You may have noticed from December’s puzzle that if Professor Rando 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, for the identity of the two integers. 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 this, and there is no collusion. Once a student gives an answer (which will be correct), that student wins and the game ends.

Will this game always eventually end? If so, submit “yes”. If not, submit the smallest N such that the game might not necessarily ever end.