It isn’t a given that all possible configurations are solvable, only that this problem takes place within a solvable configuration. We aren’t told how many logicians there are, or how many colors, or how many of each color. So it isn’t problematic to say that some configurations would leave the logicians unable to deduce their color, while the problem itself is still solvable. Just like a configuration where all the logicians had the same color headband should not be considered because it would violate the premise that there are many colors, so too would an unsolvable configuration not be considered for violating the premise that the problem is not impossible to solve.
To you hypotheticals:
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I think that’s an interesting shade of purple you chose; I certainly wouldn’t have used it. My mental idea of the color wheel has a much deeper purple. I might be tempted to fill in the last dot with orange, but are you and I thinking of the same orange? I’d say this problem is unsolvable, because there is still an infinite (though bounded) set of oranges to choose from to complete the pattern.
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I’m on the fence about whether this one is solvable. I think there’s a problem in that, though there seems a definite pattern suggested, there are in fact an infinite number of patterns that satisfy this. 2R, 2B, 2Y, 2R, 2B, 2Y, 2R, 2B, 1Y, 1G is a pattern, there doesn’t seem any logical reason to reject it (except that G would have no way to guess that she G and not O or P).
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Let’s assume the empty circle is Green, and ask ourselves, what would Yellow see? 2O, 3R, 4B, 5G. Is Yellow thinking, “I must be 1 Purple!” Or maybe “I must be G, so that all of the groups have a number that is divisible by 3 or 4!” It looks to me like this one is unsolvable too. We can therefore rule it out.