the dots and lines game (thing)

To play, first draw a grid of dots (I usually make it square, but it's not a requirement).
Then, the players take turns connecting two vertically or horizontally adjacent dots. The object of the game is to capture more grid cells than your opponent(s). This is done by being the one who draws the final line enclosing a square. When a player captures a square, s/he must move again.

This is a pretty simple (and fun) game, but by no means a trivial one. Here's part of a little example game between A and B (I start it in the middle of the game for brevity's sake):

A:
*--*--*
|  
*  *--*
|
*--*--*


B: "Oh no!"
*--*--*
|  
*--*--*
|
*--*--*


A:
*--*--*
|A |A |
*--*--*
|A |A |
*--*--*

Cascades like the the one that allowed A to capture all the squares on the board are very common. On larger grids, the game usually ends up consisting of several such cascades. The tricky part is the point in the game where B said "Oh no!". Here, no matter where B moves, s/he will be filling in the third side of a square, so that A will be able to capture it on the next move. In larger games, like the one below, one has to choose wisely which cascades to "open up" for the opponent in order to force them to open up a larger cascade:

*--*--*  *
|  1  | 2|
*  *  *  *
|     |  |
*--*--*  *
  3   |  
*--*  *--*

Here, there are three independant cascades. Notice how if you captured all the squares in region 1, you would then still be required to move and thus open up one of the other cascades for the other player. There lies the complexity hidden behind this game's apparent simplicity.