One question to know if the number is 1, 2 or 3

"I am thinking of a number which is either 0 or 1. Is the sum of our numbers greater than 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 $m$.

First of all I ruled out indirect ways of using reference to either of the numbers 1, 2 ,3 to frame a question, as I thought it's implicit in the question that it should challenge your thinking, not your cleverness. If she answers I don't know, compared to yes or no, it is more likely that she is confused between two numbers, ruling out one possibility. If the boy thinks of a number, the most common way to link it up to 1, 2, or 3 will be by divisibility. Also, I thought out of 1, 2, and 3, if I can rule out one number by the way I frame the question, I will be left with two options. The most common way of describing a number is whether it's odd or even.

So how about asking: "I am thinking of an odd number. Is it perfectly divisible by your number?"

If she says no, clearly the number is 2. If she says yes the number is 1 because only with 1 can you be sure that any number is divisible by 1. If she says I don't know the number is 3, because an odd number can or cannot be divisible by 3.