Tangle

For 2 or more players, each with a different colored pencil.
Start with a large square (or any convex polygon) drawn on paper.
Player 1 starts by drawing a straight line-segment, using a
straight-edge, from any corner of the square/polygon through the
interior of the polygon until hitting an edge of the polygon.
Players together draw a path by alternatingly drawing line segments,
using a straight-edge (and making each segment the color associated
with the player drawing the segment), each segment which is from the
end of the last segment drawn by the previous player, in any
direction(with restrictions*), and ending at the first previously
drawn line-segment intersecting the segment being drawn, or ending at
a boundry of the square/polygon.
(*) If the last line-segment drawn ended at the edge of the
square/polygon, the path must "bounce", ie. the next segment must be
towards the interior of the polygon.
If the last segment drawn ended at a previously drawn segment (of any
color), the path must cross this older segment, ie. the new segment
must be on the opposite side of the older crossed segment from the
immediately previous segment.
Segments cannot be drawn to vertexes (where the path crosses itself or
bounces off the edge of the polygon.)
The players draw a predetermined number of segments (say, a total of
50).
Each player gets a point every time the path crosses a pre-drawn
segment of his/her color.
The winner has the most number of points.

A clarification:
The "path" refered to above is the total collection of line-segments.
The line-segments are drawn end-to-end and alternate in color.
I will attempt some ascii-art.
(View with fixed-width font.)
If the game is like so:
\< older line-segment
.\
--\ < last line drawn by previous player
.. \
... \
you continue on the opposite side of the older line-segment in any
direction:
\.. /< your new segment
.\ /
--\
.. \
... \
And your new segment continues only until the first line-segment or
edge of the square/polygon encountered, where your line-segment then
terminates.
If the game is like so:
\
. \ < last line drawn by previous player
.. \
======== < edge of square/polygon
You may bounce in any direction (either to left or right), as long as
the path stays within the square/polygon:
.\ .../
. \ ./ < your new segment
.. \/
==========
One final note:
No segment is allowed to be drawn along a previous segment.
(No co-linear segments.)

I noticed that the first player has an advantage, at least when
this game is played by 2 people.
A simple fix: Play an even number of rounds, and have each player take
turns being player 1, then add up each player's scores of the rounds
to get each player's grand score.
If there are more than 2 players, I do not know what kind of advantage
player 1 has.
If the players want a fair game, and there are m players, just play a
multiple of m rounds, alternating the order the players play, and add
up the scores for each player's grand score.
(Maybe the m players should play m! rounds, with each permutation of
player-order represented once.)

4 more things:
1) It is evident after I played this that it would probably be better,
rather than making a path of a predetermined number of line-segments,
that the game simply continue until one player first gets a
predetermined score, say 10 or 20.
In this way, players do not have to worry about keeping track of the
number of moves made. And there is also, with this new rule, no chance
of a tie.
2) One purpose of playing this game, aside from winning, is to
hopefully get a cool abstract design.
:)
3) I was a little unclear about when a point is scored
(when I wrote "Each player gets a point every time the path crosses a
pre-drawn segment of his/her color.").
Whenever a line-segment is drawn by anyone to another segment of color
c, and player c is using pencil/pen color c, then player c gets a
point.
By "path crosses", I mean 2 segments come together at the previously
drawn line-segment.
Like so:
-- previously drawn segment of color c,
\ < path crosses this segment.
. \ _______
../ \
./ .. \
/ .... \
4) As is noted elsewhere in this thread, there is an advantage to
being player 1 (and so players should play multiple rounds, trading
off who is player 1).
There is a likely set of opening moves. As an easy puzzle, try to
give the best next move for player 2:
Playing on a square "board".
Moves:
1) Player 1 draws segment from lower-left (Southwest) corner
East-Northeast to right edge of square.
2) Player 2: NW to upper edge of square.
3) Player 1: South to player 1's first segment.
4) Player 2: SE to right edge of square.
5) Player 1: West to player 1's first segment.
Now, where should player 2 move?
(Easy, but this will help you familiarize yourself with the game, if
you choose to try to solve this "puzzle".)

Thanks,
Leroy Quet