6.13 There are 20 teams participating in the World Championship and the European Championship. Among these teams, there are $k$ European teams, and the results of their matches in the World Championship are counted in the European Championship. The championships are played in a round-robin format. If the matches are played according to:
(1) the ice hockey system (draws are allowed, the winning team gets 2 points, the losing team gets 0 points, and in case of a draw, each team gets 1 point),
(2) the volleyball system (no draws, the winning team gets 1 point, the losing team gets 0 points),
what is the maximum value of $k$ such that the team with the highest score in the European Championship can be the team with the lowest score in the World Championship?
[Solution] We study a more general case, i.e., when $n (\geqslant 7)$ teams participate in the World Championship.
(1) When $n$ teams participate in the World Championship, the total score is $n(n-1)$, and when $k$ teams participate in the European Championship, the total score is $k(k-1)$. Let the scores of the European champion team in the World Championship and the European Championship be $x$ and $y$, respectively, then $x \geqslant y$. Thus, the scores of the other teams in the World Championship are at least $x+1$ points, and the scores of each European team in the European Championship are at most $y-1$ points. Therefore, we have
$$
\begin{array}{l}
x+(n-1)(x+1) \leqslant n(n-1), \\
y+(k-1)(y-1) \geqslant k(k-1).
\end{array}
$$

Solving these, we get $x \leqslant n-2+\frac{1}{n}$ and $y \geqslant k-\frac{1}{k}$. Since $x$ and $y$ are integers, we have $k \leqslant y \leqslant x \leqslant n-2$.

The figure below shows an example of a round-robin scoring table, where $n$ teams participate in the World Championship, and then $n-2$ teams participate in the European Championship. In the European Championship, the 3rd team scores $n-2$ points, and the scores of the other teams do not exceed $n-3$ points, so the 3rd team is the European champion. It scores $n-2$ points in the World Championship, but the scores of the other teams are at least $n-1$, meaning the 3rd team scores the least. 

In summary, the maximum value of $k$ is $n-2$.
(2) When $n$ teams participate in the World Championship, the total score is $\frac{1}{2} n(n-1)$, and when $k$ teams participate in the European Championship, the total score is $\frac{1}{2} k(k-1)$. Similarly, we can derive
$$
\begin{array}{l}
x+(n-1)(x+1) \leqslant \frac{1}{2} n(n-1), \\
y+(k-1)(y-1) \geqslant \frac{1}{2} k(k-1).
\end{array}
$$

From this, we get $x \leqslant \left[\frac{n-3}{2}\right]$ and $y \geqslant \left[\frac{k}{2}\right]+1$. Therefore,
$$
\left[\frac{k}{2}\right] \leqslant \left[\frac{n-3}{2}\right]-1 = \left[\frac{n-5}{2}\right].
$$

Dividing $n$ into even and odd cases, we get
$$
k \leqslant \left\{\begin{array}{l}
n-5, \text{ when } n \geqslant 8 \text{ is even,} \\
n-4, \text{ when } n \geqslant 7 \text{ is odd.}
\end{array}\right.
$$

We still use the example of a round-robin scoring table to show that the equality in the above formula can be achieved:

In the left figure above, $n=8, k=3$, the 6th, 7th, and 8th teams participate in the European Championship, and the 6th team is the European champion. Teams $A$ and $B$ are supplementary teams, i.e., when the scoring table for $n=2l$ has been constructed, for $n=2(l+1)$, they can be added in the form shown in the figure. In the right figure above, $n=7, k=3$, the 5th, 6th, and 7th teams participate in the European Championship, and the 5th team is the European champion. Teams $A$ and $B$ are supplementary teams. It is easy to see that the scoring tables in both figures meet the requirements of the problem.
In summary, when the game is played according to the volleyball rules, the maximum value of $k$ is
$$
k_{\max} = \left\{\begin{array}{l}
n-5, \text{ when } n \geqslant 8 \text{ is even,} \\
n-4, \text{ when } n \geqslant 7 \text{ is odd.}
\end{array}\right.
$$

In particular, when $n=20$, under the ice hockey rules, the maximum value of $k$ is 18; under the volleyball rules, the maximum value of $k$ is 15.