Puzzle: There are four dogs each at the corner of a unit square. Each of the dogs starts chasing the dog in the clockwise direction. They all run at the same speed and continuously change their direction accordingly so that they are always heading straight towards the other dog. How long does it take for the dogs to catch each other and where?
Solution:
Each dog started at the corner and moving symmetrically. So each dogs start moving perpendicular to the adjacent dogs. Lets assume v.
So each one start moving with speed v towards the next dog.
If we see realtive speed of the dog1 (v1), w.r.t dog 2, it changes perpendicularly. So it’ll not affect the time taken along the direction of the dog1 to dog2 and the speed will be only v always.
So if they have started at corners with the distance of the length of the square (d).
Time = d/v. They’ll meet at the center.
Each dog can move in either two directions clockwise and anticlockwise for not colliding.so prob = (0.5*0.5*0.5*0.5)+(0.5*0.5*0.5*0.5)