Friday, May 6, 2011

Solution to the Tiger and the Lake

See the original puzzle

You can swim in two directions: the radial direction (towards the edge), and the tangential direction (around the center).  You need to swim in the radial direction to get out of the lake, but you also need to swim in the tangential direction in order to avoid the tiger.

The problem is, the further you get from the center of the lake, the less effective it is to swim in the tangential direction.  Near the center, you can just swim in little circles, and the tiger running around the edge of the lake will never be able to catch up.  But near the edge, the tiger can catch up.  At one fourth the radius of the lake, you can swim around the center just as fast as the tiger can run around the lake.

So here's the strategy.  Swim partly in the tangential direction and partly in the radial direction such that the tiger is always on the opposite side of the lake.  Eventually this spiral path should reach one fourth the radius, or at least very close.  And then you make a break directly for the edge of the lake.

It turns out you can reach the shore before the tiger catches up to you.  However, it turns out that you can't outrun the tiger after you reach the shore.  Also, in real life, tigers can swim.  So you can't really escape safely.

If you're curious what the spiral path looks like exactly, you're in luck!  This is exactly the kind of problem that physicists are good at solving.  It turns out that the correct spiral path is a semi-circle.  And you can actually reach one fourth the radius exactly.  See spoiler image