[Guest]

One cold winter day, a young woman finds herself alone near a frozen lake. Suddenly, a hooded thug appears from behind the trees holding a knife and demands all her money. In panic, she runs out over the frozen lake but finds that the lake is too slippery to run on - she can only walk fast...

Luckily for her, the thug is afraid to step on the ice (probably because of some traumatic childhood incident, I imagine). But he waits patiently around the edge of the lake always moving towards her.

Now begins the puzzle:

Assume the lake is perfectly circular with a radius of 1 mile exactly. The thug can run exactly 4 times as fast as she can walk on the ice. If she can get to the edge of the lake before the thug gets there, she can outrun him (she's in her college track team, you see). At the dead center of the fully frozen lake, there is a buoy.

Part 1 - Easy

a) How can she reach the edge of the lake quickest before the thug gets there?

b) How far will the thug be when she gets there?

Part 2 - Advanced

a) What strategy should she adopt to maximize the distance between her and the thug when she gets to the edge?

b) How much is this maximum possible distance (along the edge of the lake) ?

PS: She doesn't have a cell phone - so no wisecracks about calling 911 :-P And there is nobody else around to hear her scream for help!

[bonanova]

Identical to this post.

With added questions ... keeping it open.

[Guest]

Don't you need to know where she is on the ice to start? If she has only gone a few feet when she realizes she can't run, it changes the answer from, say, if she were in the center of the ice.

[akaslickster]

Ice is thick or can it break?

Edited May 8, 2008 by akaslickster

[Guest]

Don't you need to know where she is on the ice to start? If she has only gone a few feet when she realizes she can't run, it changes the answer from, say, if she were in the center of the ice.

It seems to me that's another difference between this problem and the original. However, based on the OP, I would assume the girl and thug start at the same position on the perimeter of the lake, and the girl gets to move first. So far, it seems to me like she'll end up running to the center anyhow, but it's possible there's a faster method that involves changing direction before that.

[Guest]

If she starts to slowly walk away from the thug, then the thug will go to the other side. Once he's at the other side just start to walk back the way you came. Because she is walking, she won't get tired nearly as quickly as the thug. Once the thug get's so tired that he will collapse, just walk away from him and she will be fine. The thug will be too tired to run.

[bonanova]

Don't you need to know where she is on the ice to start? If she has only gone a few feet when she realizes she can't run, it changes the answer from, say, if she were in the center of the ice.

It seems to me that's another difference between this problem and the original. However, based on the OP, I would assume the girl and thug start at the same position on the perimeter of the lake, and the girl gets to move first. So far, it seems to me like she'll end up running to the center anyhow, but it's possible there's a faster method that involves changing direction before that.

This is not a solution, just an observation, so I won't "spoil" it.

The previous riddle simply asked how it was possible to escape.

That answer has already been posted there.

Now, since in this riddle the girl has complete freedom on the ice, and since the center is marked by a buoy,

the girl is free to walk to the center [the thug won't go onto the ice] and use the already posted strategy to escape.

But this riddle asks for more than that - we are asked to find an optimal solution.

That's the added challenge that makes this thread worth keeping open.

[Guest]

Incidentally, because I wanted to be able to test and prove different theories, I wrote a program to simulate this. I alternate moves, with the girl moving first, and I assume that the thug will always move as fast as possible towards the current angle of the girl. Using the previously recommended solution, the girl makes it to safety in 640 steps of .01 radius lengths. The nice thing is that if anybody thinks they can come up with a faster way, or specific values which are optimal, I can test it.

Edit: I hope to answer the harder questions after I can test with a few different options. Incidentally, so far, I've found that she gets out faster (186 steps) if she starts moving along the arc at .15 radians from the center. At .1, she never makes it out. Hmmm. Not sure why yet.

Edited May 9, 2008 by Duh Puck

[Guest]

If she moves to the middle of the lake and then looks for the purp, then exits in the opposite direction of where he is standing at that point he will have a very long run to her walk and as stated she can easily outrun him on land.

[Guest]

If she moves to the middle of the lake and then looks for the purp, then exits in the opposite direction of where he is standing at that point he will have a very long run to her walk and as stated she can easily outrun him on land.

Actually, no. If you review the other thread you'll see that she gets caught if she does that. His "very long run" is (radius x pi), while her walk is just radius, for a ratio of 3.142:1. However, he walks four times as fast as her, so he'll be waiting at the edge when she gets there. The previously suggested solution allows her to get directly opposite the thug while being closer to the edge than the center, so she can escape.

