## Task 4 - 140834

Achim, Bernd, Christian, and Detlef were the four participants in the final round of a chess tournament. Each had to play against each other exactly twice. One point was awarded for each win, half a point for each draw, and zero points for each loss.

A wall newspaper article about this tournament contains the following information:

- Bernd and Christian together scored exactly one point more than Achim and Detlef together.
- Christian and Detlef together scored exactly 7 points.
- Achim and Christian together could achieve exactly 5 points less than Bernd and Detlef together.

The question is, how many points did each of the four participants receive? Determine all answers that correspond to the given information!
Exactly two games of the following combinations were played: AB, AC, AD, BC, BD, CD. Therefore, a total of 12 games were played, and thus 12 points were awarded.

If now, in a point distribution that matches the given information, Achim, Bernd, Christian, and Detlef scored $a, b, c, d$ points respectively, then the following equations hold:

$$
\begin{aligned}
a+d+1 & =b+c \\
c+d & =7 \\
b+d & =a+c+5
\end{aligned}
$$

Since a total of 12 points were awarded, i.e., $a+b+c+d=12$, one can sequentially eliminate the unknowns from the resulting system of 4 equations and then determine them in sequence.

It turns out that $a=1, d=4.5, b=4$, and $c=2.5$. Therefore, the answer that Achim received 1 point, Bernd 4 points, Christian 2.5 points, and Detlef 4.5 points, must be correct.

Indeed, this answer satisfies all the conditions of the problem. As the following table shows, there are examples of a distribution of game outcomes that result in the point distribution mentioned in the answer.

|  | $\mathrm{A}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{D}$ |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathrm{A}$ | - | 1 | 0 | 0 | 1 |
| $\mathrm{~B}$ | 1 | - | 2 | 1 | 4 |
| $\mathrm{C}$ | 2 | 0 | - | 0.5 | 2.5 |
| $\mathrm{D}$ | 2 | 1 | 1.5 | - | 4.5 |

Furthermore, in this distribution, Bernd and Christian together scored 6.5 points, which is exactly one point more than the 5.5 points scored by Achim and Detlef. Additionally, Christian and Detlef together scored exactly 7 points. Finally, Achim and Christian together scored 3.5 points, which is exactly 5 points less than the 8.5 points scored by Bernd and Detlef.

In each case, the number of points that the player in the respective row scored against the player in the corresponding column is shown.