Puzzle: Infinite Hats Problem
February 24, 2014 § 2 Comments
For those of you who are unaware, I have been in Budapest for a little over a month, and I have been quite a bit busier than I expected. I know I have been much worse than I had intended about keeping up with this blog, but I heard a great puzzle the other day, and it is quite exciting. Here it is:
There are infinitely many logicians (one for each integer), and an adversary tells the logicians that he will place a hat on each of their heads. He tells each logician that each hat will either be red or blue. He then tells each logician that, once he has placed a hat on each of their heads, the logicians may see the color of all the hats that are not their own, and then they must attempt to guess their own hat color. The logicians win if only a finite number of them guess incorrectly. Is there a strategy that will allow the logicians to win if the adversary knows their strategy?
Just to make sure the problem is clear, each logician gets a different hat. At no point do they get to see their own hat color nor do they receive any information from any of the other logicians once they have received their hat. All they can see is the color of the hat placed on each other logician’s head.
Thanks for reading!
-A Student of Logic
A Student of Logic 
You got to spend a month in Budapest! Wow I envy you. The three days I spent there a couple of years ago were absolutely wonderful.
About the puzzle, can’t only a finite number of people make a guess at all, let alone an incorrect guess?
As of this point, I have gotten to spend almost four months in Budapest, though my time here will come to an end in a few weeks. I agree, Budapest is quite wonderful.
I am not sure why you believe that only a finite number of people can make a guess. The idea is that each of the logicians guess simultaneously, and we would like to ensure that only a finite number of these guesses are incorrect.