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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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