49 replies to this topic

[phaze]

Spoiler for smart maiden outwits ogre!

Assuming that the maiden's goal is merely to get out of the lake without encountering the ogre, that she has no particular direction she is obliged to go.

Wouldn't it behoove the maiden to perform continuous reevaluations of her course and steer to the spot currently opposite the ogre at every moment? I'll bet she'd have to be terribly slow oarswoman to lose this race. (However, I don't have the numbers...   

Spoiler for Wouldn't it be..

rather than the spot opposite from the ogre more basically put "away from the ogre" ?

The reason I ask is that this could be translated in this manner

a maiden that is on the brink of winning the contest sees the ogre is 2 or 3 meters away choose instead of grasping victory cross almost the entire length of the lake once more to get the opposite side of the lake from the ogre

At a guess maiden needs to be able to go over 1 1/3 miles per hour travels pi * lake radius/2 while the ogre runs 3/4 of the way around the lake

[CaptainEd]

Yes, phaze, I was way off base.

Spoiler for fuzzy thoughts

M should move to the Prime location, then head toward the closest land she can reach before the Ogre does.

If there is none, but she's almost fast enough, I can advise to keep going until the Ogre has committed to a side of the lake, and then make a heading change, but the proper time and direction are a matter of simulation rather than closed form for me right now.

[Prime]

I have a cold, my head is stuffed, I feel like an Ogre. I am not figuring out the differential equations to deny hungry Ogre his meal. However, here is some food for thought for those Brain Denizens who took the maiden's side. Indeed, she could go slower. Mayhap, someone will try carrying out the required calculations, make an error and give the maiden a wrong advice.

Spoiler for maiden voyage

To make the contest fair, the maiden should have only so much time for rowing. Having exhausted it, she falls asleep. Whereafter, the Ogre should start blowing at the boat pushing it towards the shore...

Edited by Prime, 22 February 2013 - 02:09 AM.

[Yoruichi-san]

Spoiler for Are you maiden or uber-maiden?

So clever and head-ache inducing feinting aside, I think that her best strat is to go in S-shaped movements that slowly drift backwards while the ogre is kept going in a 'pendulum' type motion on the opposite side of the lake.

Basically she needs to turn around before the ogre is committed to one direction each time...so her range of motion would have to decrease as she gets farther away from the Ogre, so it'd look something like a dampened harmonic...

 

I shoveled somewhere between 150-200 cubic feet of snow today so I don't really feel like doing the math right now tho...hooray for the snow-pocalypse...I think I prefer zombies >.<

[bonanova]

Prime has got the answer, at least as far as I have analyzed it.

That being true, I will post my analysis and people can take pot shots at improving it.

 

Spoiler for On a tangent

The large circle, radius R, is the lake. The smaller circle, radius r, contains the escape starting point for the boat. The ogre's speed multiple is f > 1, so R = f.r, using the period to mean multiplication. The rower has thus achieved her greatest possible separation from the ogre for which she can maintain the lake's center directly between them. We now compare two escape strategies.

[1] Radial escape.

The three black dots are, from right to left, the Ogre's and the rower's starting points, and the point on the shore that each heads for as the chase begins.  The black circular arc is the ogre's path, and the black leftward arrow is the rower's path. The equation of simultaneous arrival is simply

f.x = R.pi = f.r.pi
x = r.pi

Noting that x = R-r = r(f-1) we obtain

f = 1 + pi = 4.14159...

as the maximum ogre speed factor advantage that can be overcome using radial escape.

She must be able to row 0.966 mph.

[2] Tangential escape.

The red dot now is the point on the shore that each is headed for. The sum of the black and red circular arcs is the ogre's new path, and the red upward arrow is the rower's new path. The equation of simultaneous arrival is a little more complicated, but not much.

First, we need to find the extra path length for the ogre. It's simply a.R [the red arc] where a is the angle from horizontal to the new destination point. Thus

f.y = R(pi + a) = f.r(pi + a)
y = r(pi + a)

Noting that tan(a) = y/r

tan(a) = pi + a which gives a = 77.45 degrees.

Noting that cos(a) = y/R = 1/f, we obtain f = 1/cos(77.45)

f = 4.6033

This is an 11% increase in the ogre's speed that can be overcome using tangential escape compared to radial escape.

Thus the rower can row 11% slower and still escape. 

She need only row 0.869 mph.

--------

Two comments.

First, this is also only a sufficient condition. This analysis does not rule out a more efficient escape strategy. Spirals are good next steps to investigate. Piecewise linear escape paths would be easier to investigate.

