Author Topic: Kong Lake  (Read 796 times)

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Offline krehztim

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Re: Kong Lake
« Reply #15 on: July 12, 2017, 03:20:50 PM »
Before I delve more deeply - do we need to involve polar coordinates or not?
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Offline Weehawk

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Re: Kong Lake
« Reply #16 on: July 12, 2017, 03:46:08 PM »
Before I delve more deeply - do we need to involve polar coordinates or not?

No. But feel free.

Offline Weehawk

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Re: Kong Lake
« Reply #17 on: July 12, 2017, 03:55:14 PM »
The path is not going to be an infinite spiral but that's sort of on the right track.  The problem with that is that as Mario meanders close to the shore Kong will be able to just stroll around on the shore and Mario will never be able to safely disembark.

I believe the correct path will be an arc such that at every moment in time Mario is travelling in a direction that is directly opposite from Kong's position (instantaneously -- feels like calculus to me!).  So, if Kong starts out deciding to run in a clockwise direction from 12 o clock to 1 o clock, then by the time he reaches 1 o clock Mario has curved his downward path, starting out by rowing towards 6 o clock but gradually curving his path so that he is now pointed in the same direction as a vector from the center point to 7 o clock would point.  This makes it so that at every moment in time, Kong is indifferent about whether or not to continue running clockwise or counterclockwise, and if he switches then Mario just switches the direction of his arc.  By the time Mario reaches the coast, he must be travelling fast enough so that this strategy still yields an arc that is outward as compared to an orbital path.  (My hunch is that Mario could use this strategy but travel so slowly that Kong could simply run around in circles and Mario would never make outward progress beyond a certain radius).

As for calculating the actual number . . . that just hurts my brain!   <confused>

Instant Edit:  I think the above explanation is incorrect but I left it there to show the idea of what we are trying to achieve (Kong's indifference).  We don't travel in the opposite "direction" from Kong (the opposite of his vector from the circle's center point to his position).  Instead, we constantly travel in a direction towards the POSITION that is the opposite of Kong's POSITION.  So, if Kong has travelled to the 1 o clock position, Mario has now curved his arc so that he is now pointed towards 7 o clock!  Note that this direction of travel is NOT the opposite of the direction of the vector from the circle's center to Kong's position since Mario is no longer at the circle's center, nor is he positioned anywhere along the line from the circle's center to 7 o clock (he is on a curve towards that position -- the shape of the curve is NOT an arc of a circle . . . it's curvature is instantaneously becoming more -- "curvey?" as Mario travels farther and farther from the circle's center.

Calculating the number still hurts my brain!   <confused>

Dean makes a couple of key points here, but I won't specify.

Also, I should point out that "my" answer establishes the minimum speed that Mario needs to row to escape regardless of what strategy Kong uses, and the mathematics involved are not that complicated. An 11th grader that has had a year of Geometry and a year or two of Algebra should be able to handle it.

There is however a solution/strategy (from an eminently credible source) that establishes a lower speed which will enable Mario to escape, but it makes certain assumptions about Kong's choices that I don't completely understand, and the math involved is much more complicated.

I will accept either solution.

Offline krehztim

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Re: Kong Lake
« Reply #18 on: July 12, 2017, 07:07:01 PM »
My last go at this one:

You can spiral outward and keep Kong directly opposite you until you are radius times Mario velocity divided by ten mph from center and then can go straight to coast at shorter distance (r- rv/10) and Mario only needs to go 2.4145 mph. You can go slower if DK doesn't have godlike reflexes but this is minimum regardless of strategy I think.
« Last Edit: July 12, 2017, 07:11:35 PM by krehztim »
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Offline Weehawk

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Re: Kong Lake
« Reply #19 on: July 12, 2017, 07:51:00 PM »
My last go at this one:

You can spiral outward and keep Kong directly opposite you until you are radius times Mario velocity divided by ten mph from center and then can go straight to coast at shorter distance (r- rv/10) and Mario only needs to go 2.4145 mph. You can go slower if DK doesn't have godlike reflexes but this is minimum regardless of strategy I think.

And that's how simple it is.

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If I can make it a little clearer: Imagine that Kong runs 4 times as fast as Mario rows. Mario could row in a circle around the center of the pond with radius R/4 (where R is the radius of the pond) all day, keeping the center directly between himself and Kong, because the circumference of his circle is exactly 1/4 that of the pond. (The circumference of a circle is 2*Pi*R, directly proportional to the radius). Mario can make smaller circles keeping Kong opposite the center even easier, that is, without rowing as fast. Therefore Mario can row in spiral fashion out to a distance of R times (1/N), if Kong runs N times as fast as Mario rows. He can't do that any farther out because Kong would be too fast. But this allows Mario to place himself at leisure 1/Nth of the way to the opposite shore from where Kong currently is. This means Mario must travel R times ((N-1)/N)) in the time that Kong makes the half circle around the pond, distance Pi times R. Doing the algebra reveals that Mario can make it unless Kong runs (Pi + 1) times as fast as Mario. It was given that Kong runs 10 miles per hour, so Mario must row (10/4.14159) or ~2.41 miles per hour.

As I stated in a previous post, this allows Mario to escape no matter what Kong's strategy is.

Later I will post a link to analysis which assumes Kong's strategy and takes advantage of it. It's pretty deep.

Offline jwade614

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Re: Kong Lake
« Reply #20 on: July 12, 2017, 07:59:39 PM »
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Offline danman123456

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Re: Kong Lake
« Reply #21 on: July 13, 2017, 05:35:44 AM »
I looked at what jwade posted. It says

"The maximum value for K is 4.6033388489

That's right, we can outrun a monster that can run just over 4.6x the speed we can row!"

So using 4.6 that means you only need to row 2.173913043478261 + .000000000000001 = 2.173913043478262 MPH :D

I was close haha.

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