Incidentally, I haven't had a chance to review the program logic, but it seems like there must be a bug, because she obviously couldn't escape if she bolted straight for the edge, starting from a distance .15R from the center. Also, since the fastest solution would seem to be whatever gets you opposite the thug while at least .21R from the center, I really doubt that the current method (straight toward center, move on arc, straight toward edge) is the most efficient way to achieve that.

[Guest]

She goes to the middle and then spirals outward towards land.

[Guest]

She goes to the middle and then spirals outward towards land.

That doesn't work, because once she's more than .25R from the center, the thug can always stay lined up with her angle.

[bonanova]

[1] The girl starts at center

[2] Keeping the thug at her back, she moves radially a distance r(4-pi)/4. This requires both radial and "angular" movement.

[3] At r(pi/4) distance to the edge, she moves radially [directly] to the edge, arriving simultaneously with the thug.

[4] She outruns the thug on land.

There are two issues with this:

[1] The amount of angular movement required in step [2] depends on the thug's movement.

The worst case is the thug immediately moves at top speed CW or CCW throughout the escape attempt.

Then we can calculate the minimum distance the girl must move in step [2], altho it's not a trivial calculation.

[2] Simultaneous arrival at the end of step [3] might not constitute escape.

However, assuming the girl and the thug are point objects, the girl need only move an infinitessimally greater distance in step [2] to arrive at the edge ahead of the thug.

[Guest]

Yes. I can simulate that. I'm in the process of upgrading the development tools on my notebook so it might be a day or two.

In my current simulation, in each iteration 1) the girl moves, then 2) thug always moves at top speed toward the new angle of the girl. I believe this is the optimal approach for the thug, and we obviously need optimal movement from him to have an accurate solution. Can you envision any scenario in which it would be advantageous for the thug to act otherwise?

In any case, I've currently provided the following available moves for the girl:

1. Move toward center.

2. Move toward edge.

3. Move on arc (maintain current distance from center and move away from thug angle)

4. Move to opposite edge(moves toward point on edge opposite thug's current position)

5. Move away on tangent (moves straight perpendicular to radial angle, away from thug. e.g., if facing away from center and thug is behind her to her left, she makes a right turn)

I will add "Move Opposite Thug," which will mean that she will move away from the thug along the line drawn between their locations.

In response to your suggestions ...

[1] The girl starts at center

Since the girl starts at the edge, I have a suspicion that turning before reaching the center will be faster, but I'll test it.

[2] Keeping the thug at her back, she moves radially a distance r(4-pi)/4. This requires both radial and "angular" movement.

I don't currently move her along an adjusted curve, but using small increments of "Move Opposite Thug" should create that effect. However, I suspect she will be doing a lot of angular motion (spiraling) as she approaches the r(4-pi)/4.

[3] At r(pi/4) distance to the edge, she moves radially [directly] to the edge, arriving simultaneously with the thug.

[4] She outruns the thug on land.

There are two issues with this:

[1] The amount of angular movement required in step [2] depends on the thug's movement.

The worst case is the thug immediately moves at top speed CW or CCW throughout the escape attempt.

Then we can calculate the minimum distance the girl must move in step [2], altho it's not a trivial calculation.

[2] Simultaneous arrival at the end of step [3] might not constitute escape.

However, assuming the girl and the thug are point objects, the girl need only move an infinitessimally greater distance in step [2] to arrive at the edge ahead of the thug.

It's just like establishing a calculus limit. Any amount longer on step [2] and she'd be farther away when she reaches the edge, so I would think that solution is good enough.

However, we still need the second part of the solution. How to maximize distance upon reaching the edge?

[Guest]

[1] The girl starts at center

[2] Keeping the thug at her back, she moves radially a distance r(4-pi)/4. This requires both radial and "angular" movement.

[3] At r(pi/4) distance to the edge, she moves radially [directly] to the edge, arriving simultaneously with the thug.

[4] She outruns the thug on land.

There are two issues with this:

[1] The amount of angular movement required in step [2] depends on the thug's movement.

The worst case is the thug immediately moves at top speed CW or CCW throughout the escape attempt.

Then we can calculate the minimum distance the girl must move in step [2], altho it's not a trivial calculation.

