The wizard has placed hats on the heads of three logicians. He explains that he has written a positive integer on each hat, and that one of the numbers is the sum of the other two numbers. Each person can see only the numbers on the other people's hats.

The wizard offers a prize to the first person who can logically determine the number. The wizard starts questioning the logicians in order, starting over again if nobody can state the number with certainty. No guessing is allowed.

(1) Prove that someone will always determine the number.
(2) Prove that will always be the person with the highest number.
(3) Find an explicit formula for the number of rounds needed until the number is found.