Caveat: this post has nothing to do with mice as in set theory…
I ran into the following puzzle lately and it took me a while to come up with a solution:
Suppose there are countably many mice standing in the following sequence: Mouse 0, Mouse 1, Mouse 2, and so on. Each mouse is given a hat which is either red or blue. Each mouse can see the hat color of every mice standing after it, but not their own, nor anyone standing before. Now, each mouse have to guess their own color. The mice collectively win the game if all but finitely many of them guesses correctly. Find a winning strategy for the mice. (Hint: Axiom of Choice.)
If you haven’t seen this, think a bit on your own!
Ever since the release of ChatGPT, I got interested in observing how generative AI interacts with logical and mathematical tasks. So I want to see ChatGPT’s performance in solving this problem. I am using ChatGPT3.5, and here is what I get.
If you read til the end of our conversation, you already know the solution, which is rather transparent by hindsight — modulo the equivalence relation of “equality at all but finitely many points”, each mice get the same information, namely the equivalence class of the input sequence, and they win if they collectively output a sequence in the same class. Now, before the game they fix a representative for each class (by an application of the Axiom of Choice). After the hats are given, everyone observes that they are in some class \([r]\), where \(r\) is the common representative chosen beforehand. They then win the game by collectively playing \(r\), i.e. Mouse \(N\) plays the \(N\)th digit of \(r\).