Partisan Game states that are Impartial

Partisan games are those where the different players do not have the same options for moves (such as in Hex or Chess, but not like Nim). Some states of partisan games, however, mimic impartial games.

This is demonstrated clearly in the game of Domineering, where players alternate placing dominoes on a checkerboard, each taking up two spaces. The Left player must play dominoes vertically (meaning they take up one space and the space directly below that) while the Right must play them horizontally. No pieces are allowed to overlap on the same square of the board.

Some situations in Domineering behave like impartial games, however. Suppose the only free space to play on a board is three squares arranged in the following way: (forgive my ascii art)

XXXX
XOOX
XOXX
XXXX

(Legend: X means this space is covered by a domino. O means this space is uncovered.)

Here, either player may play, but afterwards there are no available plays on the board for anyone. This is much different from

XXXX
XOOX
XOOX
XXXX

where after one of the two players makes a move, they will then be the only one left able to play there. For example, if Left plays on the above board, the situation then becomes:

XXXX
XXOX
XXOX
XXXX

leaving Left able to play again but Right with no such options. With our top example, however, either may play here, but only that one play is left. This is equivalent to the Nim state where there is one pile with only one stick in it. Either player may move by taking that stick, but then no more moves are available.

In 2005, Gabriel Drummond-Cole found positions in Domineering (and Chess!) equivalent to a Nim pile with two sticks. These Domineering states are quite a bit more complicated than those above. The most simple he finds is:

XXXXXXXXX
XXXOXOXXX
XXOOOOOXX
XOOXOXOOX
XXOOXOOXX
XOOXOXOOX
XXOOOOOXX
XXXOXOXXX
XXXXXXXXX

The ascii art here shows its inability to describe what is really happening here (see the paper for better pictures). Gabriel goes on to describe more such situations and even detail the patterns that cause this. One interesting note in this instance is that the covered square in the exact middle can also be uncovered: the value is still equivalent to the Nim-heap with 2 sticks.

Gabriel points out a problem with his constructions, however: there is no way to reach this game state from an initial board! Notice the blocked spaces (X's) in the middle of the diagram: they are each just one blocked square surrounded by unblocked squares. Thus, no domino could have been placed there.

Constructing these situations is a large task, aided often by evaluating software such as Aaron Siegel's Combinatorial Game Suite. Gabriel laments he has "not been able to find any ordinary Domineering positions" equivalent to the Nim-heap of size 2.

Finding other partisan situations equivalent to Nim-heaps is a continuing study. A Nim-heap of size 3 is equivalent to a game with two heaps, of sizes 1 and 2. Thus, we see that we can already have an equivalent domineering game by appending our two boards together:

XXXXXXXXXXXX
XXXOXOXXXOOX
XXOOOOOXXOXX
XOOXOXOOXXXX
XXOOXOOXXXXX
XOOXOXOOXXXX
XXOOOOOXXXXX
XXXOXOXXXXXX
XXXXXXXXXXXX

Work on finding partisan states equivalent to a Nim heap of size 4 is the next big task, and I have heard of some potential results... :)