Editing Domineering (section)

===More complex grids===
[[Image:20x20square.png]][[Image:20x20square.png]]<br>
[[Image:20x20square.png]][[Image:20x20square.png]]

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 −1. Thus the [[surreal number]] notation is {1|−1}.  However, this is not 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.

[[Image:20x20square.png]][[Image:20x20square.png]][[Image:20x20square.png]]<br>
[[Image:20x20square.png]][[Image:20x20square.png]][[Image:20x20square.png]]

This is a 2×3 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 two 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]]:

[[Image:20x20square.png]][[Image:20x20square.png]][[Image:20x20square.png]]<br>
[[Image:20x20square.png]]

This game is not a hot game (also called a [[cold 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} = −{{frac|1|2}}.

Our 2×3 grid, then, is {2|−{{frac|1|2}}}, which can also be represented by the mean value, {{frac|3|4}}, together with the bonus for moving (the "temperature"), {{frac|1|1|4}}, thus: <math>\textstyle\left\{2 \left| -\frac{1}{2}\right.\right\} = \frac{3}{4} \pm \frac{5}{4}</math>