Setup

Mathematician S is thinking of a number which is either 1, 2, or 3. Mathematician P can ask S only one question to determine which number S is thinking of. S can only answer "Yes", "No", or "I don't know" to the question posed by P.

Question

What question might P ask to determine which number S is thinking of?

Answer

There are many questions P might ask to determine S's number. I'll share what I'd ask S: If n is an odd integer larger than any you've divided any number into, is n divisible without remainder by the number you're thinking of? Let m be S's number. There are three options:

1. S knows n is divisible without remainder by m iff m is 1
2. S knows n is not divisible without remainder by m iff m is 2
3. S does not know whether n is divisible without remainder by m or not iff m is 3

To see other question P might ask, check out the forum where I came across the puzzle here.