A pen-and-paper game with surprising complexity invented by John Conway.
Designer: John Conway, Michael Stewart Paterson
Publisher: (Public Domain), A. K. Peters, Ltd.
In the image shown, the two black spots each have only two curves coming out of them, but nonetheless cannot be used because no curve can be drawn between them without crossing another already-drawn curve. Thus this game is over. The two black spots are called "survivors", and the number of survivors is what can effectively be controlled in the game.
Sprouts is one of many combinatorial games analyzed in Winning Ways for Your Mathematical Plays as well as other math books on the subject of combinatorial games.