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Triangle Puzzle

Article: 122 of geometry.puzzles
Newsgroups: geometry.puzzles
From: sander@geom.umn.edu (Evelyn Sander)
Subject: Triangle Puzzle
Organization: The Geometry Center, University of Minnesota
Date: Wed, 24 Nov 1993 16:14:55 GMT
Lines: 16

Pick an arbitrary point in the interior of an equilateral triangle. Now go halfway towards one of the three vertices (you get to pick which vertex). Now you are at a new point in the triangle. Again go halfway towards any vertex. Continue this process.

What is the long term behavior? In other words, what points can you reach after repeating this process an arbitrarily large number of times?

I am interested in using this problem to demonstrate the use of computers in teaching. Please let me know if you solve it using computers and what software you use. I will post an answer next week.

Happy Thanksgiving!


  1. When I ask "what points can you reach after repeating this process an arbitrarily large number of times," I had in mind: "what points are the limit of infinite sequences of applications of this process?" Thank you to Dan Asimov for pointing this out. This is also the answer to Art Mabbot's question: I do not want you to save points along the way; just look at where they tend to cluster if you could apply the process forever.

  2. At each stage, you are free to choose which vertex from which to halve your distance. You do not have to do so in cyclic order.

  3. In accordance with 1 above, it is not important whether your point starts in the interior or on the boundary of the triangle. However, I do not allow for points in the exterior of the triangle.


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Created: May 15 1994 --- Last modified: Jun 18 1996