NY Times math problem

yes, this straight line may not be a radius, but it is unlikely to differ greatly from a radius.

or

2 a

at

nt.

th

ath

Using your strategy, what is the answer to part 2 of the puzzle?

Dave

Us Part 2...what's the fastest speed the agent can run where the rabbit can still escape ?

Oh...yeah thanks for that.

Well...I have the rabbit escape at an azimuth of 259.25649 when the agent is at an azimuth of 535. And so the rabbit got out at a azimuth of 619.25649 relative to the agent .

So the agent got around 1.48611 loops but needed to get around 1.72016 loops to catch the rabbit with this systematic method...

Okay...I have the radius at 10 so the arc distance around the circle is

62.832 . Divide by a agent rate of 4 and that's 15.708 time units for the agent to go 360 degrees around the circle. So the agent went 535 degrees in 23.3438 time units. But the agent needs to go 619.25649 degrees (or 1.72016 loops) in 23.3438 time units. Okay...1.72016 loops * 62.832 circle arc distance = a distance of 108.0811. Rate * Time = Distance so Rate = 108.0811 / 23.3438 or a rate of 4.6300 . Tie goes to the runner...

But now does the rabbit make the same path with this systematic method ?

Since the agent is moving around the circle faster...the rabbit is making directional changes at a quicker pace even as the rabbit speed stays the same. Well the rabbit speed is slower relative to the agent at 21.5983% verus the previous 25%...so I have to change the rabbit distance (per degree of agent arc distance) in the KBH Code to 0.037696233 and run the code for the answer.

And now the rabbit gets out at an agent azimuth of 786 degrees or 2.18333 loops...but the rabbit gets out at about the same spot as before.

So I'll just conclude that the rabbit always gets out no matter what the agent speed ? Or try a slower agent speed ?

I change the rabbit distance (per degree of agent arc distance) to

0.049570229 and the rabbit gets out at an agent azimuth of 468 degrees. And beginning here the exit coordinates are different than the previous two...

Okay change the rabbit distance (per degree of agent arc distance) to 0.06 and the rabbit gets out at an agent azimuth of 238 degrees.

Okay change the rabbit distance (per degree of agent arc distance) to 0.07 and the rabbit gets out at an agent azimuth of 171 degrees.

Change the rabbit distance (per degree of agent arc distance) to 0.10 and the rabbit gets out at an agent azimuth of 108 degrees.

Change the rabbit distance (per degree of agent arc distance) to 0.20 and the rabbit gets out at an agent azimuth of 51 degrees.

Change the rabbit distance (per degree of agent arc distance) to 0.30 and the rabbit gets out at an agent azimuth of 34 degrees.

Change the rabbit distance (per degree of agent arc distance) to 0.40 and the rabbit gets out at an agent azimuth of 26 degrees.

Change the rabbit distance (per degree of agent arc distance) to 0.50 and the rabbit gets out at an agent azimuth of 21 degrees.

Change the rabbit distance (per degree of agent arc distance) to 0.60 and the rabbit gets out at an agent azimuth of 17 degrees.

Change the rabbit distance (per degree of agent arc distance) to 0.70 and the rabbit gets out at an agent azimuth of 15 degrees.

Change the rabbit distance (per degree of agent arc distance) to 1.00 and the rabbit gets out at an agent azimuth of 11 degrees or less.

Change the rabbit distance (per degree of agent arc distance) to 10.00 and the rabbit gets out at an agent azimuth of 1 degrees. So each time the agent moves 0.174533333 the rabbit moves 10.00 ! So here the rabbit speed is

57.2956 times the agent speed instead of the given rabbit speed of 0.25 times the agent speed.

Part 2...what's the fastest speed the agent can run where the rabbit can still escape ?

The fastest speed...not relative to the given speed...that the agent can run...with the rabbit still escaping...is...the smallest possible speed greater than zero ?

is

ops

in

16 108.0811

ree

r

nd

. 6 7 d

ent

It looks like you have had the rabbit running faster and faster, but you should be increasing the ratio of speed of the agent to the speed of the rabbit, until the rabbit can't escape.

