Numerical notation

Some further notation will help us describe the possibilities a little better.

(0)

If neither player has a legal move in a given position, then we denote that position by

$$0=\{\ \vert\ \}=\{\emptyset \vert\emptyset \},$$

where $\emptyset$ denotes the empty set. In the game of Domineering, this occurs if there is only one square remaining:

In this position, the player to move loses.

(1)

If Left has exactly one legal move in a given position, and Right has none, then we denote that position by

$$1=\{0 \vert\ \}=\{\{\emptyset \vert\emptyset \} \vert\ \}.$$

In the game of Domineering, this occurs if there are two or three consecutive squares in a horizonal row remaining (and any number of isolated squares):

In this position, Left wins no matter who plays first.

(-1)

If Right has exactly one legal move in a given position, and Left has none, then we denote that position by

$$-1=\{\ \vert \}=\{\ \vert\ \{\emptyset \vert\emptyset \} \}.$$

In the game of Domineering, this occurs if there are two or three consecutive squares in a vertical row remaining (and any number of isolated squares):

In this position, Right wins no matter who plays first.

(*)

If Left and Right each have exactly one legal move in a given position, then we denote that position by

$$*=\{\ 0\vert \}=\{\{\emptyset \vert\emptyset \} \vert\ \{\emptyset \vert\emptyset \} \}.$$

In the game of Domineering, this occurs if

In this position, whoever plays first wins.