Alice, Betty, and Charlie are all wearing a hat, and each hat is either red or blue. They also know that overall there are **two blue hats** and **one red hat**.

Alice cannot see her own hat, but sees that Betty is wearing a red hat and Charlie is wearing a blue hat. What color hat is Alice wearing?

Alice, Betty, and Charlie are all wearing a hat, and each hat is either red or blue. They also know that there were originally two red hats and two blue hats (so one hat is unused). Each person can see the hats everyone else is wearing, and not their own.

Assuming they don't speak, which of the following must be true?

In the previous problems, the people were able to infer information about their hats from just seeing the hats of others. Usually, this will not be the case, and the people will have to have a coordinated strategy before the hats are placed.

Hat puzzles are generally divided into two main categories: puzzles in which *at least one* person must guess their hat color, and puzzles in which *everyone* (or almost everyone) must guess their hat color. Usually, in the first category people will guess simultaneously (so that they cannot encode any information in their guesses), while in the second category, the people will guess in some order (so that they can get further information from others' guesses).

Alice and Bob are each wearing a hat, and each hat is either white or black. They can both see each other's hats, but cannot see their own. **In order**, Alice will guess her hat color, then Bob will guess his hat color. They win if either Alice or Bob correctly guesses their hat color. With a perfect strategy, what is the probability they win the game?

NOTE: Alice and Bob can plan their strategy before the game starts.

Alice and Bob are each wearing a hat, and each hat is either white or black. They can both see each other's hats, but cannot see their own. **At the same time**, Alice will guess her hat color, and Bob will guess his hat color. They win if either Alice or Bob correctly guesses their hat color. With a perfect strategy, what is the probability they win the game?

NOTE: Again, Alice and Bob can plan their strategy before the game starts.

This illustrates a general strategy for "at least one" problems (problems in which only one person needs to guess their hat color). The possible sets of hat colors are divided up into different categories, and each player guesses according to their category.

For instance, in the previous problem the two categories were "Alice and Bob have the same hat color" and "Alice and Bob have different hat colors." Since one of these must be the case, if Alice guesses accoring to the first category (meaning she guesses the same as Bob's hat color) and Bob guesses according to the second category (meaning he guesses the opposite of Alice's hat color), one of them must be right (and, of course, one of them must be wrong!).

Alice, Bob, and Charlie are each wearing a hat, and each hat is either red, green, or blue. They can all see each other's hats, but cannot see their own. At the same time, the three people guess their hat color. They win if any of them correctly guesses their hat color. With a perfect strategy, what is the probability they win the game?

(Note: There is not necessarily one of each color; red, green, and blue are just the possible color choices.)

The final class of hat puzzles are **competitive** versions, in which a person wins if they are the *first* person to correctly guess their hat color. This opens up a new element to the strategy, as now the *absence* of a guess (i.e. silence) can give information as well.

Alice, Betty, and Charlie are each wearing a hat, each hat is either black or white, and each speaks simultaneously: "yes" if they see a black hat, and "no" if they do not (as usual, each person can see all hats except their own and know what hats are possible).

All three people say "yes."

They are then asked if they know their hat color, and again speak simultaneously: "yes" if they know, "no" if they do not.

All three people say "no."

What is the distribution of hats?

Alice, Betty, and Charlie are each wearing a hat, and the hats are taken from a pool of 3 white and 2 black hats (so two hats are unused). The three people sit in a line, so that each person can see the hats of the people in front of them but not the hats of the people behind them (or their own). More specifically, Charlie can see Betty's and Alice's hat, Betty can see Alice's hat, and Alice cannot see any hats. The winner is the first person to correctly guess their hat color.

Alice declares she doesn't know her hat color, followed by Charlie and Betty (in that order).

Given that each of these 3 has made their declaration, one of them is able to conclude what color hat they are wearing, and win the game . Who is it? And what color is their hat?

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