I came across this article https://jaydaigle.net/blog/what-is-the-axiom-of-choice/#fnref:7 and was wondering how would one mathematically express and explain the Uncountable Hat Puzzle (under the same heading on the page) since there are no definite ordered "steps", just a couple of assumptions? I have only started studying about equivalence relations and real coordinate spaces so i am not very sure how to go about this.
I would appreciate any and all help on this >:)