Koan meaning

kō'än'
A puzzling, often paradoxical statement, anecdote, question, or verbal exchange, used in Zen Buddhism as an aid to meditation and a means of gaining spiritual awakening.
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In Zen Buddhism, a verbal puzzle put to a student as a means to enlightenment.
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(Zen Buddhism) A story about a Zen master and his student, sometimes like a riddle, other times like a fable, which has become an object of Zen study, and which, when meditated upon, may unlock mechanisms in the Zen student’s mind leading to satori.
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A riddle with no solution, used to provoke reflection on the inadequacy of logical reasoning, and to lead to enlightenment.
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Origin of koan

Japanese kōan public (from Middle Chinese kəwŋ) (also the source of Mandarin gōng) an matter (from Middle Chinese ʔan`) (also the source of Mandarin àn)