Hackenbush is a two-player partisan mathematical game that consists of several colored line segments connected to the ground. More precisely, there is a ground (usually, 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 or indirectly, via one or more paths. In combinatorial game theory, a game is partisan or partizan if it is not impartial. ... Mathematical games include many topics which are a part of recreational mathematics, but can also cover topics such as the mathematics of games, and playing games with mathematics. ...

On his turn, a player "cuts" (erases) a line segment of his choice (from those which he is allowed to select--see below for more details). Every line segment which is no longer attached to the ground by any path "falls" (also gets erased). As usual, 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. Note that 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, given that there are only finitely many line segments directly "touching" the ground. This article may be too technical for most readers to understand. ...

Hackenbush is a partisan game, meaning that the options (moves) available to one player would not necessarily be the ones available to the other player if he were given the same exact board. In common practice, there are two versions of the game: In combinatorial game theory, a game is partisan or partizan if it is not impartial. ...

• Blue-Red Hackenbush: Each line segment is colored either red or blue. One player (usually the first, or left, player) is only allowed to cut blue line segments, while the other player (usually the second, or right, player) is only allowed to cut red line segments.

• Blue-Red-Green Hackenbush: Each line segment is colored either red, blue, or green. The rules are the same as for Blue-Red Hackenbush, with the additional stipulation that green line segments can be cut by either player.

Clearly, 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.

Hackenbush is fundamental to the study of combinatorial game theory and, indeed, is often one of the very first games taught to students in this field (see Combinatorial game theory (pedagogy)). This article may be too technical for most readers to understand. ... The teaching of combinatorial game theory normally uses the following examples: Blue-Red Hackenbush -- At the finite level, this partisan combinatorial game allows constructions of games whose values are dyadic rational numbers. ...

Further analysis of the game can be done using graph theory by considering the board as a collection of vertices and edges and examining the paths to each vertex which lies on the ground (which should be considered a property of the vertex, rather than an actual line on which the points are drawn). In mathematics and computer science, graph theory studies the properties of graphs. ... This article just presents the basic definitions. ... This article just presents the basic definitions. ... In mathematics, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the successor vertex. ...

Finally, Hackenbush is directly related to the study of surreal numbers. Finite Blue-Red Hackenbush boards can construct dyadic rational numbers, while the values of infinite Blue-Red Hackenbush boards account for the entire set of real numbers. Furthermore, Blue-Red-Green Hackenbush allows for the construction of additional values, such as star and all other nimbers. In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... In mathematics, a dyadic fraction or dyadic rational is a rational number that when written as a fraction has denominator a power of two, i. ... The text or formatting below is generated by a template which has been proposed for deletion. ... Star, written as * or *1, is the value given to the combinatorial game {0 | 0}, where zero is the zero game. ... In mathematics, the proper class of nimbers is introduced in combinatorial game theory. ...

In the movie A Day at the Races Groucho Marx played a vet called Dr. Hugo Z. Hackenbush. Code book scene A Day at the Races A Day at the Races (1937) is the seventh movie starring the three Marx Brothers, with Margaret Dumont, Allan Jones and Maureen OSullivan. ... Groucho Marx poses for an NBC promotional photograph Julius Henry Marx, known as Groucho Marx (October 2, 1890 – August 19, 1977), was an American comedian, working both with his siblings, the Marx Brothers, and on his own. ...