fivethirtyeight wrote:You and six friends are on a hit game show that works as follows: Each of you is randomly given a hat to wear that is either black or white. Each of you can see the colors of the hats that your friends are wearing but cannot see your own hat. Each of you has a decision to make. You can either attempt to guess your own hat color or pass. If at least one of you guesses correctly and none of you guess incorrectly then you win a fabulous, all-expenses-paid trip to see the next eclipse. If anyone guesses incorrectly or everyone passes, you all lose. No communication is possible during the game — you make your guesses or passes in separate soundproof rooms — but you are allowed to confer beforehand to develop a strategy.

What is your best strategy? What are your chances of winning?

Extra credit: What if instead of seven of you there are 2N−1?

It seems sort of obvious that no strategy can beat assigning one person to guess arbitrarily and the rest to pass, giving a 0.5 probability of winning. After all, if hats are assigned randomly and no communication is allowed, it is impossible to get any information about your own hat whatsoever. If more than one person guesses, the odds get strictly worse. What am I missing?

I've seen problems similar to this before, but generally there is *some* sort of communication involved, like allowing one player's decision to be known to other players. But here, there is *nothing*. How can seeing other hats be useful if they are all random?