Sunday, September 21, 2008

Vertex/Intersection Polygon Game

For 2 players, each with a different color of pencil/pen.
First, using a straight-edge, draw a large n-gon
(say, n = 6) on a piece of paper.
(The n-gon does not need to be regular.)
Each player draws straight line segments (with a
straight-edge and their color pencil/pen) within
the n-gon as follows:
The first player draws a line segment from any vertex
of the n-gon to any other.
Players thereafter on each move draw a line segment
between a vertex (of the n-gon) or an intersection point
(made by the crossing of two previously drawn
line-segments) and another vertex/intersection.
Players start each line-segment where the other player
ended his/her latest segment (point A).
Players end each segment at any vertex/intersection
(point B) such that:
*No previously drawn line-segment connects the points
A and B.
*Aside from the 2 previously-drawn crossing segments
(in the case of an intersection) or the two edges of
the n-gon (in the case of a vertex), point B does
not have any other line segments already drawn to it.
*The 2 previously-drawn crossing line segments (in the
case of an intersection) are of two different colors
(ie the lines are made by different players).
The last player able to move is the winner.
(If a player wrongly believes he/she cannot move,
then this player still loses.)
Either player may draw a line-segment to/from the
vertex where the first player started their first
Alternative rule: Perhaps a player should be required
that any intersection they draw their segment to be of
the SAME color previously-drawn segments, instead of a
different color. Or maybe one player should draw to
same-color intersections, the other player to
different-colored intersections.
I wonder which rule is the most fun.
I suggest that after a line-segment is drawn to an
intersection/vertex, then the player doing so draws a
dot at the vertex so as to signify that the
vertex/intersection not be drawn to again.

If someone programs this game on a computer, I suggest
allowing the players to zoom in on any part of the game,
and I suggest storing the coordinates of each inersection
with a relatively higher level of precision. This is
because many intersections of some games tend to bunch up
in small areas within the polygons.
Also, you can play this game without any rule at all
requiring intersections drawn to to be of certain colors.
I just had this rule to help prevent absurdly long (and
possibly infinite) games.

Leroy Quet

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