Second, one could ask whether the ogre, once he sees the direction taken by the rower, might not help himself by switching directions and taking the shorter route to the red dot. The answer is no. If he does, the rower fires up photoshop, constructs a new, larger inner circle, and heads tangentially for shore in the other direction, the result of which is actually to the rower's net advantage. The ogre's best strategy is to stay the course [clockwise in this case] for the red dot.

The drawing is for a value of f in the general range we are considering. That is, a is not exact, but close to 77.45 degrees. Thus it is valid to perceive the advantage of tangential escape by noting that y is not significantly larger than x, while a + pi is significantly larger than pi. Tangential escape makes the ogre's task more difficult.

Edited by bonanova, 22 February 2013 - 05:38 AM.
Giving proper attribution to Prime

[Prime]

Phaze has got the answer, at least as far as I have analyzed it.

That being true, I will post my analysis and people can take pot shots at improving it.

 

Spoiler for On a tangent

The large circle, radius R, is the lake. The smaller circle, radius r, contains the escape starting point for the boat. The ogre's speed multiple is f > 1, so R = f.r, using the period to mean multiplication. The rower has thus achieved her greatest possible separation from the ogre for which she can maintain the lake's center directly between them. We now compare two escape strategies.

[1] Radial escape.

The three black dots are, from right to left, the Ogre's and the rower's starting points, and the point on the shore that each heads for as the chase begins.  The black circular arc is the ogre's path, and the black leftward arrow is the rower's path. The equation of simultaneous arrival is simply

f.x = R.pi = f.r.pi
x = r.pi

Noting that x = R-r = r(f-1) we obtain

f = 1 + pi = 4.14159...

as the maximum ogre speed factor advantage that can be overcome using radial escape.

She must be able to row 0.966 mph.
boat escape.gif

[2] Tangential escape.

The red dot now is the point on the shore that each is headed for. The sum of the black and red circular arcs is the ogre's new path, and the red upward arrow is the rower's new path. The equation of simultaneous arrival is a little more complicated, but not much.

First, we need to find the extra path length for the ogre. It's simply a.R [the red arc] where a is the angle from horizontal to the new destination point. Thus

f.y = R(pi + a) = f.r(pi + a)
y = r(pi + a)

Noting that tan(a) = y/r

tan(a) = pi + a which gives a = 77.45 degrees.

Noting that cos(a) = y/R = 1/f, we obtain f = 1/cos(77.45)

f = 4.6033

This is an 11% increase in the ogre's speed that can be overcome using tangential escape compared to radial escape.

Thus the rower can row 11% slower and still escape. 

She need only row 0.869 mph.

--------

Two comments.

First, this is also only a sufficient condition. This analysis does not rule out a more efficient escape strategy. Spirals are good next steps to investigate. Piecewise linear escape paths would be easier to investigate.

Second, one could ask whether the ogre, once he sees the direction taken by the rower, might not help himself by switching directions and taking the shorter route to the red dot. The answer is no. If he does, the rower fires up photoshop, constructs a new, larger inner circle, and heads tangentially for shore in the other direction, the result of which is actually to the rower's net advantage. The ogre's best strategy is to stay the course [clockwise in this case] for the red dot.

The drawing is for a value of f in the general range we are considering. That is, a is not exact, but close to 77.45 degrees. Thus it is valid to perceive the advantage of tangential escape by noting that y is not significantly larger than x, while a + pi is significantly larger than pi. Tangential escape makes the ogre's task more difficult.

Tangential escape does not seem the optimal even for the straight line escapes. The straight line escape with a 45-degree angle seems to perform better.

[CaptainEd]

Spoiler for doubting Ogre gives up so quick

so you mean the Ogre looks South, and the Maiden takes off to the North, and the Ogre (a Zen master apparently) sees the inevitability, and continues the long way around, like Mark Twain's "Cooper Indian". If the Ogre does turn and go north, he can go four times further north than M can. Exactly when should M recalculate her new tangent? Will it really require as long a road for Ogre?

[Prime]

Spoiler for doubting Ogre gives up so quick

so you mean the Ogre looks South, and the Maiden takes off to the North, and the Ogre (a Zen master apparently) sees the inevitability, and continues the long way around, like Mark Twain's "Cooper Indian". If the Ogre does turn and go north, he can go four times further north than M can. Exactly when should M recalculate her new tangent? Will it really require as long a road for Ogre?

