Seen from a distance, two inchworms that approach each other and then turn around when they come in contact are indistinguishable from two inchworms that pass each other unobstructed.

Therefore, we can get an upper bound on the time required by assuming the 1000 inchworms all pass each other unobstructed. The time for all of the inchworms to get off the rubber band is no greater than it is for one of them.

Andrew Johnson of the Colorado School of Mines had the following interesting observation. Suppose the inchworm repeats this process: "Walk one inch, see if another inchworm is in the same place as me, and, if so, turn around." If two inchworms start out in the same place, they can go into an infinite loop, turning around and re-encountering each other at every iteration.

Before you conclude that something this ridiculous could only happen in an idealized world cooked up by mathematicians, consider an experiment conducted by the 19th century French naturalist, Jean Henri Fabre on processionary caterpillars. These are caterpillars that have an instinct to follow each other in long lines, with each caterpillar walking in lock step with the one in front of it. Fabre put a chain of them around the rim of a pot that was filled with caterpillar food. The caterpillars followed each other round and round the rim of the pot for seven days, at which point they dropped off the pot due to exhaustion and starvation.

Amusingly, a Google search on "processionary caterpillars" turns up many sites offering up Fabre's experiment as a lesson about human existence.

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