## Friday, September 19, 2008

### Even/Odd: Another Game Of Contiguousness

Here is yet another grid-based game.
For 2 players, each player with a different colored pencil (or pen or
crayon).
Players take turns filling in any of the empty squares with their
colors, one square per move.
Play continues until every square is filled in.
I give two possible variations on scoring:
1) Player 1 gets a point for every 'contiguous region', of either
color, which is formed from an odd number of squares.
Player 2 gets a point for every 'contiguous region', of either color,
which is formed from an even number of squares.
2) Player 1 gets a point for every square in a 'contiguous region', of
either color, which is formed from an odd number of squares.
Player 2 gets a point for every square in a 'contiguous region', of
either color, which is formed from an even number of squares.
By "contiguous region", I mean:
A connected region of one color completely surrounded by squares of
the other color and/or the grid's border (surrounded in the directions
of up, down, left, or right,but not diagonally).
Example completed game:
* * # # # *
* * # # * *
# * * * # #
# # * * # *
* * # * # *
# # # * # #
So, we have contiguous regions of:
11 *'s, 5 #'s, 3 #'s, 3 *'s, 2*'s, 6 #'s, 2*'s, and 4 #'s.
So, under scoring method 1,
player 1 gets 4 points,
player 2 gets 4 points also.
Under scoring method 2,
player 1 gets 22 points,
player 2 gets 14 points.
Is there a bias under either scoring method for one player or the
other?