Alice: Let's play a game, Bob!
Bob: (sighing) What is it this time?
Alice: I'm thinking of a polynomial $P$ with integer co-efficients. For any integer $n$, you can ask me what $P(n)$ is. Then, if, after a while, you guess correctly what $P(2.018)$ is, you win! Otherwise, I win.
Bob(oquack?) searches up the ever-handy PSE.
Bob: We did this just over two years ago!
Alice: Did we now? Oh well, I suppose you'll have an advantage. So what are you complaining about? Let's play!
Bob: (sighing again) Okay… what's $P(1)$?
Alice: It's 1.
Bob: Hmm… what's $P(10)$?
Alice: It's 10.
Bob: Then using my astonishing powers of deduction, I conclude that $P(2.018)=2.018$.
Alice: You're wrong! I choose $P(x)=x^2-10x+10$, giving $P(2.018)=-6.10768$.
Bob: But you said they were pos... oh, you didn't, did you.
Alice: Want to play again?
Can Bob still be guaranteed to win? If so, what is the fewest number of moves in which winning is always possible?