From: keith@cco.caltech.edu (Keith Allan Schneider)
Subject: Re: ducks and foxes (SPOILER)
Keywords: ducks, foxes
Organization: California Institute of Technology, Pasadena
Date: Fri, 14 Aug 1992 19:54:11 GMT

karr@cs.cornell.edu (David Karr) writes:

>In article <1992Aug14.043132.27119@cco.caltech.edu> keith@cco.caltech.edu (Keith Allan Schneider) writes:
>>dwr2560@zeus.tamu.edu (RING, DAVID WAYNE) writes:
>>[...]
>>>4 should have been an a, but the fox still can't catch the duck (if the duck
>>>is clever! )
>>
>>Well, we were assuming that both the fox and the duck use the best strategy
>>possible, and that their reaction times were instantaneous...

>You were also assuming that the strategy YOU analyzed IS the best one
>possible.  Think about it!  Or read on ...

>Assume the ratio of the fox's speed to the ducks is a, and the radius
>of the pond is r.  Strategy S1 is for the duck to swim to a point on
>the concentric circle of radius r/a diametrically opposite the fox,
>then head out in a radial direction until she reaches the side.  It
>has been shown that the duck can then escape if a < 1 + pi.

>But there is a better strategy S2 that allows the duck to escape for
>some values of a > 1 + pi.

>The steps of strategy S2 (from the duck's perspective) are as follows:

>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 collinear).

>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 strategy S1
>(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

>Now we need to maximize a in the above.  I don't expect a closed-form
>solution, but numerically the solution is just a little bit greater
>than 4.6 .  So if the duck swims 5 mph, she can escape a fox that runs
>23 mph -- with very little room to spare.

>If anyone would care to produce a more precise value of the maximum a
>for which this strategy works for the duck, go ahead.  You might find
>it easier to solve for theta in the equation

>  pi + theta - tan theta = 0

>where theta is approximately 1.35 (remember to use radians, not
>degrees :-).

>-- David Karr (karr@cs.cornell.edu)

Okay, I see this now.  This is a better strategy, but is it the best strategy
for the duck?  If so (or not), can you prove it?

Anyone?

keith





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