1. **Stating the problem:** We need to find how many different routes can be chosen following the indicated arrow directions while avoiding the black stones. 2. **Understanding the problem:** This is a path counting problem on a directed graph with forbidden nodes (black stones). 3. **Approach:** We count all possible paths from the start node to the end node that do not pass through any black stones. 4. **Step-by-step solution:** - Identify all possible routes from the start to the end. - Exclude any routes passing through black stones. - Count the remaining valid routes. 5. **Result:** After enumerating and excluding invalid paths, the total number of valid routes is $6$. **Final answer:** B) 6