Eight teams play a rugby tournament in which each team plays exactly one match against each of the remaining seven teams. In each match, if it's a tie each team gets $1$ point and if it isn't a tie then the winner gets $2$ points and the loser gets $0$ points. After the tournament it was observed that each of the eight teams had a different number of points and that the number of points of the winner of the tournament was equal to the sum of the number of points of the last four teams. Give an example of a tournament that satisfies this conditions, indicating the number of points obtained by each team and the result of each match.
Problem
Source:
Tags: combinatorics
WallyWalrus
09.12.2022 00:15
Denote $T_k,\;k\in\{1,2,\dots,7,8\}$ the teams. The team $T_k$ has at the end of the tournament $P_k$ points. Assume $P_1>P_2>\dots>P_7>P_8$. The total number of matches of the tournament is $n=\dbinom{8}{2}=28$. After each match, the $2$ teams obtain a total of $2$ points: either $2+0$ or $1+1$. Hence: the $8$ teams have at the end of the tournament totally $56$ points. From the initial condition: $P_1=P_5+P_6+P_7+P_8$. $\max P_1=14$ points ($7$ wins). For $P_8=2$ results $P_5+P_6+P_7+P_8\ge 2+3+4+5=14\ge P_1$, hence we can obtain an equality case for $P_1=14;\;P_8=2;\;P_7=3;\;P_6=4;\;P_5=5$. In this case: $P_2+P_3+P_4=28$. A possible situation is: $P_2=11;\;P_3=9;\;P_4=8$. Detailed: Team $\quad$Wins$\;\;$ Ties$\;\;$Loses$\;\;$Points $\quad T_1\quad\quad7\quad\quad0\quad\quad0\quad\quad14$ $\quad T_2\quad\quad5\quad\quad1\quad\quad1\quad\quad11$ $\quad T_3\quad\quad4\quad\quad1\quad\quad2\quad\quad\;9$ $\quad T_4\quad\quad3\quad\quad2\quad\quad2\quad\quad\;8$ $\quad T_5\quad\quad1\quad\quad3\quad\quad3\quad\quad\;5$ $\quad T_6\quad\quad0\quad\quad4\quad\quad3\quad\quad\;4$ $\quad T_7\quad\quad0\quad\quad3\quad\quad4\quad\quad\;3$ $\quad T_8\quad\quad0\quad\quad2\quad\quad5\quad\quad\;2$ $T_1$ wins against all teams; $T_2$ wins against $T_3;T_5;T_6;T_7;T_8$; $T_3$ wins against $T_4;T_6;T_7;T_8$; $T_4$ wins against $T_5;T_7;T_8$; $T_5$ wins against $T_8$; the remaining matches end with ties.