## Task 3 - 210713

For the preparation of a sports festival, a sequence should be determined for the students Andrea, Beate, Christine, Doris, Eva, Frauke, and Gerda. It should always be the case that among two students of different heights, the taller one stands in front of the shorter one. However, if two students are of the same height, the one whose first name starts with a letter that comes earlier in the alphabet should stand in front of the other. The organizer, who is supposed to determine such a sequence, does not know the students but believes he can remember the following information:

(1) It is true that Doris is exactly 2 cm shorter than Christine.

(2) It is false that Andrea is not the same height as Gerda.

(3) It is not true that none of the students is shorter than Frauke.

(4) It is true that Eva is shorter than Doris but taller than Frauke.

(5) It is false that Frauke is taller than Christine.

(6) It is not false that Christine is exactly 2 cm taller than Gerda and that Christine is taller than Eva.

Investigate whether there is more than one, exactly one, or no possible sequence of the students that satisfies all the information (1) to (6)!

If they exist, determine all possible sequences that meet the given conditions!
If for each student, her height measured in centimeters is denoted by the first letter of her first name, the information is as follows:

(1) It holds that $D=C-2$.

(2) It holds that $A=G$.

(3) At least one of the inequalities $A>F$ or $B>F$ is true.

(4) It holds that $B>F$.

(5) It holds that $F \leq C$.

(6) It holds that $C=G+2$ and $C>E$.

I. Assuming that in a certain order of the students, all information from (1) to (6) is true.

(7) From (1) and (6), it follows that $D=G=C-2$.

(8) From (2) and (7), it follows that $A=D=G$.

(9) From (1), (4), and (8), it follows that $C>A=D=G>E>F$.

(10) From (3) and (9), it follows that $B>A=D=G>E>F>B$ all information from (1) to (6) is true.

II. If this order is present and additionally $A=D=G=C-2$, then:

It is $D=C-2$, so (1) is satisfied.

It is $A=G$, so (2) is satisfied.

It is $B>F$, so (3) is satisfied.

It is $B>F$, so (4) is satisfied.

It is $F \leq C$, so (5) is satisfied.

It is $C>G+2$ and $C>E$, so (6) is satisfied.

Thus, it is proven that there is exactly one order in which all information from (1) to (6) is true. It is as stated in (10).