Round Robin draw for 8 teams on 4 greens

Question

A lawn bowls problem: The first part is pretty simple, we have an 8 team round robin draw over 7 rounds. The problem is that we have four greens, A B C D, and that no team can stay on the same green back to back. So if team 1 & 2 play on A for there first game then they must move to either B,C or D for there second game and so on through the seven rounds. We are trying to do this so each team gets at least one game on each of A B C and D green. Is this possible?

Answer 1

Yes, it's possible. You can even require that each team play on each court at most twice. I wrote a script that produced all $5496$ such tournaments, after prescribing the matches for the first round, and also requiring team $1$ to play on courts A, B, C, and D, in that order, in the first four rounds.

Here is the first solution:

                Court
Round    A     B     C     D
   1   1 2   3 4   5 6   7 8   
   2   3 5   1 7   2 8   4 6   
   3   2 4   5 8   1 6   3 7   
   4   3 8   2 6   5 7   1 4   
   5   6 7   1 5   4 8   2 3   
   6   4 5   2 7   1 3   6 8   
   7   1 8   3 6   4 7   2 5  

The first line means that in round $1,$ teams $1$ and $2$ play on court A, teams $3$ and $4$ on court B, and so on. Looking across the rows you can check that each team plays in every round. Looking down the columns, you can check that no team plays on the same court in consecutive founds. You can also check that each team plays on each court, and that no team plays three times on the same court. It's probably easier to check the last part in the next table, though.

              Round
Team   1  2  3  4  5  6  7
  1    2A 7B 6C 4D 5B 3C 8A 
  2    1A 8C 4A 6B 3D 7B 5D 
  3    4B 5A 7D 8A 2D 1C 6B 
  4    3B 6D 2A 1D 8C 5A 7C 
  5    6C 3A 8B 7C 1B 4A 2D 
  6    5C 4D 1C 2B 7A 8D 3B 
  7    8D 1B 3D 5C 6A 2B 4C 
  8    7D 2C 5B 3A 4C 6D 1A 

Each team's schedule is found by reading across its row. For exam, team $1$ meets team $2$ on court A in round $1,$ team $7$ on court B, and so on. Reading across the rows, you can check that each team plays each other team exactly once, and that it play one each court at least once, but not more than twice.

If there is some additional criterion you would like to use in order to select the best solution, let me know.

EDIT

I noticed one annoying asymmetry in the solution. The two teams that play court A only once, namely $6$ and $7,$ play each other on that court. This doesn't happen on the other courts. I looked for a solution where it happens on all courts, but there is none. (In $544$ of the $5496$ solutions, it happens on one court; in the remainder, it happens on no court.) Here is a symmetric solution, where it never happens:

                Court
Round    A     B     C     D
   1   1 2   3 4   5 6   7 8   
   2   3 5   1 7   2 8   4 6   
   3   2 4   5 8   1 6   3 7   
   4   3 8   2 6   5 7   1 4   
   5   6 7   1 5   4 8   2 3   
   6   4 5   3 6   2 7   1 8   
   7   6 8   4 7   1 3   2 5

               Round
Team  1  2  3  4  5  6  7

  1  2A 7B 6C 4D 5B 8D 3C 
  2  1A 8C 4A 6B 3D 7C 5D 
  3  4B 5A 7D 8A 2D 6B 1C 
  4  3B 6D 2A 1D 8C 5A 7B 
  5  6C 3A 8B 7C 1B 4A 2D 
  6  5C 4D 1C 2B 7A 3B 8A 
  7  8D 1B 3D 5C 6A 2C 4B 
  8  7D 2C 5B 3A 4C 1D 6A

Answer 2

You can almost get there using the standard Howell Movement for eight pairs at bridge. It is given in page 23 of this file. It fails your request by having team 8 stay at table 1 all the time and having the team sitting NS at table 4 switch to sitting EW at table 4. It should be easy to move the matches among the tables to get every team playing at least one match at each table.