One question to know if the number is 1, 2 or 3
Solution 1:
"I am thinking of a number which is either 0 or 1. Is the sum of our numbers greater than 2?"
Solution 2:
"I'm also thinking of one of these numbers. Is your number, raised to my number, bigger than $2$?"
Let $n$ be the girl's number (unknown to me), and let $m$ be my number (unknown to her).
- $n = 1 \implies $ NO: $1^m = 1 \not > 2$ for all $m \in \{1,2,3\}$.
- $n = 3 \implies $ YES: $3^m \geq 3 > 2$ for all $m \in \{1,2,3\}$.
- $n = 2 \implies $ I DON'T KNOW: Whether $2^m > 2$ depends on $m$.