new to puzzles here. A real life situation led me to consider this. We have a small league of 6 people playing in teams of two. I would like to rotate the teams in such a way that everyone gets an equal chance to play with every other person and against every other person.
e.g. in the first game, it's A+B vs C+D and E+F sit out.
How do I arrange rotations in such a way that everyone gets an equal number of chances to play with every other player and against every other player?
Answer
Each person plays the same number of matches with each of the other 5, so the number of matches each person plays must be a multiple of 5.
If each person plays 5 matches, then there are a total of 30 players across the matches. But this is impossible because 4 players are in each match and 30 isn't divisible by 4. So each player must play at least 10 matches, with a total of 15 matches.
You can make five sets of three games each: for each set, divide the players into three pairs, and have each pair play the other two once. This is one possible way to divide the pairs:
- AB/CD/EF
- AC/BE/DF
- AD/BF/CE
- AE/BD/CF
- AF/BC/DE
You can check that for every two players, they are together in two games (in the set where they are paired) and against each other in four games (once in each of the other sets).
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