Grid Game

Date: 07/20/2001 at 10:43:05
From: Chad Warnimont
Subject: Finding a rule or pattern

Dear Dr. Math,

I have been playing with this problem for a while and I can't seem to 
find a consistent rule that always works.
The problem is:

This is a game for two players on a rectangular grid with a fixed 
number of rows and columns. Play begins in the bottom left-hand square 
where the first player puts his mark. On his turn a player may put his 
mark into a square directly above, or directly to the right, or 
diagonally above and to the right of the last mark made by his 
opponent. Play continues in this fashion, and the winnner is the 
player who gets his mark in the upper right-hand corner first.  

Find a way of winning that an average player could understand and use.

I have come up with a lot of conjectures, but all of the them are 
false, so I need a new perspective on the problem.

Thank you,
Chad

Date: 07/20/2001 at 15:41:53
From: Doctor Rob
Subject: Re: Finding a rule or pattern
Thanks for writing to Ask Dr. Math, Chad!

Number the rows from top to bottom, 1 through r, and the columns from 
right to left, 1 through c. Denote the square in row m and column n by 
the pair (m,n).

It turns out that whoever first puts a mark in an (odd,odd) square can 
force a win. Say it is player 1. When player 2 puts a mark next, 
player 1 duplicates his play: if 2 goes up, right, or diagonally, so 
does 1.

That means that if the dimensions of the rectangle, r and c, are both 
odd, the first player wins, but if either or both are even, the second 
player wins. This is because the first mark is put in square (r,c).

I leave it to you to prove that this works.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/