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.