of moves and v, e, f denote the number of vertices, edges, and faces of the planar graph obtained at the end of the game, respectively. We have: e = 2m since at each move, the player adds 2 edges. v = n + m since at each move, the player adds one vertex and the game starts with n vertices. f = 4n since there is exactly one free end in each face at the end of the game, and the number of free ends does not change during the game. The Euler characteristic for planar graphs is 2, so hence A combination of Standard Sprouts and Brussels Sprouts can also be played. The game starts with an arbitrary number (n) of dots or crosses. At each turn, the player chooses to add either a dot, or a cross, along the line they have just drawn. The duration of the game lays between (2n) and (5n - 2), depending on the number of dots or crosses having been added. For n = 1 , starting with a dot, the game will end after 2 moves. Starting with a cross, it will end after 2 moves if the first player adds a dot, after 3 moves if they add a cross: hence the first player has a winning strategy for both the normal and the misère version. For n > 1 , the analysis is not completed. References Bibliography Elwyn R. Berlekamp, John Conway and Richard K. Guy, Winning Ways for your Mathematical Plays, 1992. . External links The Complete (?) List of References for the Game of Sprouts World Game of Sprouts Association., Danny Purvis, association of Sprouts players The Game of Sprouts at University of Utah, with an interactive applet for human-vs-human play. (Requires Java) SproutsWiki, web site of Julien Lemoine and Simon Viennot, with the source code and binaries of their program Mathematical games Paper-and-pencil games John Horton Conway