Domineering is a mathematical game played on a sheet of graph paper, with any set of designs traced out. For example, it can be played on a 6x6 square, a checkerboard, an entirely irregular polygon, or any combination thereof. Two players have a collection of dominoes which they place on the grid in turn, covering up squares. One player, Left, plays his tiles vertically, while the other plays his horizontally. As in most games in combinatorial game theory, the first player who cannot move loses.

Basic examples

Single box

Other than the empty game, where there is no grid, the simplest game is a single box.

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In this game, clearly, neither player can move. Since it is a second-player win, it is therefore a zero game.

Horizontal rows

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The leftmost game is a 2-by-1 grid. As in most games, we as the mathematician are "rooting" for Left, so when he is winning, the game has a positive value. 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. The second box is also {|0} = -1, because a single box is unplayable.

The last 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 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.

Heating up

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is a more complicated game. If Left goes first, either move leaves a 1x2 grid, which is +1. Right, on the other hand, can move to -1. Thus the surreal number notation is {1|-1}. But wait! This isn't a surreal number, because 1 > -1. This is a Game, but not a number. The notation for this is ±1, and it is a hot game, because each player wants to move here.

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is a 2x3 grid, which is even more complex, but just like any Domineering game, it can be broken down by looking at what the various moves for Left and Right are. Left can take the left column (or equivalently, the right column), and move to ±1, but it is clearly a better idea to split the middle, leaving 2 separate games, each worth +1. Thus Left's best move is to +2. Right has four "different" moves, but they all leave the following shape in some rotation:

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This game is not a hot game, because each move hurts the player making it, as we can see by examining the moves. Left can move to -1, Right can move to 0 or +1. Thus this game is {-1|0,1} = {-1|0} = -1/2.

Our 2x3 grid, then, is {2|-1/2}, also written as the mean value, 3/4, and the bonus for moving, 5/4, also called the "temperature."