The king gave his word to the wise men that at least one of them was wearing a blue hat in other words, there could be one, two, or three blue hats, but not zero. He placed a hat on each of their heads, such that each wise man could see all of the other hats, but none of them could see their own. The King called the three wisest men in the country to his court to decide who would become his new advisor. The King's Wise Men Hat Puzzle Description It can be proven that X times the minor penalty is still better than the big penalty. If Charly sees that Alice and Bob are muddy and not stepping forward at the second stroke, they all together will step forward at the third stroke. If there are less than 3 muddy children, the puzzle evolves like in the case with 2 children. Let's assume that there are just 3 children: Alice, Bob, and Charly. If Alice sees Bob not stepping forward at the first stroke, she must conclude that he certainly sees another muddy child and they will step forward simultaneously at the second stroke. If only Alice is dirty, she will step forward at the first stroke, because she does not see any other dirty faces. Let's assume that there just 2 children: Alice and Bob. Hereafter, the custodian starts to count and after every stroke, every muddy child has an opportunity to step forward. The children are each told that they should step forward if they know that their own face is muddy. A custodian tells the children that at least one of them has a muddy face. But, every child knows the state of all other children's faces. None of the children can determine the state of their own face themselves. The children consider it possible to have a muddy face. 13 Countably Infinite Hat Problem with Hearing.12 Countably Infinite-Hat Variant without Hearing. In other cases, hat puzzles are described in the context of wise men. In many variations, hat puzzles are described in the context of prisoners. Hat puzzles are induction puzzle variations that date back to as early as 1961. Muddy children puzzle is a variant of the well known wise men or cheating wives/husbands puzzles. In February 2020, 437 hits on Google scholar mentioned the muddy children puzzle. Muddy children puzzle is the most frequently appearing induction puzzle in scientific literature on epistemic logic. Also, the inaction of a participant is a non-verbal communication of that participant's lack of knowledge, which then becomes common knowledge to all participants who observed the inaction. Also, usually, some kind of hint is given to suggest that the participants can trust each other's intelligence - they are capable of theory of mind (that "every participant knows modus ponens" is common knowledge). Typical tell-tale features of these puzzles include any puzzle in which each participant has a given piece of information (usually as common knowledge) about all other participants but not themselves. Once the simplest case of the induction puzzle is solved, the whole puzzle is solved subsequently. According to the principle of induction, a solution to the simplest case makes the solution of the next complicated case obvious. Ī puzzle's scenario always involves multiple players with the same reasoning capability, who go through the same reasoning steps. Induction puzzles are logic puzzles, which are examples of multi-agent reasoning, where the solution evolves along with the principle of induction.
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