Naked Science Forum
Non Life Sciences => Geek Speak => Topic started by: varsigma on 16/11/2023 23:32:50
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Since this is a problem in logic I thought this place would be appropriate.
So here goes.
The traditional version of this puzzle is as follows.
A certain king wishes to determine which of his three wise men is the wisest. He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their foreheads but that at least one spot will be white. He then offers his favor to the one who will first tell him the color of his spot. After a while, the wisest announces that he knows the color of his spot. How does he do this?
In my variation, the three men are blindfolded first, the spot of paint is applied, then the blindfolds are removed. One of the three men keeps his eyes closed and "after a while" announces he knows the color of his spot. How, using similar logic for the traditional version (which I have intentionally not supplied) does it work out?
A hint, it relies strongly on reasoning about what other wise men would reason, given the situation described.
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White.
We assume it is a fair trial.
Therefore knowing only that at least one spot is white must be sufficient information to deduce what color I am wearing.
Therefore I must be wearing white.
No need to think about what the other guys are thinking. Indeed I could have told you even before you applied the paint.
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Yeah.
Ultimately, the first of the three to reason that the only fair trial is if all of them have the same color dot can say they know. Assuming the king wouldn't waste their time with an unfair trial.
So the logic is, there must be one, two, or three white dots. But the first case isn't fair because the man with a white dot will see two black dots and immediately know his color.
Two white and one black dots isn't fair either, for similar reasons. In fact all three men should see that it isn't much of a test. . .
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So here's an old one in a similar vein.
Watson: "I say, Holmes, there's an intriguing chess problem in The Times today. White to play and mate in one move."
Holmes "Then the move is...."
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So I thought about a variation of the puzzle, which demonstrably isn't much of a test (for wise men), such that the wise men are tested. The test is that they have to figure out why the king has summoned them, sat them in a room facing each other, blindfolded them and then painted a dot on their forehead.
Let's assume that the blindfolds are left on "for a while" before being removed. The king says nothing to any of them apart from welcoming them to his palace. So each wise man has to figure out what's going on.
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Easy.
No absolute ruler likes to have too many wise men around. As can be seen from the current COVID enquiry, the Post Office Horizon scandal, Boeing black boxes, and the Church's treatment of Bruno, Galileo, et al, expert opinion can be embarrassing so must be suppressed. You can suborn one wise man if he disagrees with you, but it is the essence of N > 1 wise men that they argue with one another in public*, so the truth will out eventually.
So when N = 3, you set them a test by blindfolding them, quietly removing the two who disagree with you, then blaming the third for their disappearance. Nobody will believe him when he says he was blindfolded, and why would he, if he is wise, alive, and promoted to the House of Lords?
*Old Jewish proverb: if you want three opinions, ask two Jews.