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The warden listens in. Based off your algorithm, he can change which prisoner goes in when. However, every prisoner will visit an infinite number of times in an infinite period of time.
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My solution works for this 
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"Life is possible only with illusions. And so, the question for the science of mental health must become an absolutely new and revolutionary one, yet one that reflects the essence of the human condition: On what level of illusion does one live?"
-- Ernest Becker, The Denial of Death
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