## Task 1 - 070811

Three students from a class, Thomas (T), Rainer (R), and Bernd (B), had qualified for the final in high jump at a sports festival and won the first three places. Klaus, who had to compete in another discipline, later asked Elke about the outcome of the high jump. She could not remember exactly and said:

"Thomas did not come in first, Rainer did not come in second, but Bernd came in second."

Later it turned out that Elke had said one thing correctly, but was wrong in the other two cases. It is also known that all three students cleared different heights.

Which student came in first, second, and third?
The three statements can be represented with the notation Thomas $(T)$, Rainer $(R)$, and Bernd $(B)$ as follows:

$$
T \neq 1 \quad(1) \quad ; \quad R \neq 2 \quad(2) \quad ; \quad B=2
$$

If Elke made exactly one statement correct and exactly two incorrect, then the following three cases must be distinguished:

1. Case: (1) is correct and (2), (3) are incorrect: $T \neq 1 \quad(1 *) \quad ; \quad R=2 \quad(2 *) \quad ; \quad B \neq 2 \quad(3 *)$

From $\left(1^{*}\right)$ and $\left(2^{*}\right)$ it follows $\left(4^{*}\right) T=3$. From this, it follows $\left(5^{*}\right) B=1$. All three statements $\left(1^{*}\right),\left(2^{*}\right),\left(3^{*}\right)$ are satisfied.

2. Case: (2) is correct and (1), (3) are incorrect: $T=1 \quad(1 *) \quad ; \quad R \neq 2 \quad(2 *) \quad ; \quad B \neq 2 \quad(3 *)$

From $\left(1^{*}\right),\left(2^{*}\right)$, and $\left(3^{*}\right)$ it follows that no one can be second. This is a contradiction, so this case is not feasible.

3. Case: (3) is correct and (1), (2) are incorrect: $T=1 \quad(1 *) \quad ; \quad R=2 \quad(2 *) \quad ; \quad B=2 \quad$ (3*)

From $\left(2^{*}\right)$ and $\left(3^{*}\right)$ it follows that there are two seconds. This is a contradiction, so this case is not feasible.

Therefore, there is exactly one solution: Bernd was first, Rainer second, and Thomas third.