Here is another one of my games. It isn't as fun, probably, as some of

the other games I have posted. But maybe someone will find it

enjoyable anyway.

Start with an n-by-n grid (n-by-n lines, (n-1)-by-(n-1) row/columns)

on graph-paper. (n should be at least 5, maybe in the range of 10 or

more.)

There are two players who switch roles after each round, an offensive

player and a defensive player.

Say the grid is n lines wide((n-1) columns) and n lines high ((n-1)

rows). The defensive player starts the round by writing the integers 1

through (2n) in any order to the left of the left-most vertical line

(each integer lined up with a different horizontal line of the grid)

and above the top-most horizontal line (each integer lined up with a

different vertical line of the grid).

Whether a particular integer is written either along the left side of

the grid or along the top of the grid is up to the defensive player.

Example: 6 lines -by- 6 lines:

. 2 6 11 7 1 8

9 -------------

4 | + + + + +

10| + + + + +

12| + + + + +

3 | + + + + +

5 | + + + + +

(Note: In case ascii art doesn't look correct, the pluses are the

intersections of the grid, and are supposed to each be directly below

an integer of the top row of numbers and directly to the right of the

left column of numbers.)

The two players each take turns drawing a straight line-segment (with

a straightedge) from the intersection of the grid last drawn-to by the

opposing player to an intersection determined by the order of the move

within the round.

If the move is move m -- the defensive player moves on even-numbered

moves, and the offensive player moves on odd-numbered moves -- then

the player can move to any intersection in the same row/column lined

up with the m along the grid's edges. In other words, if the value m

is written along the top of the grid, then the player on the mth move

can move to any one of the n intersections in the same COLUMN as the

value m. And if the value m is written along the left side of the

grid, then the player on the mth move can move to any one of the n

intersections in the same ROW as the value m.

Players cannot draw line-segments along already drawn line-segments.

(ie Line-segments can only intersect at most at one point.)

Line-segments cannot cross intersections that are already the

endpoints of other line-segments.

After 2n total moves (n moves for each player), the round is over.

The offensive player gets a point for every time a line-segment (drawn

by either player) crosses another segment.

If k line-segments intersect at a common point, the offensive player

gets k(k-1)/2 points for that intersection.

This is the same as counting the number of line-segments intersected

by a line-segment AS the line-segment is being drawn. (If your line

segment is crossing an intersection with (k-1) line-segments already

intersecting there, then add (k-1) to the offensive player's score. Or

wait until after the game is over to enumerate the crossings, and give

the offensive player k(k-1)/2 points for those k line-segments

intersecting at a common point.)

So, the defensive player moves so as to try to keep the lines from

crossing. The offensive player moves to try to get as many crossings

as possible.

After a round is complete the players switch who is the offensive

player and who is the defensive player. The highest score wins, after

playing an even number of rounds, of course.

What is a good strategy for this game?

Thanks,

Leroy Quet

## Monday, September 22, 2008

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