SEARCH HOME
 Math Central Quandaries & Queries
 Subject: player combinations Name: Ken Who are you: Teacher I coach a 5th grade basketball team with 12 players. I'd like to come up with a way to get as many player combinations as possible with two teams of 6 so that the makeup of the teams is different each time. An example would be: Players 1-6 on one team and players 7-12 on the other. Then I could have 1,3,5,7,9,11 on one team and the evens on the other. Then 1,2,6,7,9,10 on one team and the rest on the other. Can you help set something like this up?

Ken,

The following table gives a way to organize two teams of six players for each of 13 sessions.

 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 0 3 4 5 6 7 8 9 10 11 12 0 1 2 9 10 11 12 0 1 2 3 4 5 6 7 8

The table has the nice property that each pair of numbers (chosen from zero to 12) appears in a column exactly once.  (For example, the numbers 2 and 4 appear together in column two (which is headed by the number 1).
Number your players from 1 to 12.  For session 'zero', the first team will consist of the players in the column headed 0 and the first column after that with a zero in it (namely 1, 3, 9, 4, 5, 7) and the remaining two columns with 0 constitute the second team (10, 11, 6, 12, 2, 8)
For session 'one' choose the column headed by 1 together with the next column with 1 in it for the first team (2, 4, 10, 5, 6, 8), and the remaining two columns with a 1 will form the second team (replacing the 0 by player 1: 1, 3, 9, 11, 12, 7).
Continue in this manner: in session 'k' the first team consists of the players in the column headed by k and those in the first column to the right that has a k in it, replacing the 0 by player k.  Session 8 for example has players 9, 11, 4, 12, 8, 2 on the first team.
If you play more that 13 sessions you can combine other pairs of columns to form the first team.

Chris

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.