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A koan is a puzzle-like question or story that's designed to help one obtain enlightenment when worked on in meditation when assigned a koan by one's teacher. A classic example is "What is the sound of one hand clapping?"
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A Koan is used to help the student circumvent his dependance on logic to arrive at answers. It promotes the instantaneous appreciation of the whole.
"Standard" answers are available for many Koans (Q: Does a dog have a Buddha nature? A: Mu) but the answers are as much Koans as the questions. In this case Mu (not thinking) may apply to the dog implying it is not a sapient being, that a dog achieves the Buddha nature because he is one with the flow of life without thinking, that the question/answer is without value, or that you don't even have to think about the answer.
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Koan means public record. Koans became teaching tools collected and recorded in Sung Dynasty China. The encounters happened earlier usually in Tang Dynasty China between teacher and student, usually resulting in the student having an insight (satori).