## Task 3 - 130813

In a mathematics competition organized by the student magazine "alpha," three students from a school won prizes. When asked about their first names, the following seven answers were given:
(1) Christian, Uwe, Iris
(2) Eva, Elke, Uwe
(3) Roland, Marion, Bernd
(4) Iris, Heike, Uwe
(5) Roland, Heike, Bernd
(6) Eva, Marion, Christian
(7) Christian, Eva, Elke.

It turned out that in exactly one of the answers, all three first names were correct, in exactly two answers exactly two first names were wrong, and in exactly three answers all three first names were wrong.

Determine the first names of the three students who won the prizes!
First, it follows from the task that in the remaining answers, exactly two names were correctly specified. Therefore, in an answer, all three first names are correct only if in exactly two other answers exactly one of these first names and in exactly one other answer exactly two of these first names occur.

This does not apply to

(1) because in the three answers (2), (6), (7) each of the names from (1) appears exactly once,

(2) because in the three answers (1), (4), (6) each of the names from (2) appears exactly once,

(6) because in the three answers (1), (2), (3) each of the names from (6) appears exactly once,

(3) because only in the answer (6) does exactly one of the names from (3) appear,

(5) because only in the answer (4) does exactly one of the names from (5) appear,

(7) because only in the answer (1) does exactly one of the names from (7) appear.

Therefore, only the answer (4) can be correct; the first names of the prize winners are thus Iris, Heike, and Uwe.