Professor Rando is running a smaller version of his game with only three students.
For some positive integer N, he picks two integers independently and uniformly at random from 1 to N (inclusive). The two may be equal. He then privately tells each student one fact about the two numbers:
- Max is told the maximum of the two integers,
- Mindy is told the minimum of the two integers,
- Tim is told the product of the two integers.
Each day until the game ends, he gathers the students and asks them, in the order Max, Mindy, Tim, for the identity of the two integers (which two values were drawn; their order does not matter). Each student gets one chance per day, when called on. A student answers only when he or she knows the two integers for certain, and otherwise says nothing that day. All three are perfect logicians, all of these rules are common knowledge, and there is no collusion. The first student to answer (always correctly) wins, and the game ends.
For small N this game always ends. For some N it can go on forever, depending on which integers Rando draws.
What to submit: the smallest N for which the game might never end.