Recently I encountered the following puzzle:
Consider a game for two players which can only result in a win of one of the players (no ties). Now $n$ players decided to play this game each with each; $n(n-1)/2$ games total, $n\geq 3$. For each won game a player gets $1$ primary point. After the tournament the score of each player is calculated as the sum of primary points of all the players he've beaten. Turned out that everyone have the same score. Is it possible that amount of primary points wasn't the same for everyone?
I brainstormed this one several times in the past month but haven't come up with anything useful. Hypothetically the answer is no, checked for $n\leq 6$.