Now, if the agent can run 10 times as fast as the rabbit can swim, he can position himself at the closest point on the shore to the rabbit. The rabbit then will swim away from the shore. Thus the rabbit cannot escape from the pond. We've already established that the rabbit can escape when the agent runs 4 times as fast as the rabbit can swim. What number between 4 and 10 is the fastest the agent can run and still let the rabbit escape? We are assuming that the rabbit follows your strategy and the agent follows his best strategy.

Dave

Oh...at an agent rate of 4 the rabbit gets out at an azimuth of 619.25 with the agent at an azimuth of 535 . But relate this as to one loop with a rabbit azimuth of 259.25 and an agent azimuth of 175.

And at an agent rate of 4.63 the rabbit gets out at an azimuth (related as to one loop) of 259.25 with the agent azimuth of 393 (related as going in to the second loop).

So there might be some agent rate between 4 and 4.63 where the agent run and the rabbit run catch...with this systematic method ?

Okay...that's the slowest speed that the agent can run with a rabbit escape. The fastest speed might be unlimited unless a note that I added above works out...

Well I know the output of the systematic program...in the first case from the plot...so I will go back to ultimate values as I expect them.

At an agent rate of 4 the rabbit gets out at an azimuth of 619.25 with the agent at an azimuth of 535 ...

And at an agent rate of 4.63 the rabbit gets out at an azimuth of about

619.25 with the agent azimuth of 786...

So there might be some agent rate between 4 and 4.63 where the agent run and the rabbit run catch...

e

If I were the rabbit, I'd sit in the center for a while to consider the situation. Or maybe I'd run out near the edge, taunt the Secret Service agent, then take a nap.

To discuss what the rabbit "will do" one needs to constrain the problem further. I would add the following objective function F (which the rabbit seeks to maximize and the agent to minimize):

Let a =3D 1 if the rabbit escapes, 0 otherwise b =3D the time it takes the rabbit to reach the edge c =3D the angle separating the rabbit from the agent when the rabbit reaches the edge

Then let F =3D a x^2 - b x + c, where "x =3D infinity" (or F =3D (a,-b,c) with dictionary ordering).

With these constraints, we can say, for example, that when the agent's speed is less than pi times the rabbit's, the rabbit will make a beeline for the point opposite the agent's initial position.

When the agent/rabbit speed ratio is the critical value given by Dave, I think that the family of optimal solutions has the following form. Both agent and rabbit move at maximal speed the entire time. The agent reverses direction arbitrarily until the rabbit leaves the critical disc, and heads toward the rabbit's exit point thereafter. The rabbit increases his distance from the center while remaining opposite to the agent until he leaves the critical disc, then travels along a tangent to the disc thereafter.

ent

ar

ic

s,

I made a small spreadsheet to use your rabbit's strategy. Correct me if I am wrong, but isn't it true that after the agent has gone about

295 degrees around the pond, he is approximately on the radius containing the rabbit? If, at that point, he keeps running at full speed, as in your program, he will be running away from the rabbit. This is hardly an optimal strategy for the agent. In fact, once the agent gets on the rabbit's radius, he should stop. The rabbit will swim back through the center and the cycle will repeat. By your strategy, the rabbit never will reach the shore even at a speed ratio of only 4, assuming that the agent uses his best strategy.

Dave

Dave:

I made a small spreadsheet to use your rabbit's strategy. Correct me if I am wrong, but isn't it true that after the agent has gone about

295 degrees around the pond, he is approximately on the radius containing the rabbit? If, at that point, he keeps running at full speed, as in your program, he will be running away from the rabbit. This is hardly an optimal strategy for the agent. In fact, once the agent gets on the rabbit's radius, he should stop. The rabbit will swim back through the center and the cycle will repeat. By your strategy, the rabbit never will reach the shore even at a speed ratio of only 4, assuming that the agent uses his best strategy.

KBH wrote:

It's a systematic method only...

But we now know what happens if the rabbit takes one straight line to shore and we know what happens if the agent only loops around the circle in one direction...with the rabbit correcting direction with each movement of the agent. In one case the rabbit has the wrong strategy and in the other case the agent has the wrong strategy.

Really I was interested in the rabbit path plot with the agent looping around the circle in one direction...and with the rabbit correcting direction with each movement of the agent.

