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An "Oldie" with a twist of lemon


Best Answer Prime, 22 February 2013 - 02:08 AM

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...

Go to the full post


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49 replies to this topic

#31 bonanova

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Posted 22 February 2013 - 05:48 AM

 

 


Spoiler for doubting Ogre gives up so quick

 

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?

 

Good catch Prime, and I have no health excuse. Apologies.

I've edited my post 25 to properly attribute you as the solver.

 

Edit:

We can't continue indefinitely increasing the angle however.

At some point it will become to the ogre's advantage to reverse his direction.


Edited by bonanova, 22 February 2013 - 05:56 AM.
Add final statement

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#32 Prime

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Posted 22 February 2013 - 06:12 AM

...

Edit:

We can't continue indefinitely increasing the angle however.

At some point it will become to the ogre's advantage to reverse his direction.

For a straight line escape, it seems clear that the optimal angle is between the zero (straight line to the nearest point on the shore) and the tangential line escape. Past the tangential escape, maiden re-enters her inner circle going towards the Ogre causing him to change the direction.

You have calculated the precise rate O:M rate (f) for the tangential escape. And it is better than the nearest point escape.

However, for the other angles in between those two, calculation of maiden's trip is a bit more messy.

I think this problem still has a suspense left and warrants further investigation.


Edited by Prime, 22 February 2013 - 06:14 AM.

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Past prime, actually.


#33 bonanova

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Posted 22 February 2013 - 07:17 AM

...

Edit:

We can't continue indefinitely increasing the angle however.

At some point it will become to the ogre's advantage to reverse his direction.

For a straight line escape, it seems clear that the optimal angle is between the zero (straight line to the nearest point on the shore) and the tangential line escape. Past the tangential escape, maiden re-enters her inner circle going towards the Ogre causing him to change the direction.

You have calculated the precise rate O:M rate (f) for the tangential escape. And it is better than the nearest point escape.

However, for the other angles in between those two, calculation of maiden's trip is a bit more messy.

I think this problem still has a suspense left and warrants further investigation.

 

If re entering the small circle is the criterion that gives advantage to ogre to change his direction then I think we know the optimal straight-line escape path.

 

Reason: I postulate the advantage increases initially with angle from radial, and then does one of two things:

  1. It increases monotonically with angle from radial to tangential.

    If so, we're done. A very quick and dirty analysis shows the advantage is still increasing with angle at tangential.

     
  2. It reaches a maximum at some angle between radial and tangential.

    If this is true, the advantage would be decreasing with angle at tangential. But analysis says it's increasing. It seems clear from the diagram that every degree of angle added to the escape path adds proportionately to the ogre's path. While the rower's path increases to a much lesser degree. In particular, the 45 degree case was analyzed rigorously and shows an advantage intermediate to those of radial and tangential. Only when the ogre can advantageously change direction would further increase in angle be ill advised.

It would be interesting to show that going beyond tangential incents ogre to reverse direction. Rower would then reverse as well, putting us in zig zag mode.


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#34 Prime

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Posted 22 February 2013 - 09:19 AM

Then the problem is solved tor a straght line escape and it is the tangential line off the inner circle's perimeter. The new problem becomes how to feed the Ogre. Perhaps, if we convinced the maiden that there is such a thing as zig-zag mode, the Ogre could get his dinner after all.

Spoiler for the formula

 

Although, the straight line escape, most likely is not the optimum. The optimum must be some curve dictated by Ogre's movement. To prove (or disprove) that we could let her row half way along the tangent line and then figure out new optimal path from that position.


Edited by Prime, 22 February 2013 - 09:23 AM.

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Past prime, actually.


#35 CaptainEd

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Posted 22 February 2013 - 05:22 PM

Spoiler for zig-zag


 


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#36 bonanova

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Posted 23 February 2013 - 09:35 AM

Spoiler for After some more thought


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#37 Prime

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Posted 24 February 2013 - 12:51 AM

Spoiler for After some more thought

 

It seems, you have proven that the tangential straight line escape is the optimal path. I'll try to put put a simpler perspective on this whole Ogre-Maiden business using BMAD's “Dog Freedom” recent problem in this forum.

 

Spoiler for the leash

 

It appears, there is no spiral escape. But don't tell the maiden.


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Past prime, actually.


#38 bonanova

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Posted 24 February 2013 - 05:34 AM


Spoiler for After some more thought
 It seems, you have proven that the tangential straight line escape is the optimal path. I'll try to put put a simpler perspective on this whole Ogre-Maiden business using BMAD's Dog Freedom recent problem in this forum. 
Spoiler for the leash
 It appears, there is no spiral escape. But don't tell the maiden.
I'm still surprised...

Differential analysis at the tangential landing point should be able to confirm or deny it. That is, if the landing point is moved a very small amount clockwise. There might then be a tractable expression for differential change in speed ratio. First-order perturbation theory it's called. Like using only the first term in a Taylor series expansion of the various quantities. The same idea as L'Hopital's rule. As a last gasp I might look at it, but I feel a bit burned out.
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#39 Prime

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Posted 24 February 2013 - 08:32 AM

Spoiler for logical proof

 

This puzzle bears some logical inferences, which allow to dispense with extra complex math. Just what we all like in puzzles.


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Past prime, actually.


#40 bonanova

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Posted 24 February 2013 - 08:54 AM

I agree with that logic.

 

I've now also verified with differential analysis that for angles greater than a the rower's path increases more than the ogre's path.

Details in next post.


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