HMMT November 2013 U35

Consider the following 4 by 4 grid with one corner square removed:

You may start at any square in this grid and at each move, you may either stop or travel to an adjacent square (sharing a side, not just a corner) that you have not already visited (the square you start at is automatically marked as visited). Determine the distinct number of paths you can take. Your score will be max $\left\{0,\left\lfloor 20-200\left|1-\frac{k}{S}\right|\right\rfloor\right\}$, where $k$ is your answer and $S$ is the actual answer.