A boy thinks of a number that is \(1\) , \(2\) or \(3\) and a girl is then only allowed to ask one question about that number. The boy can only answer " Yes ", " No ", or " I don't know" . Through a clever questioning, the girl manages to name the correct number that the boy was thinking of after the boy has answered her. What is your question?
A possible solution is: " I'm also thinking of one of these numbers. Is your number raised by my number greater than \(2\) ? " Namely, let \(n\) be the boy's number (which the girl does not know) , and \(m\) the number of the girl (which the boy does not know). Then:
$$n=1 \Rightarrow \text{Nein: } 1^m = 1 \ngtr 2 \,\, \forall \, m \in \{ 1, 2, 3 \}$$ $$n=2 \Rightarrow \text{Ich weiß es nicht: } \text{Ob } 2^m > 2 \text{ hängt von } m \text{ ab}$$ $$n=3 \Rightarrow \text{Ja: } 3^m \geq 3 > 2 \,\, \forall \, m \in \{ 1, 2, 3 \}$$