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| Math Central | Quandaries & Queries |
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Subject: player combinations 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. |
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
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