## Task 4 - 070914

Four teams $A, B, C$, and $D$ are participating in a football tournament. Each team plays exactly one match against each of the others, and teams are awarded 2, 1, or 0 "bonus points" for a win, draw, or loss, respectively.

The day after the tournament concludes, Peter hears the end of a radio report: "...Team $D$ finished fourth. Thus, no two teams had the same point total. The match between $A$ and $B$ was the only one that ended in a draw."

Peter is disappointed that his favorite team was not mentioned in this part of the report. Nevertheless, from the information he heard and his knowledge of the tournament format, he can determine not only the placement but also the point total of his favorite team. How is this possible?
Since $A$ against $B$ is the only game that ended in a draw, $A$ and $B$ each have an odd number of points, while $C$ and $D$ each have an even number of points. The sum of these points is twelve, as exactly six games were played, each awarding two points. The number one could not have been awarded, because then $D$, as the last team and with an even number of points, would have received zero points. Thus, $D$ would have lost every game, and therefore every other team would have won at least two points.

Therefore, the point distribution is 0, 3, 4, 5, since no team has more than six points and no two teams have the same number of points. Since $D$ is the last team, $D$ has zero points. Consequently, $C$, Peter's favorite team (as it is the only one not mentioned in the report), has four points and thus took second place.