There are many longstanding unsolved problems in mathematics for which a solution has still not yet been found. The unsolved problems in statistics are generally of a different flavor; according to John Tukey, "difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." A list of "one or two open problems" (in fact 22 of them) was given by David Cox.
Read more about Unsolved Problems In Statistics: Inference and Testing, Experimental Design, Problems of A More Philosophical Nature
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“The child knows only that he engages in play because it is enjoyable. He isn’t aware of his need to play—a need which has its source in the pressure of unsolved problems. Nor does he know that his pleasure in playing comes from a deep sense of well-being that is the direct result of feeling in control of things, in contrast to the rest of his life, which is managed by his parents or other adults.”
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—Bruno Bettelheim (20th century)
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—Andrew Lang (1844–1912)