There are four dogs/ants/people at four corners of a square of unit distance. At the same instant all of them start running with unit speed towards the person on their clockwise direction and will alw

They meet in the center in 1/2 unit time.

If possible, can you give an explanation.
I think they move in spirals.. but why does it take 1/2 unit time?

It's actually slightlymore that 1/2 unit time because it's an arc. So it's really 1/4 of the perimeter of a circle with a radius of 1/2 unit. pie*1/2 unit squared /4
Sorry that the best ican type on my phone.

So let's look at bottom left and bottom right. Bl starts moving up and br starts moving left, but see how br starts curving up since Bl is going up? Bl is also curving right since tl is moving right. So they all move in a nice arc toward the center

If they travel in perfectly circular paths (which I am not yet sure about, but it seems probable) then they would travel along a circle centered at the midway point on the edge of the side of the square.
Then they would travel a distance of pi/16 units which is what aarontomsky was trying to say.
pi/16 since the whole circle is pi*r^2 r is 1/2 unit and it's 1/4 of the circle so pi(1/2)^2*1/4 = pi/16

What? Oops. That's area. It's actually pi*d/4 so pi/4 since d is the unit side. My bad.
pi/4 is the distance travelled.

Scratch all that. They would not move in perfect circular curves. Try it out. Draw the square and the circles and you can see that at the halfway point the paths would not point at each other. I'll have to think about it for a while.

If you don't believe it. Try plugging it into a calculator. Use bigger and bigger numbers for n.