The Maiden and the Ogre

[bonanova]

If you think you've heard this one before, read it carefully. It's not the standard puzzle.

A beautiful maiden sits in a boat at the center of circular lake. On shore waits an ogre anxious to have his way with her. Being an excellent sprinter she knows she can outrun and therefore escape the lumbering ogre if only she can land her boat safely. But should the ogre reach her landing point first, alas, all will be lost.

The boat is propelled by a motor capable of only a fraction f of the ogre's speed.

What is the minimum value of f that will permit the Maiden to escape?

 

 

 

[EventHorizon]

Spoiler

I have the minimum at approximately 0.2172336282.

That value is cos x, where x solves tan x = pi + x.  pi+x being how far the ogre will need to run around the lake of radius 1, and tan x = sin x / cos x = distance along the tangent line of the inner circle of radius  f  to the edge of the lake divided by  f.

As for the distance addition... it sounded more interesting when I wrote it.  It may be more interesting starting from the center of the lake (trade-off between movement inside circle of radius  f  and outside it), but still not that interesting.  Though it would be interesting to see the minimum length path for all  f.  As it approaches the minimum, you'd see more and more spiraling in the circle of radius f.

 

[EventHorizon]

I'd say that the minimum value of  f  is slightly less than...

Spoiler

1/(pi+1)

I'll play around with it a bit to see how much I can lower it.  Another interesting addition might be, once the minimum  f  is found, to find the minimum travel distance (e.g., amount of gas) needed for some  f  a little higher than the minimum.

[bonanova]

15 hours ago, EventHorizon said:

Another interesting addition might be, once the minimum  f  is found, to find the minimum travel distance (e.g., amount of gas) needed for some  f  a little higher than the minimum.

Interesting idea. How would we pose that question exactly?

Spoiler

Say the lake has radius of 1.
The boat's starting point P₀ is on a shared diameter, 1 + f  from the ogre and 1 - f  from shore.
The shortest travel distance is the length of a straight line from P₀ to the landing point P₁.
Among the landing points that make sense, one minimizes f , as asked by OP; another minimizes travel distance, namely 1 - f .
What if, between those landing points, both f  and the distance | P_{0 -} P₁ | change monotonically?

.

[The Lonewolf Brand]

The minimum value of f  is 1.0.

[bonanova]

On 8/27/2019 at 1:38 AM, EventHorizon said:

  Reveal hidden contents

I have the minimum at approximately 0.2172336282.

That value is cos x, where x solves tan x = pi + x.  pi+x being how far the ogre will need to run around the lake of radius 1, and tan x = sin x / cos x = distance along the tangent line of the inner circle of radius  f  to the edge of the lake divided by  f.

As for the distance addition... it sounded more interesting when I wrote it.  It may be more interesting starting from the center of the lake (trade-off between movement inside circle of radius  f  and outside it), but still not that interesting.  Though it would be interesting to see the minimum length path for all  f.  As it approaches the minimum, you'd see more and more spiraling in the circle of radius f.

 

Nicely done. 

As for adding some wrinkle to the problem, how about this?

Suppose we have to add some time for her to get out of the boat before she starts to run. We could say the ogre must have an angular separation of s radians from the boat when it lands, and then minimize f. Hmmm. I'm guessing the same path just minimizes f to a slightly larger value.

On 8/28/2019 at 6:23 AM, The Lonewolf Brand said:

The minimum value of f  is 1.0.

Hi @The Lonewolf Brand and welcome to the Den.

That would mean the maiden's boat and the ogre have the same speed. She can escape from a faster ogre.