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Imagine a perfectly smooth sphere the size of the Earth with a ribbon tied snugly around its "equator." If you cut the ribbon and add another 12 inches of ribbon, then shake the newly elongated ribbon to distribute it evenly around the sphere, it clearly would be off the sphere's surface; but how far off the surface? If you slid a stack of as many free pizza coupons as you could between the ribbon and the sphere's surface (say, 100 coupons to the inch), after redeeming the free pizza coupons, how long will it take you to eat all the pizza?
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