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Puzzled: Ant Alice’s Adventures

These three puzzles involve my favorite ant, Ant Alice. Like all ants on this page, Alice moves at exactly one centimeter per second in whichever direction she happens to be facing; if she meets another ant head on, both immediately reverse direction and walk away from each other, each still at speed 1 cm/sec. Figuring out how Alice and her friends behave is surprisingly easy if viewed the right way; here's a tip: Certain physical principles may play a role in your reasoning.
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What is the probability Alice ends up where she started?
  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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UF1 Figure. What is the probability Alice ends up where she started?

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DOI

10.1145/2447976.2447998

May 2013 Issue

Published: May 1, 2013

Vol. 56 No. 5

Page: 104

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