Odd # teams, Don't want any byes

[dalessbo]

I am scheduling a basketball league with four divisions (based on grade).  Two of the divisions have an odd # of teams (7 teams & 5 teams).  Games take place once a week.  The season is 10 weeks long.

Each team MUST play a game a week so NO TEAM can get a BYE (which means one team per week will get two games - which they'll be happy to play!).

In this case, I cannot figure out how to equitably give the "BYE" team a game each week (and ONE TEAM two games each week) in an even fashion so no team plays another team too many times AND no team gets significantly MORE games than another over the course of the season.

I tried just dropping teams in to play the "BYE" team and doing a different team each week, but it never works out right.  

HELP!

[Ian Wakeling]

I think you can do this by rearranging a standard schdule.  For example with 7 teams:

 G1    G2    G3   Bye
(B G) (C F) (D E)  A
(C A) (D G) (E F)  B
(D B) (E A) (F G)  C
(E C) (F B) (G A)  D
(F D) (G C) (A B)  E
(G E) (A D) (B C)  F
(A F) (B E) (C D)  G

take the games from the last two rounds minus (B E) and form the 4th game in each week.

 G1    G2    G3    G4   Plays Twice
(B G) (C F) (D E) (A F)  F
(C A) (D G) (E F) (B C)  C
(D B) (E A) (F G) (C D)  D
(E C) (F B) (G A) (A D)  A
(F D) (G C) (A B) (G E)  G

Now make a suitably chosen tranformation of the letters so that ABCDEFG becomes FAEGDBC.  This gives the next 5 weeks of play.

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

in order to complete the double round robin you would need an 11th week with the matches (B E) and (A D).

[dalessbo]

Ian - this is great - I will study this and see if I can make it work.  How would it work for a 5-man team?

[dalessbo]

I'm sorry - how would it work for a 5-team league?

[Ian Wakeling]

Using the same principle take 3 copies of the standard 5 team round robin to give 9 weeks of play.  The 10th week are the three games that were missed out from each block of 3 weeks.

(A D) (B E) (C D)  D
(E C) (D B) (E A)  E
(D E) (A C) (B A)  A

(C D) (E A) (C B)  C
(E C) (B D) (A B)  B
(B E) (A C) (D E)  E

(C E) (D A) (C B)  C
(D C) (B E) (A B)  B
(B D) (A C) (E D)  D

(C B) (D A) (E A)  A

[dalessbo]

Ian - THANK YOU!   I'm sorry, but this is a science - one that I'm not sure I'll ever completely understand.  You are a genius, and I can't thank you enough for your help.  HAPPY THANKSGIVING WEEKEND TO YOU!