## Task 2 - 310622

A football tournament was held between four teams $A, B, C, D$. Each team played exactly one match against each of the others.

For a win, two points were awarded, for a draw, one point, and for a loss, no points. The match between teams $C$ and $D$ was the only one that ended in a draw. No two teams achieved the same number of points. Team $B$ finished last.

a) Investigate whether this information uniquely determines which place team $A$ finished in and how many points they scored!

If this is the case, state both!

b) Is the given information also sufficient to determine the exact final standings (placements of the individual teams and the points they each scored) of the tournament?

Justify your answer!
a) First, the draw game $C$ against $D$ should be entered into a table. From the information that this was the only draw, it follows: All other point totals are 0 or 2.

If $A$ had lost to $C$ and $D$ (Table 1), then due to the different point totals achieved by $C$ and $D$, $B$ would have played differently against $C$ and $D$ and would have thus achieved at least as many points as $A$; this case is therefore ruled out.

If $A$ had lost to $B$ and one of the teams $C, D$ (without loss of generality against $C$; Table 2), then $B$ would have already achieved at least as many points as $A$.

Table 1

|  | $\mathrm{A}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{D}$ |
| :---: | :---: | :---: | :---: | :---: |
| $\mathrm{A}$ | - |  | 0 | 0 |
| $\mathrm{~B}$ |  | - |  |  |
| $\mathrm{C}$ | 2 |  | - | 1 |
| $\mathrm{D}$ | 2 |  | 1 | - |

Table 2

|  | $\mathrm{A}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{D}$ |
| :---: | :---: | :---: | :---: | :---: |
| $\mathrm{A}$ | - | 0 | 0 |  |
| $\mathrm{~B}$ | 2 | - |  |  |
| $\mathrm{C}$ | 2 |  | - | 1 |
| $\mathrm{D}$ |  |  | 1 | - |

Therefore, $A$ must have won at least two games.

If $A$ had won against $C$ and $D$ (Table 3), then $B$ would have played differently against $C$ and $D$ and would have thus achieved more points than the team beaten by $B$ among these two. Therefore, it follows: $A$ has won against $B$ and exactly one of the teams $C, D$, without loss of generality against $C$ (Table 4).

| Table 3 |  | A | B | C | D |
| :---: | :---: | :---: | :---: | :---: | :---: |
|  | A | - |  | 2 | 2 |
|  | B |  | - |  |  |
|  | C | 0 |  | - | 1 |
|  | D | 0 |  | 1 | - |

Table 4

|  | $\mathrm{A}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{D}$ |
| :---: | :---: | :---: | :---: | :---: |
| $\mathrm{A}$ | - | 2 | 2 | 0 |
| $\mathrm{~B}$ | 0 | - |  |  |
| $\mathrm{C}$ | 0 |  | - | 1 |
| $\mathrm{D}$ | 2 |  | 1 | - |

Further, it follows: $B$ has fewer points than $A$, so at most one game won. If $B$ had achieved the same results against $C$ and $D$ as $A$ against $C$ and $D$, that would result in 2 points for $B$ and 1 point for the losing team. If, however, $B$ had achieved opposite results against $C$ and $D$ compared to $A$, then both $C$ and $D$ would have 3 points.

Therefore, $B$ cannot have won any game at all, one of the teams $C, D$ has 3 points, the other 5 points. Thus, it has been uniquely determined: $A$ took second place with 4 points.

b) Since both Table 5 and Table 6 correspond to all the information but provide different final standings, it follows: The information is not sufficient to determine the exact final standings of the tournament.

| Table 5 |  | A | B | $\mathrm{C}$ | $\mathrm{D}$ | Table 6 |  | A | B | C | D |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  | $\bar{A}$ | - | 2 | 0 | 0 |  | $\overline{\mathrm{A}}$ | - | 2 | 2 | 0 |
|  | B | 0 | - | 0 | 0 |  | B | 0 | 0 | 0 | 0 |
|  | $\mathrm{C}$ | 0 | 2 | - | 1 |  | C | 2 | 2 | - | 1 |
|  | $\mathrm{D}$ | 2 | 2 | 1 | - |  | D | 0 | 2 | 1 | - $\quad$ - |