 

It is a lot simpler than that for the zen master Ogre. The Ogre ignores where the maiden looks, where she goes, and where she intends to go. Ogre simply draws an imaginary straight line from the center of the lake, through the boat, to the shore; and heads for that spot using the shorter arch. Insofar as the optimum is concerned, it is the Ogre who determines the path -- not the maiden. The maiden reacts to Ogre's moves, not other way around.

If the maiden wanders back into her inner circle r, that could make the Ogre change the direction when the boat crosses straight line drawn from the Ogre to the center of the lake. Then the young lady would be going towards Ogre rather than away from him. (Not an optimal strategy.)

Presently, we are solving the sub-problem, where the maiden after leaving her inner circle goes in a straight line towards some point on the shore. We must find, that point. Bonanova suggested going off the inner circle on a tangent line. That gives a better ratio than going in a straight line to the nearest point on the shore. However, I don't believe it is the optimum.

The numbers I have given in the post #23 inside the spoiler are off. But then, as everyone knows, I am siding with Ogre.

[bonanova]

Spoiler for doubting Ogre gives up so quick

so you mean the Ogre looks South, and the Maiden takes off to the North, and the Ogre (a Zen master apparently) sees the inevitability, and continues the long way around, like Mark Twain's "Cooper Indian". If the Ogre does turn and go north, he can go four times further north than M can. Exactly when should M recalculate her new tangent? Will it really require as long a road for Ogre?

 

It is a lot simpler than that for the zen master Ogre. The Ogre ignores where the maiden looks, where she goes, and where she intends to go. Ogre simply draws an imaginary straight line from the center of the lake, through the boat, to the shore; and heads for that spot using the shorter arch. Insofar as the optimum is concerned, it is the Ogre who determines the path -- not the maiden. The maiden reacts to Ogre's moves, not other way around.

If the maiden wanders back into her inner circle r, that could make the Ogre change the direction when the boat crosses straight line drawn from the Ogre to the center of the lake. Then the young lady would be going towards Ogre rather than away from him. (Not an optimal strategy.)

Presently, we are solving the sub-problem, where the maiden after leaving her inner circle goes in a straight line towards some point on the shore. We must find, that point. Bonanova suggested going off the inner circle on a tangent line. That gives a better ratio than going in a straight line to the nearest point on the shore. However, I don't believe it is the optimum.

The numbers I have given in the post #23 inside the spoiler are off. But then, as everyone knows, I am siding with Ogre.

 

Blame it on the cold, Prime.

 

I am not convinced that tangential is optimal - it may not be enough!

It's easy to overlook that r is one of the unknowns.

[Prime]

 

Spoiler for doubting Ogre gives up so quick

so you mean the Ogre looks South, and the Maiden takes off to the North, and the Ogre (a Zen master apparently) sees the inevitability, and continues the long way around, like Mark Twain's "Cooper Indian". If the Ogre does turn and go north, he can go four times further north than M can. Exactly when should M recalculate her new tangent? Will it really require as long a road for Ogre?

 

It is a lot simpler than that for the zen master Ogre. The Ogre ignores where the maiden looks, where she goes, and where she intends to go. Ogre simply draws an imaginary straight line from the center of the lake, through the boat, to the shore; and heads for that spot using the shorter arch. Insofar as the optimum is concerned, it is the Ogre who determines the path -- not the maiden. The maiden reacts to Ogre's moves, not other way around.

If the maiden wanders back into her inner circle r, that could make the Ogre change the direction when the boat crosses straight line drawn from the Ogre to the center of the lake. Then the young lady would be going towards Ogre rather than away from him. (Not an optimal strategy.)

Presently, we are solving the sub-problem, where the maiden after leaving her inner circle goes in a straight line towards some point on the shore. We must find, that point. Bonanova suggested going off the inner circle on a tangent line. That gives a better ratio than going in a straight line to the nearest point on the shore. However, I don't believe it is the optimum.

The numbers I have given in the post #23 inside the spoiler are off. But then, as everyone knows, I am siding with Ogre.

 

Blame it on the cold, Prime.

 

I am not convinced that tangential is optimal - it may not be enough!

It's easy to overlook that r is one of the unknowns.

I did overlook that r is one of the unknowns because of cold and my partiality to Ogre's cause.

Still I don't see that the problem is solved even for a straight line escape.

By the way, I did not see the answer by Phaze to which you referred. Is BD not displaying some of the posts to me?

Edited by Prime, 22 February 2013 - 05:27 AM.