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Puzzled: Ant Alice's Adventures

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Ant Alice's Adventures, illustration

What is the probability Alice ends up where she started?

Credit: Peter Winkler

  1. Ant Alice is the middle ant of 25 ants on a meter-long stick, some facing east, some facing west. (We may assume ants are tiny compared to the distances between them, so they can be thought of as moving points.) At a signal, all begin to march in whichever direction they are currently facing, bouncing and reversing direction whenever two collide. Those reaching the end of the stick fall off and float gently to the ground (no ants were harmed in the creation of these puzzles). How long must we wait before we are sure Alice has fallen off the stick?
  2. Suppose the ants' initial positions, and the directions they face, are uniformly random. What is the probability that when Alice falls off the stick, she falls off the end she was initially facing?
  3. Suppose Alice is one of only 12 ants, each initially placed uniformly at random on a circle of length (circumference) one meter (see the figure here). Each ant initially faces clockwise or counterclockwise with equal probability. At a signal, they begin marching (and bouncing off one another) according to the usual rules. What is the probability that 100 seconds later Alice will find herself exactly where she began?

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Peter Winkler ( is William Morrill Professor of Mathematics and Computer Science at Dartmouth College, Hanover, NH.

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Readers are encouraged to submit prospective puzzles for future columns to

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UF1Figure. What is the probability Alice ends up where she started?

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©2013 ACM  0001-0782/13/05

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