King and Hats famous puzzle

BLUE
The King’s Wise Men: This is one of the simplest induction puzzles and one of the clearest indicators to the method used.

• Suppose that there was one blue hat. The person with that hat would see two white hats, and since the king specified that there is at least one blue hat, that wise man would immediately know the color of his hat. However, the other two would see one blue and one white hat and would not be able to immediately infer any information from their observations. Therefore, this scenario would violate the king’s specification that the contest would be fair to each. So there must be at least two blue hats.
• Suppose then that there were two blue hats. Each wise man with a blue hat would see one blue and one white hat. Supposing that they have already realized that there cannot be only one (using the previous scenario), they would know that there must be at least two blue hats and therefore, would immediately know that they each were wearing a blue hat. However, the man with the white hat would see two blue hats and would not be able to immediately infer any information from his observations. This scenario, then, would also violate the specification that the contest would be fair to each. So there must be three blue hats.

Since there must be three blue hats, the first man to figure that out will stand up and say blue.

Source : https://en.wikipedia.org/wiki/Induction_puzzles

The answer is blue.

The king hasn’t decided which he wants to make his advisor so all the sages have to see the same thing to make it a fair test.

If his hat were blue both B and C would see two people with blue hats and cannot say which colour their hat would be due to the fact that the king did not specify about how many hats would be of each colour.

If his hat were white then, again, neither B nor C can deduce what colour their own hat is as the king didn’t specify as I said above.

If sage A had a white hat, then using your words, Sage A would be “Unique” but since the King cannot decide which Sage to chose it would have to be a fair test.

I may be wrong, but after thinking about it, this is the only logical explanation I could reach.

A’s hat must be blue.

If it was white then either of B or C (who we are told are wise sages and presumably also reasonable logicians) would be able to work out that they have a blue hat.

Here’s how: If A’s hat is white then B can reason that if his own hat was also white, then C would see two white hats. Since they have all been told that at least one of them will be given a blue hat, C would know immediately that his must be the blue one. Since C has said nothing “for hours” B should be able to conclude quite quickly that his own hat was also blue.

Since B has also not said anything for hours, A must eventually conclude that the premise that his own hat is white is false and that he also has a blue one (although if these sages were any good they would reach this conclusion after a few minutes).

The logic fails if the reason for the other sage’s silence is because they have fallen asleep with their eyes open.

Sage A’s hat is white.
It cannot be blue.

1. The puzzle says that “Sage A sees that the other 2 sages are wearing blue hats.” So both sage B and sage C wear blue hats. That’s a given.

2. Neither B or C can deduce the color of their own hats, and therefore, they remain silent.

Here’s why:
If A’s hat were white, B would see C’s blue hat, and see A’s white hat. B’s hat could be either blue or white.

If A’s hat were blue, B would see C’s blue hat, and A’s blue hat. B’s hat could still be either blue or white.

If A’s hat were white, C now would see B’s blue hat, and see A’s white hat. C’s hat could be either blue or white.

If A’s hat were blue, C would see B’s blue hat, and see A’s blue hat. C’s hat could still be either blue or white.

So Sage B and C cannot speak up for sure.

3. After a while, when Sage A sees that neither sage B or sage C speak up, (and they speak up if they could solve the ridde), sage A reasons that neither sage B nor sage C, based on what they see, can deduce the color of their own hats.

Sage A reasons further that since one of the three sages should be able to speak to the king, sage A alone must be the one able to figure it out, since the other two cannot.

There must be something that makes him unique, something that he sees that the other two don’t see, something that he sees that makes the riddle solveable for him, while the other two cannot solve it. What does he see?
He sees two blue hats (B’s blue hat and C’s blue hat). So Sage A again reasons correctly that only he sees two blue hats. Since sage B’s hat is blue and sage C’s hat is blue, Sage A correctly reasons that his own hat cannot be blue. (Or else all three sages would see the other two sages wear blue hats).

So A’s hat must be white, and so A speaks up.

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Note:
If A’s hat were blue, all three sages would be wearing blue, each sage would see the other two wearing blue hats, and there would be nothing to differentiate them, with all three silent. Nobody would speak up and they would all go home.