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Three bugs on a line 2002-02-12
From Murray:
  1. Three bugs are crawling on the coordinate plane. They move one at a time, and each bug will only crawl in a direction parallel to the line joining the other two.
    • If the bugs start out at (0,0), (3,0), and (0,3), is it possible that after some time the first bug will end up back where it started, while the other two bugs switch places?
    • Can the three bugs end up at (1,2), (2,5), and (-2,3)?


  2. A single peg is placed at the bottom left-hand corner of a grid that extends infinitely far up and to the right. You play a game in which you are allowed to make the following move: if the hole immediately above and the hole immediately to the right of a peg are both empty, you can remove the existing peg and place pegs in those two holes instead.
    • Show that, no matter how you move, you can never remove all the pegs from the 3-by-3 square at the bottom left-hand corner of the grid. (b)
    • Is it possible to remove all the pegs from the six holes closest to the bottom left-hand corner of the grid (the region indicated in the picture below)?

Answered by Claude Tardif.



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