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OrphanWilde comments on The Doubling Box - Less Wrong Discussion
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As long as you comprehend the number in the box, you're not in an infinite loop. The input is different. Once the number is no longer meaningful, you're potentially in an infinite loop; the input is the same.
I'm pretty sure I could stay out of an infinite loop much longer than I could comprehend what was in the box. The contents of the box are growing exponentially with the number of days. If I just count the number of days, I can stay in the realm of small numbers much longer.