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Seven Heads, Three Colors

题目

7 people sit in a circle, each wearing a hat colored red, green, or blue (assigned independently and arbitrarily). Everyone sees all hats except their own. They must all SIMULTANEOUSLY write down a guess of their own hat color (no information exchange after seeing the hats). They agree on a strategy beforehand. Using the optimal modular-sum strategy, what is the maximum number of the 7 people they can GUARANTEE will guess correctly, no matter how the hats are assigned?

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