Countably infinite prisoners, numbered 1,2,3 etc. are told that tomorrow they will be given hats, colored either black or white, and lined up so that each prisoner can only see the prisoners whose hats are labeled with a greater number. The prisoners then get to guess which color hat they wear, in order from 1 onwards. If all but finitely many of them guess which color hat they have, the prisoners win and they are let free. The challenge: Describe a strategy which lets the prisoners go free. Assume as with all such challenges that the prisoners can't communicate in secret once they get their hats, they can see infinitely far to all the prisoners that are supposed to be visible to them, they have perfect memory and are perfectly logical and all those other things that people can't do in real life. Bonus points if you can create a strategy that places a specific upper bound on how many prisoners get their hat wrong.