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Hats with numbers on them

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Each of $5$ prisoners, numbered $0$ , $1$ , $2$ , $3$ , $4$ , gets a hat showing a whole number from $0$ to $5$ , and all $6^{5} = 7776$ assignments are equally possible. Each prisoner sees the other $4$ hats but not his own. Prisoner $i$ guesses that his own number is the one that makes the total of all $5$ hat numbers leave remainder $i$ when divided by $6$ . For how many of the $7776$ assignments is at least one guess correct?

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Hint

A guess is correct exactly when the true total has a particular remainder. For which remainders of the total can someone be right?

Solution

Prisoner $i$ is right exactly when the true total of the $5$ numbers leaves remainder $i$ on division by $6$ .
So someone is right exactly when that remainder is one of $0$ , $1$ , $2$ , $3$ , $4$ , that is, when it is not $5$ .
The remainders of the total are equally spread over the $7776$ assignments, with $1296$ assignments for each of the $6$ remainders.
The number of assignments with at least one correct guess is therefore $5 \times 1296 = 6480$ .

Answer: $6480$

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