I have to arrange a card game tournament, in which the players will play in tables of 4. I want to arrange as little games as possible, but to prevent biases I want every 2 players to have played by each other by the end, but only once. Also, as the game has a more or less constant duration, the games should take place in N rounds, and in each of those no player should be idle.
Therefore the number of rounds should be $(N-1)/3$ and the number of tables $N/4$
It's easy to check that all possible solutions for N are of the form $4 + 12·k$, where $k$ is an integer. I managed to convince myself that there is no solution for $N = 16$, but I don't want to check manually for $N = 28$ and above. Can you find the general solution?
Note: This is not a theoretical problem, I actually want to arrange a card game tournament.