Maths olympiad, minimum possible number of points

Question

Six players compete in a tournament. Each player plays exactly two games against every other player. In each game, the winning player earns 2 points and the losing player earns 0 points. If the game results a draw (tie), each player earns 1 point. What is the minimum possible number of points that a player needs ti earn in order to guarantee that he/she will be champion (i.e he/she has more points than every other player)?

Answer 1

In order to figure this out, we need to assume that 2 different players are basically dominating the other 4 and then play each other. Each player versus the other 5 players twice. Thus, each player plays 10 games. Max number of points is 20.

Let's say Player 1 plays Players 3-6 (4 different players) and wins all the games. Player 1 would then have 16 points (4 players * (2 games / player) * (2 points / game)

Let's say Player 2 does the same. Plays Players 3-6 (4 different players) and wins all the games. Player 2 would then have 16 points (4 players * (2 games / player) * (2 points / game)

This means, that going into the final 2 games (which are between Players 1 & 2), both players 1 & 2 have 16 points... The outcomes of their 2 games will crown the winner. Players 3-6 cannot compete any longer since they lost both matches to 2 players it is impossible for them to have more than either player 1 or 2.

Now between the last 2 games, there are only 4 points that can be given out. In order to ensure victory (and not end the contest in a tie of 18 to 18 points), either player 1 or 2 must get at least 3 points, resulting in a 19-17 or a 20-16 victory.

Thus, the minimum amount of points required to guarantee winning the tournament is 19 points.

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You can also think about it like this... If you Win every match except for 1 match, which you tie, then noone can have as much or more than you since everyone person would have lost matches to you and not been able to beat you at all. Winning 9 games and tieing 1 gives 19 points.