9. (40 points) In each game of bridge, four players play together. It is known that the number of games played is exactly equal to the number of players, and any two players have played at least one game together. Try to find the maximum number of players.


9. There are $n$ players, and each game can produce 6 pairs of players.
From $\mathrm{C}_{n}^{2} \leqslant 6 n$ we get
$$
n \leqslant 13 \text {. }
$$

Next, we prove that $n=13$ is feasible.
As shown in Figure 13, number the 13
players from $0 \sim 12$.
The players in each game are as follows:
$$
\begin{array}{l}
(0,2,3,7), \\
(1,3,4,8), \\
(2,4,5,9), \\
(3,5,6,10),(4,6,7,11),(5,7,8,12), \\
(6,8,9,0),(7,9,10,1),(8,10,11,2), \\
(9,11,12,3),(10,12,0,4),(11,0,1,5), \\
(12,1,2,6) .
\end{array}
$$

The conditions are satisfied.