So the rabbit...and it should be a duck...is never caught if he keeps turning away from the agent with each movement of the agent. Then assuming a strategy of reversing direction by the agent...the rabbit can only get out by picking some point where he breaks for shore. But the agent can't be sure of where the rabbit curves are going to carry and the rabbit can't be sure of his break point. It's merely possible for the rabbit to get out...

The rabbit can always move to the center of the pond.

From the center, the rabbit can move away from the agent and stay on the same diameter as the agent at least until the rabbit's angular velocity about the pond center equals the agents angular velocity about the pond center at full speed, i,e, to 1/4 of a radius from the center in the direction opposite to the agent but on the same diameter.

The rabbit is now 3/4 of a radius from the nearest point on the circumference and the agent must travel pi times that radius along the edge of the pond to get to that same point.

re

e e

In the second case, both the agent and the rabbit have the wrong strategy.

Dave

cle

ement

e g

ing a

out

sure

ure

This has been stated numerous times. Now, we're fighting down those who have suboptimal strategies for either the rabbit or the agent, especially when determining the answer to the second part of the puzzle, the fastest speed of the agent that still allows the rabbit to escape.

Dave

Virgil wrote:

Well...consider the graphics plot that I made. As the agent moves the rabbit turns away from the agent and the rabbit carries the rabbit's distance of movement. Now from any agent point on the circle a line can be scaled from the agent point to a tangent point on the rabbit path and that by knowing that the rabbit path direction...and the fact that the rabbit is moving away from the agent. And from that strategies can be developed...

Then consider the computer code that I wrote for the same situation. The code can be modified so that the agent systematically runs clockwise to a point but then changes to counterclockwise movement. I don't have time to work with it but that's interesting. For instance the agent can run clockwise for 360 degrees and then run counterclockwise 90 degrees or so to run the rabbit back toward the center.

Well...the rabbit must spiral out from the radius point as he works to get on the same radial rotation with the agent but also position on the opposite side of the radius point. If the rabbit is rotating too fast then he would just keep working outward until the rotation is correct. But if the rotation is correct but the positon is wrong then the rabbit would have to zig-zag to work into position. When in correct position on the opposite side of the radius point and on the same radial rotation with the agent...then break for the opposite shore. But break for shore hoping that it will work because the rabbit is not calculating...

It will never be a radius. If it were, he would get eaten. It will be along a radial line.

--
 [mail]: Chuck F (cbfalconer at maineline dot net) 
 [page]: 
            Try the download section.

But so long as the rabbit is less than 1/4 of a radius from center, it can continue to move outward and simultaneously maintain its position on the opposite side of same diameter that the agent is on.

Unnecessarily complicated. All the rabbit has to do is stay on the diameter thru the agent's position until it is 3/4 of a diameter from the point opposite the agent, then run straight for that point.

What the agent does cannot prevent the rabbit from ding exactly that, but if the agent at any time is not going around the pond at top speed one way or the other, it helps the rabbit.

Agent strategy: 1. While the rabbit is not on the same diameter as the agent, the agent runs as fast as possible along the boundary of the pond towards the boundary point nearest to the rabbit. 2. While the rabbit is on the same radius as the agent, stay still. 3. While the rabbit is on the opposite half of the diameter through the agent, go as fast as possible in either direction, switching directions at random.

Rabbit strategy: 1. Move as fast as possible to center of pond, then. 2. Move as fast as possible away from the agent while keeping on the same diameter as the agent until the rabbits time to the nearest point on the boundary is noticeably less than the agent's time to that point, if possible, or until it takes all the rabbits speed to maintain that distance. 3. If the rabbits time to the nearest boundary point is now less than the agent's time, run for that point, otherwise tread water.

Unless the agent is continually switching directions, the rabbit never need zig nor zag until it is ready to run for the edge.

If the speeds are correct the rabbit has .75 radii to cover but the agent has pi radii ( half a circumference) to cover, and the ratio of those distances is less that the ratio of speeds, 1 to 4 ( or about

0.2387 ) so the rabbit has a bit of a cushion.

Relative distance ratio of .75/pi is roughly 5/21 to the relative speeds

5/20, so the rabbit has a bit of a cushion.