## Task 1 - 180821

Four students with the first names Alfred, Benno, Detlev, Egon and the last names Ampler, Baumbach, Dürer, Erbe are described as follows:

(1) Egon is younger than Benno, but older than Alfred.

(2) Detlev is older than Alfred, but younger than Benno.

(3) The student Dürer is older than the student Erbe, but younger than the student Ampler.

(4) The student Baumbach is older than the student Dürer, but younger than Benno.

(5) Exactly one of these four students has a first name that starts with the same letter as their last name.

Determine whether these statements allow for a unique answer to the following questions (a), (b)! If this is the case, determine the answers!

(a) What are the full names of the four students?

(b) What is the order of the students by age, starting with the youngest student?
The ages of the four students Alfred, Benno, Detlev, and Egon are denoted by $a, b, d, e$ in this order; the ages of the students Ampler, Baumbach, Dürer, and Erbe are denoted by $A, B, D, E$ in this order.

If statements (1) to (5) are true, then:

from (1): $a<e<b, \quad$ from (2): $a<d<b \quad$ from (3): $E<D<A, \quad$ from (4): $D<B<b$

From (1) and (2), it follows that Alfred is the youngest and Benno is the oldest student. From (3) and (4), it follows that Erbe is the youngest and Dürer is the second youngest student.

The youngest student is therefore Alfred Erbe. Furthermore, since Benno is the oldest student and he cannot be Erbe or Dürer, and because of (4) he cannot be Baumbach, he must be Ampler.

From (5) it now follows: One student is named Detlev Dürer. Thus, the fourth student is Egon Baumbach. Therefore, the only possible names and their order according to age are Alfred Erbe, Detlev Dürer, Egon Baumbach, and Benno Ampler, listed in this order, which can answer questions (a), (b) in accordance with statements (1) to (5).

Conversely, it is shown: If this list gives the names and the order of the students according to their age, then statements (1) to (5) are true. Thus, with this list, the unique answers to questions (a), (b) have been determined.