# Rambling Thoughts

## A Converging Rabbit

I’ve decided to make a page of some fun problems. I’m not sure how successful this will be, but in theory, there will eventually be a nice long list of my favorite problems. Here’s the first one:

A rabbit climbs out at its hole, and walks $1$ mile in a straight line. Then, the rabbit repeatedly turns $\pi/3$ radians and walks half of the distance it just walked,
as pictured below.

How far away from the rabbit’s hole is the point at which the rabbit converges?

This is an ARML-style problem: There are lots of ways of solving this, but some methods are cleaner and faster than others. Click to see some solutions.