I think that's a great question, but I would guess the answer is no, there are no other solutions. One important axiom for this type of puzzle is that everyone

I think that's a great question, but I would guess the answer is no, there are no other solutions. One important axiom for this type of puzzle is that everyone does deduce every piece of information that they can mathematically deduce with 100% certainty. That would eliminate the "just guess" and "guess because your friend will not play along" type of answers some are giving in the thread. With this, my hand-wavy reasoning for not being any other solution is that when you try to model the knowledge the players possess after each turn (or equivalently, the state of the game), there are no branches or decisions they could make except for the STOP condition of the game. In each turn, they can (and because of the "axiom" above: they DO) eliminate some numbers from the set of possible number of statues the other player could be seeing if they passed in their round. Once the set goes down to one, they KNOW the answer. Since there are no choices [1] they could possibly make to alter the game before it ends, I don't see how there could be an alternate solution. [1] I'm discounting the possibility that they insert a constant pattern of extra passes in the game. (E.g., "We will both pass 5 times and then deduce the number based on the strategy described above.")