In order to
outsmart the wolf, consider your relative angular speed as you swim along concentric circles. Without loss of generality, we may assume the pond has radius
![1](http://latex.codecogs.com/gif.latex?1)
centered at the origin. As you swim along the path
![r = r_0](http://latex.codecogs.com/gif.latex?r = r_0)
, the wolf can cover the same central angle more quickly if
![r_0 > \tfrac{1}{4}](http://latex.codecogs.com/gif.latex?r_0 > \tfrac{1}{4})
, more slowly if
![r_0 < \tfrac{1}{4}](http://latex.codecogs.com/gif.latex?r_0 < \tfrac{1}{4})
, and at precisely the same speed if
![r_0 = \tfrac{1}{4}](http://latex.codecogs.com/gif.latex?r_0 = \tfrac{1}{4})
.
From this, we can develop a simple strategy. Swim out near to but just shy of the circle
![r = \tfrac{1}{4}](http://latex.codecogs.com/gif.latex?r = \tfrac{1}{4})
. Since we can cover the same angle more quickly than the wolf, we may swim in a circle until the wolf is directly behind us. We can cover the remaining distance of slightly more than
![\tfrac{3}{4}](http://latex.codecogs.com/gif.latex?\tfrac{3}{4})
in less than the time it takes the wolf to cover its distance of
![\pi](http://latex.codecogs.com/gif.latex?\pi)
.
To be more precise, we can successfully implement this strategy of first swimming to
![r = r_0](http://latex.codecogs.com/gif.latex?r = r_0)
exactly when
![1 - \tfrac{\pi}{4} < r_0 < \tfrac{1}{4}](http://latex.codecogs.com/gif.latex?1 - \tfrac{\pi}{4} < r_0 < \tfrac{1}{4})
. There is a tradeoff. The smaller the
![r_0](http://latex.codecogs.com/gif.latex?r_0)
we choose in this interval, the faster we can get to the edge of the pond. On the other hand, the larger we choose
![r_0](http://latex.codecogs.com/gif.latex?r_0)
, the further away the wolf is when we reach the edge.