The One Or The Plus One
You give Alice and Bob each a closed box containing a number. They both know that the two numbers are consecutive positive integers, but do not know the opponent's number. Then you give them a blank card and a pencil each, and have them play the following game:
- In each turn, they could predict what the number in their opponent's box is by writing it down on the card. Whoever can do so wins. They are only allowed to do this when they know the answer.
- Or, they could choose not to play the turn and exchange blank cards, indicating that they do not know the answer yet as of that turn.
Given that the two integers given to Alice and Bob are 16 and 17, respectively, who wins and after how many turns?
Clarification: Alice/Bob only attempts to answer if they know the correct answer. This is not a guessing game.