Eight soccer teams participate in a soccer tournament, so that each team plays with every other team exactly once. If one teams wins against another, then the winning team gets two points, and the losing team gets zero points. If the two teams tie, then both teams get one point each.

After the tournament, the teams are ranked by the total number points each team won. What is the minimum number of points a team must win, to ensure that it is in the top four teams?

I think that the four teams in the top have to have the same number of points to maximize the number of points the other team needs to get in, but I'm not sure.

  • 2
    $\begingroup$ How do you deal with equality cases ? If the best five teams have the same number of points, do you consider that they are in the top four, or not ? $\endgroup$ Oct 28 '20 at 14:50
  • $\begingroup$ If a team scores $8 \times 7 / 4 = 14 $ points, then they are guaranteed to be in the top 4 from a point basis. However, this implies that they won every game that they played, which is clearly too strict. $\endgroup$
    – Calvin Lin
    Oct 28 '20 at 15:31
  • $\begingroup$ What if the top $4$ teams tie against one another, and beat each of the bottom $4$ teams? This gives the top teams $11$ points each. Does this work? $\endgroup$
    – saulspatz
    Oct 28 '20 at 15:39

With an eight team round robin, $28$ games will be played in total (The total edges in $K_8$). enter image description here Since each game will produce two points regardless of outcome, $56$ total points will be shared among the eight teams. In the absence of a well-defined tie breaking system, we assume that ties do not qualify as a guarantee. We define a team's point total as "ensuring" fourth place when the fifth place team could not possibly have that many points.

\begin{align} Team && Points && && Team && Points\\ A && 10 && && A && 11\\ B && 10 && && B && 11\\ C && 10 && && C && 11\\ D && 10 && && D && 11\\ --- && --- && && --- && ---\\ E && 10 && && E && 11\\ F && 2 && && F && 1\\ G && 2 && && G && 0\\ H && 2 && && H && 0\\ \end{align}

Ten points are not enough and eleven points must be enough. In one extreme case, teams $A,B,C,D$ and $E$ all draw each other and win against $F,G,$ and $H$ (who also draw each other), giving a fifth place team with ten points. If we have a fifth place team with eleven points then teams $A,B,C,D,$ and $E$ share at least $55$ points, meaning teams $F, G,$ and $H$ share at most one point. However, $F, G,$ and $H$ play each other and must share at least six points, a contradiction. Therefore, $11$ points guarantees fourth place at a minimum. $\blacksquare$


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