
3. In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives 3 points and the losing team receives 0 points. If the game is a draw, the two teams receive 1 point each.

Can the final scores of the six teams be six consecutive numbers $a, a+1, \ldots, a+5$ ? If so, determine all values of $a$ for which this is possible.


3. In all, 15 matches are played. In each match, the two teams together earn 2 or 3 points. The sum of the final scores is therefore an integer between $15 \cdot 2=30$ (all matches end in a draw) and $15 \cdot 3=45$ (no match is a draw).

On the other hand, the sum of the six scores equals $a+(a+1)+\cdots+(a+5)=15+6 a$. Hence $30 \leqslant 15+6 a \leqslant 45$, which shows that $3 \leqslant a \leqslant 5$. We will prove that $a=4$ is the only possibility.

First consider the case $a=5$. The sum of the scores equals $15+30=45$, so no match ends in a draw. Because in every match the teams earn either 0 or 3 points, every team's score is divisible by 3 . Therefore, the scores cannot be six consecutive numbers.

Next, consider the case $a=3$. The scores sum up to $3+4+5+6+7+8=33$. The two teams scoring 6 and 7 points must both have won at least one out of the five matches they played.

The team scoring 8 points must have won at least two matches, because $3+1+1+1+1=7<8$. Hence at least 4 matches did not end in a draw, which implies that the sum of the scores is at least $4 \cdot 3+11 \cdot 2=34$. But we have already see that this sum equals 33 , a contradiction.

Finally, we will show that $a=4$ is possible. The table depicts a possible outcome in which teams $A$ to $F$ have scores 4 to 9 . The rightmost column shows the total scores of the six teams.

|  | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $A$ | - | 3 | 1 | 0 | 0 | 0 | 4 |
| $B$ | 0 | - | 1 | 0 | 3 | 1 | 5 |
| $C$ | 1 | 1 | - | 3 | 0 | 1 | 6 |
| $D$ | 3 | 3 | 0 | - | 1 | 0 | 7 |
| $E$ | 3 | 0 | 3 | 1 | - | 1 | 8 |
| $F$ | 3 | 1 | 1 | 3 | 1 | - | 9 |
