This is a game played by any number of people.

It is played on an n-by-n section of grid taken from graph paper. (I

suggest an n of about 12 for beginners, if there are only 2 players.)

Players take turns who is the offense player. (The other players play

defense on a round.) A round is played for each player playing the

game. An empty n-by-n grid is used each round (with the same n as in

the other rounds).

Players take turns each filling in the empty squares of the grid, one

empty square filled in each move by each player.

If there are m players, then each player fills in floor(n^2/(2m))

squares. (That is a total of m*floor(n^2/(2m)) squares filled in all

together.)

Then the offense player draws a straight line (with a straight-edge)

from any side of the n-by-n grid to any other side.

The line must not be perfectly vertical or perfectly horizontal.

The offense player gets a point for every boundary between a filled-in

square and an empty square that the line passes through.

Highest score wins.

Example:

Filled-in square = *. Empty square = o.

n = 6. (View with fixed-width font.)

\ 1 2 3 4 5 6

A o * * * o *

B o * * o o o

C * o * o o *

D * * o o o o

E o o * * o o

F * o * * * *

Let us say that the line goes from just below the upper-left corner of

the grid to just left of the lower right corner. (The line meets the

perimeter of the grid less than one square's length from each of these

corners.)

This is kind of hard to depict here, because the squares of the grid

are literally squares, while they are not in my diagram; but hopefully

it is clear anyway.

The line first crosses the boundary between 1B and 2B. Then it crosses

the boundary between 2B and 2C. Then it crosses the boundary 2C to

3C. Then 3C-3D. Then 4D-4E. Then 4E-5E. And finally, 5E-5F.

Note: Technically we are concerned about the number of times the line

crosses from a filled-in square into an unfilled-in square, or vice

versa. So if a line crosses from a square to a diagonally adjacent

square via the vertex that joins them, then what we are concerned

about is the two squares' status. In other words, the vertex is

considered the "boundary" in that case.

Thanks,

Leroy Quet

## Monday, September 22, 2008

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