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181 geometry/duck.and.fox.p


This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

181 geometry/duck.and.fox.p

A duck is swimming about in a circular pond. A ravenous fox (who cannot
swim) is roaming the edges of the pond, waiting for the duck to come close.
The fox can run faster than the duck can swim. In order to escape,
the duck must swim to the edge of the pond before flying away. Assume that
the duck can't fly until it has reached the edge of the pond.

How much faster must the fox run that the duck swims in order to be always
able to catch the duck?


Assume the ratio of the fox's speed to the duck's is a, and the radius
of the pond is r. The duck's best strategy is:

1. Swim around a circle of radius (r/a - delta) concentric with the
pond until you are diametrically opposite the fox (you, the fox, and
the center of the pond are colinear).

2. Swim a distance delta along a radial line toward the bank opposite
the fox.

3. Observe which way the fox has started to run around the circle.
Turn at a RIGHT ANGLE in the opposite direction (i.e. if you started
swimming due south in step 2 and the fox started running to the east,
i.e. clockwise around the pond, then start swimming due west). (Note:
If at the beginning of step 3 the fox is still in the same location as
at the start of step 2, i.e. directly opposite you, repeat step 2
instead of turning.)

4. While on your new course, keep track of the fox. If the fox slows
down or reverses direction, so that you again become diametrically
opposite the fox, go back to step 2. Otherwise continue in a straight
line until you reach the bank.

5. Fly away.

The duck should make delta as small as necessary in order to be able
to escape the fox.

The key to this strategy is that the duck initially follows a
radial path away from the fox until the fox commits to running either
clockwise or counterclockwise around the pond. The duck then turns onto
a new course that intersects the circle at a point MORE than halfway
around the circle from the fox's starting position. In fact, the duck
swims along a tangent of the circle of radius r/a. Let

theta = arc cos (1/a)

then the duck swims a path of length

r sin theta + delta

but the fox has to run a path of length

r*(pi + theta) - a*delta

around the circle. In the limit as delta goes to 0, the duck will
escape as long as

r*(pi + theta) < a*r sin theta

that is,

pi + arc cos (1/a) - a * sqrt(a^2 - 1) < 0

Maximize a in the above: a = 4.6033388487517003525565820291030165130674...
The fox can catch the duck as long as he can run about 4.6 times as fast as
the duck can swim.

"But wait," I hear you cry, "When the duck heads off to that spot
'more than halfway' around the circle, why doesn't the fox just double
back? That way he'll reach that spot much quicker." That is why the
duck's strategy has instructions to repeat step 2 under certain
circumstances. Note that at the end of step 2, if the fox has started
to run to head off the duck, say in a clockwise direction, he and the
duck are now on the same side of some diameter of the circle. This
continues to be true as long as both travel along their chosen paths
at full speed. But if the fox were now to try to reach the duck's
destination in a counterclockwise direction, then at some instant he
and the duck must be on a diameter of the pond. At that instant, they
have exactly returned to the situation that existed at the end of step
1, except that the duck is a little closer to the edge than she was
before. That's why the duck always repeats step 2 if the fox is ever
diametrically opposite her. Then the fox must commit again to go one
way or the other. Every time the fox fails to commit, or reverses his
commitment, the duck gets a distance delta closer to the edge. This
is a losing strategy for the fox.

The limiting ratio of velocities that this strategy works against
cannot be improved by any other strategy, i.e., if the ratio of
the duck's speed to the fox's speed is less than a then the duck
cannot escape given the best fox strategy.

Given a ratio R of speeds less than the above a, the fox is sure to
catch the duck (or keep it in water indefinitely) by pursuing the
following strategy:
Do nothing so long as the duck is in a radius of R around the centre.
As soon as it emerges from this circle, run at top speed around the
circumference. If the duck is foolish enough not to position itself
across from the center when it comes out of this circle, run "the short
way around", otherwise run in either direction.

To see this it is enough to verify that at the circumference of the
circle of radius R, all straight lines connecting the duck to points
on the circumference (in the smaller segment of the circle cut out
by the tangent to the smaller circle) bear a ratio greater than R
with the corresponding arc the fox must follow. That this is enough
follows from the observation that the shortest curve from a point on
a circle to a point on a larger concentric circle (shortest among all
curves that don't intersect the interior of the smaller circle) is
either a straight line or an arc of the smaller circle followed by a
tangential straight line.


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