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This feels like it should be a common puzzle, but I haven't found anything exactly like it searching online, so here goes. Apologies if it's a duplicate.

Three logicians, person 1, 2, and 3, along with a host, are playing a game. The host places one of three hats, each with a positive integer label A, B, or C, on the head of each player. All the player's know at the start is that A = B + C. They do not know who is wearing A, B, or C. The players can look at each other's hats, but they cannot see their own hat. The host asks each person in order(starting with person 1, then 2, then 3), what number is on their hat. If they know their own number, they say it, otherwise they say "I don't know" and the host moves on to the next player. If the host reaches person 3 and no one has stated their number, he starts again from person 1 and so on. The game ends when one of the players correctly states the number on their own hat.

Assuming no one makes guesses unless they are sure, show that the person wearing hat A can always win.

Bonus: Give a method for determining how may iterations(asking person 1, then 2, then 3, is one iteration) the host must make for a given pair B C before someone can answer correctly.

This feels like it should be a common puzzle, but I haven't found anything exactly like it searching online, so here goes. Apologies if it's a duplicate.

Three logicians, person 1, 2, and 3, along with a host, are playing a game. The host places one of three hats, each with a positive integer label A, B, or C, on the head of each player. All the player's know at the start is that A = B + C. They do not know who is wearing A, B, or C. The players can look at each other's hats, but they cannot see their own hat. The host asks each person in order(starting with person 1, then 2, then 3), what number is on their hat. If they know their own number, they say it, otherwise they say "I don't know" and the host moves on to the next player. If the host reaches person 3 and no one has stated their number, he starts again from person 1 and so on. The game ends when one of the players correctly states the number on their own hat.

Assuming no one makes guesses unless they are sure, show that the person wearing hat A can always win.

Bonus: Give a method for determining how many iterations(asking person 1, then 2, then 3, is one iteration) the host must make for a given pair B C before someone can answer correctly.