Hackenbush

According to the normal play convention of combinatorial game theory, the first player who is unable to move loses.

The existence of an infinite number of line segments does not violate the game theory assumption that the game can be finished in a finite amount of time, provided that there are only finitely many line segments directly "touching" the ground.

In the original folklore version of Hackenbush, any player is allowed to cut any edge: as this is an impartial game it is comparatively straightforward to give a complete analysis using the Sprague–Grundy theorem.

This is because Blue-Red Hackenbush is a so-called cold game, which means, essentially, that it can never be an advantage to have the first move.

Blue-Red-Green Hackenbush allows for the construction of additional games whose values are not real numbers, such as star and all other nimbers.

Further analysis of the game can be made using graph theory by considering the board as a collection of vertices and edges and examining the paths to each vertex that lies on the ground (which should be considered as a distinguished vertex — it does no harm to identify all the ground points together — rather than as a line on the graph).

In the impartial version of Hackenbush (the one without player specified colors), it can be thought of using nim heaps by breaking the game up into several cases: vertical, convergent, and divergent.