When I was a child, my mother taught me a simple pencil-and-paper game. I would like to know whether this game, or an equivalent formulation of it, has already been studied.
Let the integer $n > 1$ be given. Then, consider the following two-player impartial game played on a finite connected region $R \subset \mathbb{R}^2$, consisting of a union of unit squares (cells). The outer boundary of $R$ is fixed.
At each move, a player adds a unit-length edge along a side shared by two adjacent cells inside $R$ (i.e., a single horizontal or vertical segment of the square lattice).
If this move disconnects the remaining free region into two connected components $R_a$ and $R_b$, then the player immediately captures the smaller component (i.e., the one containing $\min\{|R_a|,|R_b|\}$ cells; in case of equality, the player may choose either component), and that component is removed from the playable region.
The game ends when no further disconnection is possible. The score of each player is given by the total number of cells they have captured, and the winner is the player with the larger score.
I am particularly interested in the case where $R(n)$ is the diamond-shaped polyomino consisting of $2 \cdot n^2 - 2 \cdot n + 1$ unit squares arranged in rows of lengths $1, 3, 5, \ldots, 2 \cdot n - 3, 2 \cdot n - 1, 2 \cdot n - 3, \ldots, 5, 3, 1$ (see the figure below for the case $R(4)$, shown in an intermediate stage of a game).
Question(s):
- Has this game, or an equivalent formulation of it, already been discussed in the literature?
- If not, what can be said about optimal play for each given $R(n)$? In particular, I would like to know which player wins, for each given $n$, under perfect play. For example, a draw clearly occurs for $n=2$, while I strongly suspect that Player II can force a win for $n=3$, possibly by a score of $8$-$4$.
