Three good friends went fishing together; and there were not two of them who caught the same number of fish. The next day they told this: $A$ : I caught the most fish, and $C$ caught the least.

$B$ : I caught the most fish, more than $A$ and $C$ combined.

$C$ : I caught the most fish, $B$ only caught half as many as I did.

Who caught the most fish, if out of the six statements made by the children, exactly three are true? Can it be determined who caught the least?
Among the children's first statements, only one can be true because they are contradictory in pairs. Therefore, exactly two of the second statements are true. Since the second statements of $A$ and $C$ are contradictory, one of them must be false, so $B$'s second statement must be true. This implies that $B$'s first statement is also true. Therefore, $B$ caught the most fish. This also means that $C$'s second statement is false, so $A$'s second statement is true, meaning $C$ caught the least fish. In summary, we can conclude that $C$'s both statements are false, $B$'s both statements are true, and $A$'s first statement is false, while the second is true.

Mogyorósi András (Vác, Sztáron S. Gimn., I. o. t.)