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Problem of the Week 996

Creeping Ants

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At a certain time, 101 ants are placed on a one-meter stick, with one of them, Alice, placed at the exact center. The positions of the other 100 ants are random, as are the directions they face. All ants start crawling in whatever direction they are facing, always traveling at one meter per minute. When an ant meets another ant or reaches the end of the stick, it immediately turns around and continues going in the other direction. What is the probability that after 1 minute Alice is at the exact center of the stick?

Source: This is a variation of a well-known problem, which appeared recently in Emissary, the MSRI newsletter, and also the FunFacts web page of Francis Su ( http://www.math.hmc.edu/funfacts/ ). The version here is due to John Guilford (Agilent).

© Copyright 2003 Stan Wagon. Reproduced with permission.

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13 November 2003