Asian Pacific Mathematics Olympiad

Cell $(x, y)$ is directly reachable from another cell if and only if $x-k \geq 1$ or $x+k \leq 100$ or $y-k \geq 1$ or $y+k \leq 100$, that is, $x \geq k+1$ or $x \leq 100-k$ or $y \geq k+1$ or $y \leq 100-k$ (*). Therefore the cells $(x, y)$ for which $101-k \leq x \leq k$ and $101-k \leq y \leq k$ are unreachable. Let $S$ be this set of unreachable cells in this square, namely the square of cells $(x, y)$, $101-k \leq x, y \leq k$.
If condition (*) is valid for both $(x, y)$ and $(x \pm 2, y \pm 2)$ then one can move from $(x, y)$ to $(x \pm 2, y \pm 2)$, if they are both in the table, with two moves: either $x \leq 50$ or $x \geq 51$; the same is true for $y$. In the first case, move $(x, y) \rightarrow (x+k, y \pm 1) \rightarrow (x, y \pm 2)$ or $(x, y) \rightarrow (x \pm 1, y+k) \rightarrow (x \pm 2, y)$. In the second case, move $(x, y) \rightarrow (x-k, y \pm 1) \rightarrow (x, y \pm 2)$ or $(x, y) \rightarrow (x \pm 1, y-k) \rightarrow (x \pm 2, y)$.
Hence if the table is colored in two colors like a chessboard, if $k \leq 50$, cells with the same color as $(1,1)$ are reachable. Moreover, if $k$ is even, every other move changes the color of the occupied cell, and all cells are potentially reachable; otherwise, only cells with the same color as $(1,1)$ can be visited. Therefore, if $k$ is even then the reachable cells consists of all cells except the center square defined by $101-k \leq x \leq k$ and $101-k \leq y \leq k$, that is, $L(k)=100^{2}-(2 k-100)^{2}$; if $k$ is odd, then only half of the cells are reachable: the ones with the same color as $(1,1)$, and $L(k)=\frac{1}{2}\left(100^{2}-(2 k-100)^{2}\right)$.