In a round robin tournament with N teams,
every 2 teams play in a head-to-head
match. Points are awarded as follows: 3
points for a win, 1 points for a tie and 0
points for a loss. What is the smallest
value of N, such that it is possible for all
the teams to have the same number of
points, but for (at least) two teams to win
a different number of matches?
To find the smallest value of N such that it is possible for all teams to have the same number of points, but for at least two teams to win a different number of matches, we can analyze the problem.
Let's start by considering the number of points a team can earn in a round-robin tournament. Each team plays against every other team exactly once.
Since a win awards 3 points, a tie awards 1 point, and a loss awards 0 points, there are three possible outcomes for each match:
1. Team A wins and Team B loses.
2. Team A and Team B tie.
3. Team A loses and Team B wins.
Now, let's assume that all teams have the same number of points. Since a team can only earn 3 or 1 points in a match, it is evident that the number of matches played by each team must be a multiple of 2. This eliminates the possibility of ties, as ties earn 1 point, which would make the number of points uneven among the teams.
Therefore, in order for all teams to have the same number of points, each team must win an equal number of matches.
Now, if we consider the case where at least two teams win a different number of matches, we can start with the minimum number of wins. Let's assume one team has 0 wins, which means that the remaining (N-1) teams must share equally the remaining wins in order to maintain equal points.
To distribute the wins evenly among (N-1) teams, each team must win at least one match. So the minimum number of wins that can be distributed evenly among (N-1) teams is (N-1) * 1 = N-1.
Therefore, the smallest value of N that satisfies the condition is N = (N-1) + 1, which simplifies to N = 2.
Hence, the smallest value of N for all teams to have the same number of points, but for at least two teams to win a different number of matches, is N = 2.