A riddle involving the axiom of choice
So i found a riddle on reddit which is a consequence of the axiom of choice.
There is a house with 100 rooms, and each room contains countably many boxes indexed with the natural numbers. Each box contains a random real number, which is the same over all the rooms (that is, box n contains the same real number in every room).
100 set theorists play a game. Each person will go into a unique room and open as many boxes as they like (perhaps countably many) as long as they leave at least one box in their room unopened. Then, each of them need to pick an unopened box in their room, and guess what real number is inside of it.
In order to win, 99 of them need to guess correctly.
The mathematicians can discuss a strategy beforehand, but after they go into their respective rooms, no more communication is allowed. What is a 100% winning strategy for this seemingly impossible task?
The solution is here: https://www.reddit.com/r/math/comments/rpbhos/predicting_random_real_numbers_with_the_axiom_of/?utm_source=share&utm_medium=ios_app&utm_name=iossmf
My question is why can the mathematicians agree on a specific representative for each equivalence class? I thought that the axiom of choice guaranteed a choice function existed, but didnt specify it. But for the solution to the riddle, they would need to know what the choice function is. Can someone explain?