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Topic: Nowhere To Run, Nowhere To Hide (Read 5977 times) |
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ThudnBlunder
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Nowhere To Run, Nowhere To Hide
« on: Dec 28th, 2010, 3:24am » |
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A hunter is searching for an undesirable entity on the surface of a planet of unit radius, say. In order to remain alive the entity must stay out of sight of the sure-shot hunter. Assuming the entity has a maximum angular velocity of one radian per unit time, at least how many times faster (and at what height) must the hunter travel to be sure of it becoming toast?
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| « Last Edit: Jan 2nd, 2011, 2:25am by ThudnBlunder » |
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towr
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Re: Nowhere To Run, Nowhere To Hide
« Reply #1 on: Dec 29th, 2010, 11:55am » |
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At first glance it seems to me that being ever so slightly faster should suffice, regardless of height.
But then it comes to mind the hunter isn't clairvoyant.
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SMQ
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Re: Nowhere To Run, Nowhere To Hide
« Reply #2 on: Dec 29th, 2010, 12:53pm » |
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If I'm visualizing this right, in the worst case, the hunter needs to be able to circumnavigate the globe faster than it takes the prey to travel the diameter of the hunter's circle of vision, otherwise an omniscient (or just extremely lucky) prey could manage to stay just out of sight. Thus a hunter of height h needs to cover 2 radians faster than it takes the prey to travel 2tan-1 h radians, for a proportional speed strictly greater than /tan-1 h, which would be 2:1 in the limiting case of an infinitely tall hunter that can see half the globe.
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alien2
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Re: Nowhere To Run, Nowhere To Hide
« Reply #3 on: Dec 29th, 2010, 2:29pm » |
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If the entity is mortal, the hunter is sure that it’ll become dead meat eventually, regardless of the outcome of the hunt.
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ThudnBlunder
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Re: Nowhere To Run, Nowhere To Hide
« Reply #4 on: Dec 30th, 2010, 2:29am » |
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on Dec 29th, 2010, 12:53pm, SMQ wrote:| If I'm visualizing this right, in the worst case, the hunter needs to be able to circumnavigate the globe faster than it takes the prey to travel the diameter of the hunter's circle of vision, otherwise an omniscient (or just extremely lucky) prey could manage to stay just out of sight. |
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That is the concept I have in mind, but my formulae and ratio are different. Could you please post a simple diagram?
on Dec 29th, 2010, 2:29pm, alien2 wrote:| If the entity is mortal, the hunter is sure that it’ll become dead meat eventually, regardless of the outcome of the hunt. |
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This entity must be specially fragged so that it does not respawn.
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SMQ
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Re: Nowhere To Run, Nowhere To Hide
« Reply #5 on: Dec 30th, 2010, 10:50am » |
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on Dec 30th, 2010, 2:29am, ThudnBlunder wrote:| Could you please post a simple diagram? |
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Not easily, but I can explain my reasoning:| hidden: | I envision the sphere as consisting of two regions: the area where the prey might be and the area where the prey is known not to be. (In my head they're yellow and black respectively.) As the hunter moves, he "drags" a circle of the radius of his vision around with him, expanding the black area as he goes. However, at the same time, outside of his current vision the yellow area encroaches on the black area at the speed of the prey's movement. By walking the perimeter of the black (excluded) area, the hunter can exclude more area, and so long as he is always able to expand the black faster than the yellow encroaches he can eventually exclude the entire planet and so capture the prey.
With that visualization, it seems to me that the worst case would be when the two regions are hemispheres. At that point if the hunter circles the perimeter in the time the prey moves one diameter of his vision he can just barely maintain the excluded region--any slower and there's a crack for the prey to slip through.
I realize that's hardly mathematically rigorous--although by calculating the rates of change of the respective areas it could perhaps be made so--but I hope it explains my thinking. |
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ThudnBlunder
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Re: Nowhere To Run, Nowhere To Hide
« Reply #6 on: Dec 31st, 2010, 12:07pm » |
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on Dec 30th, 2010, 10:50am, SMQ wrote:
I realize that's hardly mathematically rigorous--although by calculating the rates of change of the respective areas it could perhaps be made so--but I hope it explains my thinking.
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Yes, that's also what I was thinking.
If you have a pen and paper to hand, draw a circle of unit radius with centre O.
Now from O draw a line vertically upwards of length roughly 3/2. Call the end of this line point A.
Now draw a tangent to the circle from A and call this point B.
Now draw the other tangent to the circle from A and call this point C.
Let angle AOB = 
Let height of hunter above ground = h
Let circular velocity of hunter = v units/unit time
Let angular velocity of entity = 1 rad/unit time WLOG
So circular velocity of entity = 1 unit/unit time, as radius = 1
Then to escape hunter the entity must travel between B and C, ie. distance 2, before hunter can travel 2(1 + h), where cos = 1/1+h
This leads to the inequality v > (1 + h)/sec-1(1 + h), which I think has a minimum of 5.599... when h = 0.5333...
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| « Last Edit: Jan 12th, 2011, 4:01am by ThudnBlunder » |
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SMQ
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Re: Nowhere To Run, Nowhere To Hide
« Reply #7 on: Dec 31st, 2010, 12:51pm » |
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Ahh, I see the difference: I interpreted the problem statement to say that both the hunter and prey are "on the surface of a planet" and so was using the hunter's ground speed in my calculation. The interpretation where the hunter is flying is probably more realistic, though, as my version would call for a really small planet, or a really tall or really fast hunter...
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ThudnBlunder
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Re: Nowhere To Run, Nowhere To Hide
« Reply #8 on: Jan 1st, 2011, 2:31am » |
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When I asked about height previously I wasn't enquiring about shooting from the hip. 
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| « Last Edit: Jan 1st, 2011, 5:27am by ThudnBlunder » |
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ThudnBlunder
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Re: Nowhere To Run, Nowhere To Hide
« Reply #9 on: Feb 20th, 2011, 12:46am » |
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You see, you never know when these puzzles might come in useful. For example, it is now easier to calculate how far Gaddafi (and his cronies) can expect to scurry before his ass is reamed and fragged. And then reamed a bit more just for the Hell of it. (Here, a 'bit' equals the half-life of a proton, whatever it may be.) Any search for Gaddafi must leave no stone unturned, as that is probably where he will be hiding.
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| « Last Edit: Mar 14th, 2011, 3:58am by ThudnBlunder » |
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