Overview:

Initially, we have cards labeled from 1 to 2n + 1. (2n + 1 is in the range of [3, 13].) Player A and B both take n cards, and one card is left on the table. (They do not know each other's cards, and the goal is to guess the card on the table.) In each turn, one player asks the other player: "Do you have all cards from x to y?" where y >= x. The other player must answer the question honestly. The system will check.

Players alternate in asking questions. A player may guess after receiving the answer saying something to the effect "I guess the card left on the table is i (for some i up to 2n+1)" The guessing player loses the game if his/her guess is wrong; otherwise, he/she wins. Each player takes the lead in one of the two rounds. If someone wins both rounds, he/she wins. If each wins once, the one who guesses correctly in fewer turns wins.

Example:
The interesting part: player A could mislead player B by asking player B if B has a card that A has.
For example, we have cards 1, 2, 3, 4, 5.
A has card 1 and 2.
B has card 3 and 4.
Thus, 5 is left on the table.

1st turn:
A: "Do you have card 1?"
B: "No."

2nd turn:
B: "Do you have card 2?"
A: "Yes."

3rd turn:
A: "Do you have card 4?"
B: "Yes."

4th turn:
B: "I guess the card left is 5."

Because B had 3 and 4, so B knew 1, 2 and 3 might be left on the table. And After A asked the card 1 after the first turn and got a negative response, A did not claim 1 was left on the table in the third turn. Thus, B knew A asked a card he had. So, he knew A has 1 and 2. Therefore, B is confident to say 5 is on the table in the fourth turn.