riddle of the day

In a short league, five teams face each other against each other. In the end, neither pair of teams shares a score. The question is how far the fourth-placed team can go.

The tournament of ties

Strategist
Numerical territory

Riddle statement

Five teams play a round-robin league, in a single round.

Score:
- victory: 3 points;
- tie: 1 point for each team;
- defeat: 0 points.

In the end, the five teams finish with different scores.

What is the highest possible score for the fourth place team?

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Solution

Answer: the highest possible score for the fourth-place finisher is 5 points.

Let's call the final scores, from highest to lowest:

$ s_1>s_2>s_3>s_4>s_5. $

We want to maximize $s_4$.

Suppose the fourth-place finisher could have at least 6 points. As the five scores are different, the three teams above it would have to reach at least 7, 8 and 9 points respectively.

That means that the first four would add at least:

$ 9+8+7+6=30. $

Now, in a five-team league There are

$ \binom{5}{2}=10 $

matches, and each match distributes a maximum of 3 points. The maximum total points for the tournament is, therefore:

$ 10\times 3=30. $

For the first four to already have 30, the fifth team would have to have 0 points, and all matches would have to end with a victory — without any ties. But if there are no ties, each score is a multiple of 3. Scores 8 and 7 would be impossible.

It has been proven that $s_4 \le 5$.

Now it remains to be seen that 5 is achievable. A possible classification is:

$ 10,\ 7,\ 6,\ 5,\ 0. $

The following table of results is made by:

  • Team A beats B, C and D; ties with E. Ends with 10 points.

  • Team B beats C and D; tie with E; loses to A. Ends with 7 points.

  • Team C beats D and E; loses to A and B. Ends with 6 points.

  • Team E beats D; tie with A and B; loses to C. Finishes with 5 points.

  • Team D loses all of its games. It ends with 0 points.

The five scores are different and the fourth-placed team (team E) has 5 points.

Therefore, the highest possible score for the fourth-placed team is 5 points.