The logician at the back can determine whether there is an even or odd number of red hats in front of them, based solely on their own observation.
With the strategy established, each subsequent logician can then count the number of red hats visible and make an accurate deduction about their own hat color.
The first logician's answer provides a critical piece of information that allows all others to infer their hat colors based on the parity of red hats they see.
In the extended scenario with three colors, logicians can use a similar parity strategy, adjusting their responses to account for the additional color, ensuring at least nine correct answers.
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