444. Eight hockey teams play against each other in a round-robin tournament to determine the final four. What is the minimum number of points that guarantees a team's advancement to the final four?
$\triangle$ All eight teams together will score $\frac{8 \cdot 7}{2} \cdot 2=56$ points. Therefore, 7, 8, and even 9 points do not guarantee a team's advancement to the final four.

What about 10 points? Imagine that there are five teams that played all their matches against each other to a draw, while the other three teams won all their matches. Then each of these teams will score $4+6=10$ points. However, 10 points still do not guarantee a team's advancement to the final, since there are five such teams, not four.

We will prove that if a team scores 11 points, it will advance to the final. Assume the opposite: there are five teams, each of which scores at least 11 points. In this case, they will score at least 55 points together, which means the remaining three teams will score no more than 1 point together. But this is impossible, since these three teams will score at least $\frac{3 \cdot 2}{2} \cdot 2=6$ points in matches against each other.

Answer: 11.