## Task 1 - 100921

Four friends $A, B, C$, and $D$ hide a letter. One of them takes it. Then each of them makes the following three statements, at least two of which are true.

A (1) "If I don't have the letter, then $C$ has it."

(2) "I don't have the letter."

(3) "My friend has the letter."

B (1) "Either $A$ or $C$ has the letter."

(2) "All of $A$'s statements are true."

(3) "$D$ does not have the letter."

C (1) "If I don't have the letter, then $B$ has it."

(2) "I have the letter."

(3) "$B$ does not make false statements."

D (1) "I don't have the letter."

(2) "Either $A$ has the letter, or he doesn't."

(3) "$B$ invented the game."

Who has the letter?
Assume C does not have the letter. Then C b) would be false. Therefore, since according to the problem, at least two of the three statements made by C are true, C c) must be true.

Thus, all statements by B and, due to B b), also all statements by A would be true. Because of B a) and A b), C must have the letter. This contradiction proves that the assumption that C does not have the letter is false.

Therefore, the only remaining possibility is: C has the letter.

## Adopted from [5]

2. Solution:

A cannot have the letter, as in that case his statements A2 and A3 would be false. (He would not refer to himself as his own friend.) B cannot have the letter either, as in that case statement A1 and thus, along with B1, B2 would also be false. The same applies if D had the letter.

The question remains whether C can have the letter. In that case, statements A1 to A3 are correct, as C is a friend of A. Consequently, all statements B1 to B3 by B are also true, as well as all three statements by C. (Note that in C1, the antecedent of the implication, that C does not have the letter, is not satisfied in this case, making the implication true.) Finally, statements D1 and D2 (the latter being a tautology) are true; we cannot make a statement about D3. Thus, in this case, each of the four friends makes at least two correct statements, so C must indeed have the letter.