This is actually a question I find really hard to answer.any hints are appreciated. By the way feel free to edit the tags as i really do not know which category is this question is in.
Two genius mathematician have guessed two numbers which their difference is 1. Each of them asks the question "Have you guessed my number yet" repeatedly after the other one.
Prove that there will be a question which's answer would be "yes"
Suppose I know the numbers are positive integers.
If my number is $1$, and I am asked whether I know the other, I say "yes" because it is $2$. Else I say "no". If my colleague also says "no" the pair cannot be $(1,2)$ or $(2,1)$.
On the next round we confirm or exclude either $(2,3)$ or $(3,2)$ etc.
If I have $n$ it is impossible to exclude both $(n, n-1)$ and $(n, n+1)$ so the process must come to an end.