In a chess tournament, each pair of players plays exactly one game. No game is...
In a chess tournament, each pair of players plays exactly one game. No game is drawn. Suppose the player wins games and loses games. Show that
Answer & Explanation
where is the total number of players. (Each player plays games, so , and there are a total of games played, so
Consider flipping the result of one game, say between player 1 and 2.
Then changes by as , where is the number of players.
Thus is the same no matter what the results and thus is same as the extreme case when the winner wins all matches, the runner up all but one etc, for which the sum is .