In the previous quiz, we saw examples of **hat puzzles** with two or three people--i.e. puzzles in which each person has a hat they cannot themselves see, and need to guess the color of based on the hats they can see (which may be everyone else's, or only some people's). This quiz deals with the case when there are more than three people, and more generally when there are an arbitrary number of people.

Usually, puzzles with large numbers of people are cooperative rather than competitive, meaning that the people work together to guess correctly as many times as possible. **If people give responses in order, then the group can agree that a certain response has a certain meaning.**

**the hats on the people in front of them**, but not their own. One by one starting from the **back** of the line, they are called upon and asked to guess the color of their hat. If the people work together using the optimal strategy, how many people are guaranteed to guess correctly?

Up until now, we've only witnessed scenarios in which we want to *guarantee* players get their guesses correct. But in practice this is not always possible; for instance, the last few problems have shown that the first guesser usually can't be right. Thus we are often concerned with *probabilistic* methods, i.e. those in which the players try to get their guesses correct with high probability.

**At the same time** and only once, all 3 people guess the colors of their hats or choose to pass. The players win if at least one player guesses correctly and nobody guesses incorrectly. If the people work together using the optimal startegy, what is the probability they win the game?

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