If you are standing on the blue how may different routes can

If you are standing on the blue, how may different routes can you take to get to red if you cannot walk west but can walk north North and South.

Solution

Assuming that walking west is not allowed. Number of paths from blue to red are infinite, as one could indefinitely move to and fro in north and south direction indefinitely, and then finally move towards the red point. For example, one path is to make up -> down -> up -> down .... (n pairs).. up -> down, (reaching back to blue dot), and then move 2 rights and 6 ups. In this path, n is not constant, and we can make n as large as we want. Thus, total number of paths are infinite,

If we modify this question, assuming that walking both left and down directions is not allowed, then we have a genuine question in discrete mathematics. There are 6 moves in up direction, and 2 moves in right direction. The only choice we have is to decide which moves we take on nth step. Number of paths are: number of ways of selecting 6 up moves out of 8 steps = 8C6 = 8 choose 6 = 28 (same answer with 8C2)