[Darryl_P]

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6 man tourney. I have 24 players. I want to have each player play against every other player at least. I don't want anyone playing against a particular player more than twice. I want each player to play 6 games.

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I think I can prove this is impossible.

Key reason: for any 6 players playing at a given table there are 15 possible pairs, of which 10 must recur at some point later. To see this, simply assume ABCDEF are playing at the same table in the first round, and note that in all subsequent rounds you must distribute these 6 players among 4 tables, leaving at least 2 groups of 2 or 1 group of 3. That's at least 2 recurring pairs per round which makes at least 10 recurring pairs in 5 rounds. So worded another way, if you want to make sure no pair gets repeated more than once, you can have at most 5 unique pairs at a given table in a given round.

There are 4 tables per round and 6 rounds in all which makes at most 4x6x5 = 120 unique pairs in all.

But the goal of the problem is to have all of the 276 possible pairs occuring at least once in a tournament format that involves a total of 360 pairs. There is only room for 84 duplicates which means that 192 pairs must be unique. Since you cannot exceed 120 unique pairs, this cannot be done.