In an earlier version of this game Professor Rando had four students: Daphne, Max, Mindy and Sam (the same students as below, without Tim). In that version, if he selects his two integers uniformly from the range 1 to N, where N > 5, there is some chance that the game never ends.
But the professor has recently gotten a very bright new pupil, Tim, who wants to be included in the game. Rando decides the game will work similarly as before:
For some positive integer N, he generates two random integers independently and uniformly from 1 to N (inclusive), and then tells each of his students a different fact about the two numbers. He tells:
- Daphne the absolute difference of the integers,
- Max the maximum of the integers,
- Mindy the minimum of the integers,
- Sam the sum of the integers, and
- Tim the product of the integers.
Then, each day until the game ends, he congregates his students and asks them, in alphabetical order (Daphne, Max, Mindy, Sam, Tim), for the identity of the two integers (which two values were drawn; their order does not matter). Each student has only one chance to answer each day, when she or he is called upon. Each student answers ONLY when the answer is definitively known to him or her, and otherwise gives no answer that day. All of the students know N and all of these rules, all of them are perfect logicians, everything here is common knowledge, and there is no collusion. Once a student gives an answer (which will be correct), that student wins and the game ends.
What to submit: the smallest N for which there is some draw that makes the game go on forever. If the game always ends for every N, submit 0.
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