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A puzzle's scenario always involves multiple players with the same reasoning capability, who go through the same reasoning steps. According to the principle of induction, a solution to the simplest case makes the solution of the next complicated case obvious. Once the simplest case of the induction puzzle is solved, the whole puzzle is solved subsequently.
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. 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). 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.
Muddy children puzzle is the most frequently appearing induction puzzle in scientific literature on epistemic logic. In February 2020, 437 hits on Google scholar mentioned the muddy children puzzle. Muddy children puzzle is a variant of the well known wise men or cheating wives/husbands puzzles.
Hat puzzles are induction puzzle variations that date back to as early as 1961. In many variations, hat puzzles are described in the context of prisoners. In other cases, hat puzzles are described in the context of wise men.