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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