APMO

Cell $(x,y)$ is directly reachable from another cell if and only if $x-k\ge 1$ or $x+k\le 100$ or $y-k\ge 1$ or $y+k\le 100$, that is, $x\ge k+1$ or $x\le 100-k$ or $y\ge k+1$ or $y\le 100-k$ (). Therefore the cells $(x,y)$ for which $101-k\le x\le k$ and $101-k\le y\le k$ are unreachable. Let $S$ be this set of unreachable cells in this square, namely the square of cells $(x,y)$, $101-k\le x,y\le k$. 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 $x\le 50$ or $x\ge 51$; the same is true for $y$. In the first case, move $(x,y)\to (x+k,y\pm 1)\to (x,y\pm 2)$ or $(x,y)\to (x\pm 1,y+k)\to (x\pm 2,y)$. In the second case, move $(x,y)\to (x-k,y\pm 1)\to (x,y\pm 2)$ or $(x,y)\to (x\pm 1,y-k)\to (x\pm 2,y)$. Hence if the table is colored in two colors like a chessboard, if $k\le 50$, 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 $101-k\le x\le k$ and $101-k\le y\le k$, that is, $L(k)=100^2-(2k-100{)}^2$; if $k$ is odd, then only half of the cells are reachable: the ones with the same color as $(1,1)$, and $L(k)=\frac{1}{2}\left(100^2-(2k-100{)}^2\right)$.