Checkerboard ChaseDate: 12/13/2002 at 03:22:46 From: Javon Gross Subject: Checkerboard Chase Hi, My name is Javon and I attend the University of Connecticut. I have a major final next week and I need your help with this challenging math problem. Player A begins by placing a checker in the lower left-hand corner of a checkerboard (8 by 8 squares). Player B places a checker one square to the right or one square up or one square diagonally up and to the right of Player A's checker. Then A places a checker one square to the right or one square up or one square diagonally up and to the right of Player B. The players continue alternating moves in this way. The winner is the player who places a checker in the upper right corner. Would you rather be Player A or Player B? Date: 01/28/2003 at 03:36:03 From: Doctor Marshall Subject: Re: Checkerboard Chase Hi Javon, Let's number the squares a-h from left to right, and 1-8 from bottom to top. We know that the last checker played (h8) is a winner. Therefore we know that any checker played in its surrounding area (g8,g7,h7) is sure to lose, as the keen player will play in the corner next. We can draw a checker board, starting with this upper right corner, and marking an L (lose) where a W (win) is within reach, and mark a W where no possible next move exists besides L (i.e., no W touches another W). The board looks like this: +---+---+---+---+---+---+---+---+ | L | W | L | W | L | W | L | W | +---+---+---+---+---+---+---+---+ | L | L | L | L | L | L | L | L | +---+---+---+---+---+---+---+---+ | L | W | L | W | L | W | L | W | +---+---+---+---+---+---+---+---+ | L | L | L | L | L | L | L | L | +---+---+---+---+---+---+---+---+ | L | W | L | W | L | W | L | W | +---+---+---+---+---+---+---+---+ | L | L | L | L | L | L | L | L | +---+---+---+---+---+---+---+---+ | L | W | L | W | L | W | L | W | +---+---+---+---+---+---+---+---+ | L | L | L | L | L | L | L | L | +---+---+---+---+---+---+---+---+ This is player B's winning map! He just plays the W's every move. Player A loses. - Doctor Marshall, The Math Forum http://mathforum.org/dr.math/ |
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