## Task 3 - 340533

Annette, Bernd, Christiane, Dieter, and Ruth are playing the following game: the four children except Ruth agree that one of them will hide a letter and that each of these children will make three statements, at least two of which are true.

Ruth, who only knows these rules and the statements of the four, is supposed to find out who has the letter. One of the four children, Annette, Bernd, Christiane, or Dieter, came up with the game; they know who it was, but Ruth does not. The following statements are made:

Annette: I do not have the letter. Either Bernd has the letter, or Bernd does not have the letter. Christiane came up with the game.

Bernd: If I do not have the letter, then Dieter has it. I do not have the letter. Annette or Christiane or Dieter has the letter.

Christiane: Either Bernd or Dieter has the letter. Bernd made three true statements. Annette does not have the letter.

Dieter: If I do not have the letter, then Christiane has it. I do not have the letter. All three statements by Christiane are true.

Determine whether the rules and the statements uniquely determine who has the letter! If this is the case, name this player! Also, determine whether all statements comply with the rules if the letter is with the player you have identified!
I. Since Bernd's second and third statements are of equal value, they are either both true or both false. According to the rules, they cannot both be false, so they must both be true. Therefore, Bernd does not have the letter.

If Bernd's first statement is also true, it follows that: Dieter has the letter.

If, however, Bernd's first statement is false, it follows that: Christiane's second statement is false. According to the rules, Christiane's first and third statements must be true. From the first statement and the fact that Bernd does not have the letter, it follows again: Dieter has the letter.

Thus, it has been conclusively determined: If all statements comply with the rules, only Dieter can have the letter.

II. If Dieter has the letter, then Bernd, Christiane, and Dieter each made three true statements, and at least the first two statements of Annette are true. Therefore, all statements comply with the rules.