"Twelve students of one class have held a rock-scissors-paper tournament. Each competed against each one exactly once. There were two points for a win, one point for a draw and no points for a defeat.
At the end of the tournament, all participants scored different scores. The player in second place has as many points as the last five together.
How did the game end between the players on the 7th and 9th place?"
Well I know how to find out how many points together can be in this game, and I even have a possible solution, but I have to include that a draw is possible, too.
Please give me a hint! ;)
There were ${12\choose 2} = 66$ matches.
There are 2 points awarded in each match.
The bottom 5 played ${5\choose 2}$ matches among each other.
Even if they lose every match against every competitor not in the bottom 5, they score at least 20 points.
The second place finisher has at least 20 points
The first place finisher has at least 21 points.
No one scores more than 22 points
It is possible for the second place finisher to have 21 points and the first place finisher to score 22?
No, If the first place finisher scores 22 points, he won every match, including the one against the second place finisher.
The second place finisher scores 20 points The bottom 5 fail to sore a point off of anybody out of the bottom 5.
The 9th place finisher is in the bottom 5.
The 7th place finisher beat him.