Snails chasing each other

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What is this?

A number of snails are placed symmetrically around a circle. Each snail is one unit of distance from the center of the circle. At time t=0, the snails begin walking with a speed of one unit directly towards the snail they see in the anticlockwise direction. Assume the snails are infinitesimal, meaning they are the size of points.


  1. How much time t is required before the snails meet at the center, and stop?
  2. What is the distance traveled by each snail?
  3. What is the trajectory, in polar coordinates, of the snail on the right? This means find r(θ).
  4. How many times do the snails circle the origin before they meet?

The answer to the last question is: an infinite number of times!. That answer often bothers people, and it has been bothering people since ancient times. For those of us in the modern era, who have been educated in calculus and limits, the fact that an infinite number of cycles can add up to a finite distance perhaps bothers us a bit less than it did the ancients. But many students still stumble on this question, even after they find r(θ) and invert it to find θ(r). They just don't believe the answer.