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Minority colour guesses

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Each of $6$ prisoners sits at a round table and gets a red or a black hat, all $2^{6} = 64$ patterns equally likely. Each prisoner sees the other $5$ hats and guesses that his own hat has the colour that is rarer among the $5$ he sees. How many of the $64$ patterns make all $6$ guesses correct? (With $5$ hats seen there are no ties.)

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Hint

Compare what a prisoner sees when the hats are split evenly with what he sees when they are not.

Solution

Suppose the pattern has $k$ red hats. A red prisoner sees $k - 1$ red and $6 - k$ black hats, so he guesses red exactly when $k - 1 < 6 - k$ , that is when $k \leq 3$ .
A black prisoner sees $k$ red and $5 - k$ black hats, and he guesses black exactly when $5 - k < k$ , that is when $k \geq 3$ .
If $k < 3$ a black prisoner exists and is wrong, and if $k > 3$ a red prisoner exists and is wrong, so everyone is right only for $k = 3$ .
The number of patterns with $3$ red hats among $6$ is $(3 6) = 20$ .

Answer: $20$

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