[2] Simultaneous arrival at the end of step [3] might not constitute escape.

However, assuming the girl and the thug are point objects, the girl need only move an infinitessimally greater distance in step [2] to arrive at the edge ahead of the thug.

In step 2 of your solution, you say she needs to keep the thug at her back. If this means she always moves directly away from thug, then she will not actually escape, because the thug will not be on the opposite side of the circle when she reaches step 3. If this means she always keeps the center between herself and the thug (i.e., she moves to the farthest point she can reach on the line drawn between the thug and the center), then she should escape, but I'm not sure that it will be any faster to do it this way rather than the previously suggested solution. I'll test tonight.

[bonanova]

In step 2 of your solution, you say she needs to keep the thug at her back. If this means she always moves directly away from thug, then she will not actually escape, because the thug will not be on the opposite side of the circle when she reaches step 3. If this means she always keeps the center between herself and the thug (i.e., she moves to the farthest point she can reach on the line drawn between the thug and the center), then she should escape, but I'm not sure that it will be any faster to do it this way rather than the previously suggested solution. I'll test tonight.

I didn't put that very well.

Rather I meant move outward in a manner that keeps herself, the center of the ice and the thug in a straight line.

Inside the "escape radius" she's always able to do that.

She will move at her max speed with a tangential component to her velocity that matches the thug's angular velocity

and the rest of her velocity in an outward radial direction.

[Guest]

The girl walks to the centre of the lake and ducks down behind the buoy (opposite side of where the thug currently stands). The girl can then crawl away in the opposite direction from the thug, hidden from view by the buoy (however slow this may be). The thug won't see her doing this so won't know to start running again. Once she's 3/4 of a mile from the shore (her speed = 0.25 his speed, 0.25 x 3.14 = 0.785 miles minimum safe distance), she can stand up, walk, and beat him to the edge.

Edited May 15, 2008 by mgwilliams

[Guest]

Arg... this solution is complicated. Well, if we could come up with a function, we can use derivatives, and optimization to solve this. (I think?) The girl has to move in some sort of movement like an arc though, correct? Or the graph of 1/x or a log function, something like that? Or at least, that's what's coming into my head right now. >_> After I've had some more time to think about it, I'll put it up, but for now, just some initial thoughts on the solution.

Oh, but for the second part, something like running a certain angle away from the thug would get her the farthest? Or is it just running in whatever direction the thug is running? O_o

[Guest]

like mgwilliams said she'll go to the center, until she reaches the bouy. Once there she can find the purp and speed walk the other way.

[Guest]

Part 1 - Easy

a) How can she reach the edge of the lake quickest before the thug gets there?

b) How far will the thug be when she gets there?

Part 2 - Advanced

a) What strategy should she adopt to maximize the distance between her and the thug when she gets to the edge?

b) How much is this maximum possible distance (along the edge of the lake) ?

I'm going to say there is no way for her to get off the lake using the most optimized spiral with the thug travelling at four times the rate of the girl. If the curve is too shallow (straight, worst case) the thug catches up. If the curve is too steep the thug will catch up and be at the edge of the lake which represents the shortest distance for her to get off the lake directly. The only way for her to maximize her distance between her and the thug is to go back to the centre of the lake.

I'm just itching to write a program to test my hypothesis.

[Guest]

Oops, can't edit my previous post. Please refer to the spoiler for my previous spoiler edits:

Ok, now after reading the solution for the previous puzzle things are making a bit more sense, and I'm feeling a little dumb... hehe oh well, it's Friday.

Running from the edge straight to the centre we have a distance travelled = 1 mi.

We know that the straight escape path is 0.75 mi.

So far, our minimum distance travelled is 1.75mi.

The only real thing to figure out now is the optimal path from the centre to the 0.25 mi ring, which I believe will start with high angular velocity and end with a low angular velocity.

To help with the math, the thugs position can be represented on the 1 mi circle by (X,Y) = F( R ), the circle function of radius R, or R=sqrt(X^2+Y^2).

Thug -> 1=sqrt(X^2+Y^2)

Girl -> (0..0.25)=sqrt(X^2+Y^2) -- (In the domain of reals, or floats, if you will )

Our (normalized) escape condition is (Xt, Yt) = (-Xg, -Yg)

Ewww, partial derivates, I'm gonna have to go back to my text book. Good thing it's a long weekend.