Suppose that there is a unit square. There are four cats sitting at the four different corners of the square. Each of those cats start chasing the other cat in the clockwise direction. The speed of the cats are same and constant and they continuously change their direction in a manner that they are always heading straight to the other cat. How long will it take for the cats to catch each other?
Let us denote the cats with A, B, C and D in a fashion that A is chasing B, B is chasing C, C is chasing D and D is chasing A. The cats are moving in symmetry and thus the only logical answer is that they will meet at the center of the square.
At any point of time, the Cat A is perpendicular to cat B and Cat B is perpendicular to Cat C and so on. Cat A runs for Cat B but Cat B does not run towards or away from Cat A for it is moving in a perpendicular direction to Cat A. Thus the distance that need to be covered by Cat A to grab Cat B will be exactly same as the original distance between them which in this case is one unit (as it is a unit square).
Thus the speed of each cat towards the cat it is chasing will be [1 + cos (x)] where x denotes the angle on each corner.
Thus speed of the cat = 1 + cos (90) = 1 + 0 = 1
The time needed = Distance / Speed = 1/1 = 1 unit