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Checkerboard Chase

Date: 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/ 
Associated Topics:
College Discrete Math
College Probability
High School Discrete Mathematics
High School Permutations and Combinations
High School Probability

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