Prove that the boy cannot escape the teacher I'm struggling with the following problem from Terence Tao's "Solving Mathematical Problems":

Suppose the teacher can run six times as fast as the
  boy can swim. Now show that the boy cannot escape. (Hint: Draw an
  imaginary square of sidelength 1/6 unit centred at $O$. Once the boy leaves
  that square, the teacher gains the upper hand.)

Here $O$ is the center of the swimming pool. This question is a follow up on the previous one, which is solved in the affirmative in the text

(Taylor 1989, p. 34, Q2). In the centre of a square swimming
  pool is a boy, while his teacher (who cannot swim) is at one corner
  of the pool. The teacher can run three times faster than the boy can swim,
  but the boy can run faster than the teacher can. Can the boy escape from
  the teacher? (Assume both persons are infinitely manoeuvrable.)


My attempt:
Since the boy can always swim back into the small square of sidelength 1/6 centered at $O$, I can't see how to apply the hint properly. Also, since the student's path need not even be smooth (it was taken as a polygonal chain in the previous question) I'm having difficulties writing data down clearly.
Any help would be appreciated. Thanks.

The boy has no incentive to change direction since he loses time.
Which direction should he pick?  


*

*In the 6×speed case, no matter which direction the boy chooses, the teacher can get there faster.

*In the 3×speed case, there are numerous directions where the boy can escape... just not the opposite diagonal corner.




