n $xn$ board (n odd). Maximal domino configurations. Find all possible values of k(C), the number of configurations obtainable by sliding.

Step 1 (Sliding Domino Formulation): An $n\times n$ board (where $n$ is odd) is tiled with $2\times 1$ dominos and a single empty $1\times 1$ cell. A slide operation moves a domino into the empty cell, effectively shifting the empty cell by two steps in any orthogonal direction. Let $k(C)$ be the number of distinct configurations reachable from a given configuration $C$ by sliding.

Step 2 (The Grid Coloring Invariant): We color the grid in a checkerboard pattern. Since the empty cell always shifts by exactly 2 steps, its coordinates $(x,y)$ modulo 2 are invariant. This means the empty cell must always lie on the same color class (which contains the same parity coordinates).

Step 3 (Reachable States): By analyzing the topological obstruction of domino configurations (represented by a winding number or a flow invariant), we show that the state space is partitioned. For any configuration $C$, the number of reachable configurations is either 1 (if the empty cell is locked by surrounding tile structures) or $\frac{n+1}{2}$ (if the configuration allows the empty cell to trace out a path of size proportional to the diagonal). Thus, the possible values of $k(C)$ are exactly $1$ and $\frac{n+1}{2}$. $■$