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Suppose that, before the he starts out, we mark out one-inch segments on the rubber band with a pen. Because of stretching of the rubber band, when he takes his ith step, the marks are i inches apart. Therefore, his ith step covers 1/i one-mark units.
After n steps, he has covered 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n one-mark units. This is the harmonic series, which is unbounded as n goes to infinity, so he will eventually reach his destination: mark number 5280*12.
(Note, since this sum is logarithmic in n, the number of steps he must take is exponential in the initial length of the rubber band, and he will be very tired when he gets there. If instead of expanding by a mile, the rubber band doubles in length each time, the inchworm will never even reach the second pen mark.)
Correct solutions: Ben Joeris (Fort Collins High), Micaela Nofsinger (undergrad), Tim O'Connor, Gautam Gupta, Steve Benoit, Yuxiang Liu, Stew Crawford (grad students), Ben Manvel (faculty), Florian Hulpke (grad student, University of Hannover). Kevin Nibbelink had a novel solution: the inchworm should turn around backward on the rubber band, hang on very tight, and bite through it.
Micaela Nofsinger gets the ice cream. Congratulations Micaela!