Hackenbush

Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line.

The game starts with the players drawing a "ground" line (conventionally, but not necessarily, a horizontal line at the bottom of the paper or other playing area) and several line segments such that each line segment is connected to the ground, either directly at an endpoint, or indirectly, via a chain of other segments connected by endpoints. Any number of segments may meet at a point and thus there may be multiple paths to ground.

On his turn, a player "cuts" (erases) any line segment of his choice. Every line segment no longer connected to the ground by any path "falls" (i.e., gets erased). According to the normal play convention of combinatorial game theory, the first player who is unable to move loses.

Hackenbush boards can consist of finitely many (in the case of a "finite board") or infinitely many (in the case of an "infinite board") line segments. 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. Even on an infinite board satisfying this condition, it may or may not be possible for the game to continue forever, depending on the layout of the board.

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. Thus the versions of Hackenbush of interest in combinatorial game theory are more complex partisan games, meaning that the options (moves) available to one player would not necessarily be the ones available to the other player if it were his turn to move given the same position. This is achieved in one of two ways:

Blue-Red Hackenbush is merely a special case of Blue-Red-Green Hackenbush, but it is worth noting separately, as its analysis is often much simpler. 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.

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