Name: Murray

Who is asking: Student Level: Secondary

Question:
My questions are:

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)?
   
   
   
   AND
   
   
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)?

These are addmisions questions to Mathcamp 2002 and I would like HINTS only for them. Any advice you give me will be quoted in full in my solution. I have tryed to get my teachers to help but none of them know were to start. Any help at all on either or both questions would be greatly appreciated.

Thank you very much,

Murray.

1. Consider the triangle determined by the three bugs. There are some features of this triangle that are left unchanged when a bug moves as you say.
   
   
2. The starting peg has weight one, and whenever a peg is replaced by two pegs, the two pegs each get half the weight of the original one. In this way, if a peg ever appears on the point (x,y), what will be its weight?