Professor Rando picks two integers independently and uniformly at random from 1 to 4 (inclusive). All 16 ordered outcomes are equally likely. He then privately tells each of three students one fact about the two numbers:
- Hugo is told the maximum of the two integers,
- Delia is told the absolute difference of the two integers,
- Stan is told the sum of the two integers.
Each day, until the game ends, the professor gathers the students and asks them one at a time, always in the order Hugo, then Delia, then Stan, to name the two integers. "Naming the integers" means naming the unordered pair (for example, "1 and 3"); nobody has to say which number was drawn first.
Each student gets one chance per day, when called upon. A student answers only if he or she knows the pair for certain; otherwise that student stays silent for that day. Everyone hears who stays silent, everyone knows which kind of fact (maximum, difference, sum) each student was told, and everyone knows these rules; all students are perfect logicians, and there is no collusion. The first student to answer (always correctly) wins, and the game ends. The professor draws his numbers once, at the start, not each day.
What is the probability that each of Hugo, Delia, and Stan wins?
Give the three winning probabilities as exact fractions, separated by commas, in the order Hugo, Delia, Stan.
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