Here is a game for two players.
Needed, an n-by n grid drawn lightly on a piece of paper. Two pens or pencils of different colors.
(I suggest n be at least 6.)
In this game, line segments are each drawn from some vertex of the grid (along the lightly drawn grid line) to an adjacent (immediately above, right of, below, left of) vertex.
On the first move, the first player draws a line segment with one of the pens from any vertex to any adjacent vertex.
Then the other player draws a line segment with the other pen from where the first line segment started to any adjacent vertex, as long as the two line segments aren't drawn to (end at) the same vertex.
Thereafter, both players on each of their moves may draw a line segment of EITHER color from the vertex last drawn to by a line segment of that color. The line segment is drawn to any vertex such that no line-segments coincide (except perhaps at a single point). So, players may choose which end of the growing 'line' (a collection of line segments placed end to end) to extend with a line segment on each of their moves, provided that the color of any new segment matches the segment it is attached to.
The line may pass through itself. (Example: Two older consecutive vertical segments are conjoined at a vertex. Then later on, a newer line segment comes from the left and meets at that vertex, then the lastly drawn of these four segments proceeds from the vertex rightward.) (The older line segments may or may not be the same color as the newer segments.)
The line may also "bounce" off of a corner, making a new corner.(Example: An older segment proceeds upwardly to a vertex, then the next older segment proceeds to the right. Then later on, a newer segment comes from the left to meet at that vertex, and finally the last segment is drawn upwardly from that vertex.) (Again, the older line segments may or may not be the same color as the newer line segments.)
Continue the game until one of the ends of the total line cannot be drawn anymore. (This will be either at the edge of the grid or perhaps if, not necessarily when, the two ends of the line meet.)
The winner of the game has the most line segments of the color he drew his first segment in.