are 11 or 13 or 15 teams.  The one drawback is the requirement that one team play the 3-ways throughout (many teams don’t like these), but I think in most events a volunteer team could be found for this role.

The other application for PBTD’s in bridge scheduling may be in our local league of bi-monthly matches, where I have used the “circle” method of producing a schedule.  A few times the top teams have complained that the sequence of teams played sometimes gives them a run of lower-ranked teams (apparently they lose interest or something unless they have a tough match fairly often!), and PBTD’s may be a better solution.

[Ian Wakeling]

I wondered myself about the team that always play threeways, but I think it gets a lot more complicated to schedule for if you don't do this.

There will be other simpler ways to construct similar schedules to the ones based on PBTDs.  I believe there will be a cyclic method for 8n+1, or 8n+3 teams (n>=1).  For example for 9 teams:

S1 (0 4 8)  (1 2)  (3 6)  (5 7)
S2 (- - -)  (5 6)  (7 2)  (1 3)
S3 (1 5 8)  (2 3)  (4 7)  (6 0)
S4 (- - -)  (6 7)  (0 3)  (2 4)
S5 (2 6 8)  (3 4)  (5 0)  (7 1)
S6 (- - -)  (7 0)  (1 4)  (3 5)
S7 (3 7 8)  (4 5)  (6 1)  (0 2)
S8 (- - -)  (0 1)  (2 5)  (4 6)

to see the cyclic nature consider the 4 odd numbered sessions, followed by the 4 even numbered sessions.