NY Times math problem

It is not clear what the "aim point" is. If it is defined as the intersection of the tangent of the rabbit's path with the circle, then if the agent always runs toward the aim point then the rabbit can escape (even if the agent is very fast). But this assumes a stupid agent (e.g. if the rabbit is close to shore it can make the agent run away from it by swimming slowly toward the center). If the agent chooses a simpler strategy, run in the direction that decreases the angular separation (if the angular separation is 0, do not move; if the angular separation is 180 degrees, run clockwise) then the rabbit cannot cause the agent to reverse direction if the rabbit is more than 1/4 of the radius from the center. Indeed this second strategy is easily seen to be optimal for the agent.

- William Hughes

The cartoonist Sam Gross might suggest that the rabbit expire, escaping to bunny heaven. That's what the befuddled scientist's laboratory mice did, in one published work.

RL

Yes it does, which is exactly why I was arguing that it was a poor strategy for the agent to employ.

- Tim

... snip ...

You have all missed the obvious. The rabbit teases the agent, who spends his time running mightily to be on the same pond side as the rabbit. This leads to mutual exhaustion and death by shriveling. Now the only problem is "who shrivels first". Since the agent travels 4 times as far, and does not have the benefit of cooling in the pond, nor easily available drinking water, I contend the agent shrivels first.

Unless of course the agent is armed with a holy hand grenade. ;^)

And can count to three.

- William Hughes

Good point. I had an intuitive understanding for what I meant, but upon closer introspection, I see that it is not clearly stated.

What about "the intersection of the radius containing the rabbit and the edge of the pool, regardless of the direction of the rabbit's motion"? Would aiming for that be synonymous to the strategy of the agent moving to decrease the angular separation (between the rabbit and the tangent line that defines the instantaneous direction of motion of the agent?). What about just saying 'the agent moves to minimize the distance between himself and the rabbit'?

--riverman

Splash!

I've not done anything on paper, but isn't this simply a variation of the pursuit curve problem... the rabbit swims along a vector defined by the agent's position and the center-point of the pond?

...Jim Thompson

That must be a super rabbit, to keep swimming like that.

We have a jackrabbit in the neighborhood. I see him every morning running along the ridge as I'm pouring my coffee ;-)

...Jim Thompson

A useful point, but the name "aim point" seems odd as the point has no dependence on the direction of the rabbit's motion. (It is certainly not the meaning that Tim Little ascribes to "aim point")

Yes.

Note: Angular separation: Take a polar coordinate system centered at the circle. The angular separation is the angular difference between the rabbit's angular coordinate and the agent's angular coordinate.

Equivalent to minimizing the angular separation (noting that the agent's motion is constrained to be outside or on the boundary of the disk of radius r, otherwise, as Richard Heathfield points out, splash).

- William Hughes

M Rath wrote:

I writing a coordinate output computer program for an outward spiral...but my compiler is acting up.

Delphi console mode is not recognizing "Readln" after a simple loop in the main program...

Somebody wrote this:

And then William Hughes wrote this:

But I don't know who wrote the following:

Well...you people don't understand Federal police work. Let the rabbit run. Then coerce a witness into connecting the rabbit to the crime. Finally issue an APB for the rabbit...and after that it's just a matter of time before regional police are calling the Federal police with the capture.

(And a cable news columnist who is a former prosecutor sometimes points out someone saying that they can "make the case" and that when knowing that the case is not true. So the adversarial justice system sometimes gets confused with the adversarial political system.)

Well...in the movie "Day of the Jackel"...they said "get a name...then we have a passport...then we have a face...and after that it's just basic police work..."

So they get you with your papers...and the politicians love systems that make them look smart and capable.

So where did that rabbit get its shots ?

LOL. Within, of course, the constraints of the stated problem. "...a rabbit in the middle of a pond, an agent who can run around the shore four times as fast as the rabbit can swim."

Let's not let all this precision affect our accuracy, shall we? :-)

--riverman

What do you mean by 'optimal for the agent', since it leads to the rabbit escaping with the least travelled distance. I'd say its optimal for the rabbit. Strange if it turned out to be optimal for both!

I can see that this strategy will work with any size pool, as the distances and velocities are all relative. But what if the pool is reduced to a POINT (as in a limit?) Who wins then?

--riverman

It seems obvious to me that the rabbit will end up describing a spiral path, as it will constantly be trying to stay 180 degrees away from the agent. The agent will move so as to to stay at the same angle as the rabbit, & the rabbit will be trying to stay 180 degrees away from the agent. A spiral is the obvious result, ending up with the rabbit hitting the shore a whisker away from the agent. (Alternatively, the rabbit could desribe a zig-zag path, but the end-result would be the same.)

NOT FOUR!

Well, that sure sucks!