Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


David Marcus
Posts:
40
From:
Somerville, Massachusetts
Registered:
12/4/04


round robin scheduling
Posted:
Jan 11, 2003 1:15 PM


I was asked for help with a tournament scheduling problem by the Eightball Tasmania society:
http://www.eightballtasmania.org.au/
I don't know how to solve it, so I am asking for help. The question is how to put players into small roundrobin groups so that by the end of the tournament, you've played a round robin of all the players. To clarify, consider the following example.
Suppose there are 25 players and we want each player to play every other player exactly once. We have 5 eightball tables, so we decide to play the tournament in rounds. In each round, we send a group of players to each table and have them play a round robin among themselves. When we create a group, we do it so that none of the players in the group have played each other in previous rounds. We can play the entire tournament using 6 rounds using the following schedule.
Round 1 Group 1: 1 2 3 4 5 Group 2: 6 7 8 9 10 Group 3: 11 12 13 14 15 Group 4: 16 17 18 19 20 Group 5: 21 22 23 24 25
Round 2 Group 1: 1 6 11 16 21 Group 2: 2 7 12 17 22 Group 3: 3 8 13 18 23 Group 4: 4 9 14 19 24 Group 5: 5 10 15 20 25
Round 3 Group 1: 1 10 14 18 22 Group 2: 2 6 15 19 23 Group 3: 3 7 11 20 24 Group 4: 4 8 12 16 25 Group 5: 5 9 13 17 21
Round 4 Group 1: 1 9 12 20 23 Group 2: 2 10 13 16 24 Group 3: 3 6 14 17 25 Group 4: 4 7 15 18 21 Group 5: 5 8 11 19 22
Round 4 Group 1: 1 8 15 17 24 Group 2: 2 9 11 18 25 Group 3: 3 10 12 19 21 Group 4: 4 6 13 20 22 Group 5: 5 7 14 16 23
Round 6 Group 1: 1 7 13 19 25 Group 2: 2 8 14 20 21 Group 3: 3 9 15 16 22 Group 4: 4 10 11 17 23 Group 5: 5 6 12 18 24
The problem is, what should the schedule be for other values of the number of tables and players? They generally have either 5 or 6 tables, but they may have any number of players. Of course, the fewer total rounds the better. And if all the groups in a round aren't the same size, it is best if the groups are almost the same size so they will all take about the same amount of time to play their matches.
 David Marcus



