Math puzzle with a surprising answer
16 mathematicians are imprisoned by the evil emperor. They are told that in the morning they will each be assigned a hat, either white or black. Each hat is assigned randomly (50/50) and independently. Each mathematician will be able to see everyone else’s hat but not their own. After one minute to observe the hats, they will simultaneously be sent back to their private cells where they have to guess the color of their own hat. If a single mathematician guesses incorrectly, they will all be executed. If all 16 guess correctly, they go free.
What is the probability that the mathematicians will be set free, and how should they guess?
If you’d like a hint, look at this problem. Thanks to Dwight for suggesting more complex problems in that spirit; if this isn’t hard enough I’ll add another twist next week.