Crossing The Rings

Unlike most, but not all, of my earlier games, this game is NOT played
on an n-by-n grid.
Instead, carefully draw (preferably with a compass or a computer) n
equally-spaced concentric circles, where n is about 5 or more for
beginners (and when just 2 players are playing), a much larger n for
advanced players or if there are more players. (The concentric circles
should end up looking something like a target.)
(I suggest you photocopy the concentric circles, so as to make playing
multiple rounds much easier.)
Players (any number >=2) take turns being the "divider".
The divider, on his/her round, subdivides each ring between each pair
of consecutive circles into anywhere from m to 2^m sections, where m
is the order of a ring from the center of the "target". (The range of
allowable numbers of sections per ring can be modified, if players
chose.)
The line-segments dividing the rings should be perfectly straight, and
should meet the consecutive circles they connect at a right angle to
the tangent of the circles at the points where the circles and the
line-segments meet.
But it is up to the divider as to where along the rings the dividing
line-segments go exactly.
Next, the players who are not the divider take turns each drawing a
straight line-segment (with a straight-edge) from any intersection
where a dividing line-segment and a circle meet to any other such
intersection.
The player who is moving then fills in any sections (section = the
sections of the rings that are subdivided by the dividing line-
segments) that his/her line passes through using a colored pencil of a
color different from the colors of the other players' pencils.
The players' lines may intersect each other, and may pass through
already-filled-in sections. But a player can only fill in sections
that are not yet filled in.
Lastly, the divider then draws a line segment from any intersection to
any other, and fills in the sections his/her line passes through. (As
before, the players' lines may cross, but only sections not filled in
before may be filled in.)
Each player gets a point for each section of his/her color.
Rounds are played -- each round with the same number of concentric
circles -- so that each player plays the divider the same number of
times. Players also switch the order they play on each round, so that
each player plays in any given order the same number of times as every
other player does.
Scores are added from the rounds, and the player with the greatest
number of points wins, of course.
Thanks,
Leroy Quet

PS:
Clarification: Either, players should be banned from drawing their
lines to coincide with dividing
line segments, or if a game line and a dividing line segment coincide
then the player drawing the game line should not be able to fill in
either segment on the two sides of the dividing line-segment.
(The players can chose which game-rule they chose.)