Solving the Frogs & Chocolate Bar Puzzle

[AW1213]

1) Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another frog, in such a way that the "leapee" is at the midpoint of the line segment whose endpoints are the starting and ending position of the "leaper." Will a frog ever occupy the vertex of the square that was originally unoccupied?2) Two people take turns cutting up a rectangular chocolate bar that is 6 x 8 squares in size. You are allowed to cut the bar only along a division between the squares and your cut can be only a straight line. For example, you can turn the original bar into a 6 x 2 piece and a 6 x 6 piece, and this latter piece can be turned into a 1 x 6 piece and a 5 x 6 piece. The last player who can break the chocolate wins (and gets to eat the chocolate bar). Is there a winning strategy for the first or second player? What about the general case (The starting bar is m x n)?Please help me with these problems! I don't know how to even begin solving this! Any feedback is greatly appreciated!

 

[Dick]

To get to the vertex of the square that was unoccupied one of the frogs has to change it's x or y coordinate by one square unit after a number of hops. Can it do that? Get a piece of graph paper and try hopping around for a while. I don't know the answer to the second question but the strategy is the same. Try playing the game with yourself for a while with smaller rectangles. Can you see a pattern emerging?