Domineering (also called Stop-Gate or Crosscram) is a mathematical game that can be played on any collection of squares on a sheet of graph paper. For example, it can be played on a 6×6 square, a rectangle, an entirely irregular polyomino, or a combination of any number of such components. Two players have a collection of dominoes which they place on the grid in turn, covering up squares. One player places tiles vertically, while the other places them horizontally. (Traditionally, these players are called "Left" and "Right", respectively, or "V" and "H". Both conventions are used in this article.) As in most games in combinatorial game theory, the first player who cannot move loses. Domineering is a partisan game, in that players use different pieces: the impartial version of the game is Cram. Basic examples Single box Other than the empty game, where there is no grid, the simplest game is a single box. In this game, clearly, neither player can move. Since it is a second-player win, it is therefore a zero game. Horizontal rows This game is a 2-by-1 grid. There is a convention of assigning the game a positive number when Left is winning and a negative one when Right is winning. In this case, Left has no moves, while Right can play a domino to cover the entire board, leaving nothing, which is clearly a zero game. Thus in surreal number notation, this game is {|0} = −1. This makes sense, as this grid is a 1-move advantage for Right. This game is also {|0} = −1, because a single box is unplayable. This grid is the first case of a choice. Right could play the left two boxes, leaving −1. The rightmost boxes leave −1 as well. He could also play the middle two boxes, leaving two single boxes. This option leaves 0+0 = 0. Thus this game can be expressed as {|0,−1}. This is −2. If this game is played in conjunction with other games, this is two free moves for Right. Vertical rows Vertical columns are evaluated in the same way. If there is a row of 2n or 2n+1 boxes, it counts as −n. A column of such size counts as +n. More complex grids This is a more complex game. If Left goes first, either move leaves a 1×2 grid, which is +1. Right, on the other hand, can move to