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Topic: Reach the shore (Read 1295 times) |
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tseuG
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Reach the shore
« on: Jul 25th, 2007, 11:21am » |
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You are in the middle of a 2 mile wide river. The river is a straight-line and infinite. You know that the shore is 1 mile away from you on either side but you can't see and don't know your orientation. What path should you follow to reach the shore in the shortest time?
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SMQ
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Re: Reach the shore
« Reply #1 on: Jul 25th, 2007, 12:01pm » |
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We've seen several similar problems before (for example, this one and this one [edit: actually JocK proposed exactly this situation in the second half of that thread]), but I don't recall this exact variant. Based on the previous solutions, I believe the answer is:
1. walk 2/ 3 miles in any direction
2. turn 120o and walk 1/3 miles
3. continue in an arc of radius 1 mi around your starting point for 30o
4. walk straight 1 mile. (at an angle of 150o from the direction you started walking)
Wihch has a worst-case of walking 3 +  /6 + 1  3.26 miles.
--SMQ
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| « Last Edit: Jul 25th, 2007, 12:22pm by SMQ » |
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Sameer
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Re: Reach the shore
« Reply #3 on: Jul 25th, 2007, 12:29pm » |
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So would an infinitely long river have still or very fast moving water?
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"Obvious" is the most dangerous word in mathematics.
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mikedagr8
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Re: Reach the shore
« Reply #4 on: Jul 26th, 2007, 4:04am » |
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very fast, otherwise it would be an infinite body of water, so a lake?
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tseuG
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Re: Reach the shore
« Reply #5 on: Jul 26th, 2007, 10:25am » |
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Hi SMQ (or anyone else),
Could you explain semi-intuitively why that works?
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SMQ
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Re: Reach the shore
« Reply #6 on: Jul 26th, 2007, 11:49am » |
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Sure. Start off with an "obvious" answer: walk/swim/row/fly a mile in any direction, then turn 90o and "go" in a circle of radius 1 mile around the point you started. At the very worst, you just missed the shore when you turned, and will have to go half way around the circle before hitting the other shore.
If you draw two parallel lines representing the two banks of the river, and then draw the figure I just described (a radius connected to a semicircle of radius 1 mile) in the middle you'll see that it has an important property for this riddle: no matter what way you turn it, at least one point is always touching one of the edges of the river. So we know that's one solution, but + 1 4.14 miles is a long way to swim.
From there, we start optimizing. First, notice that we can straighten-out the last part and still be sure of hitting an edge, so replace the last quarter of the circle (length /2 1.57) with a straight tangent line (length 1). Next, notice that we can turn more curves into straight lines, and replace the first straight line and quarter-circle (length 1 + /2 2.57) with two sides of an equilateral triangle (length 4/3 2.31). Finally, notice that rather a lot of that equilateral triangle is outside of our original circle (and therefore potentially wasted), and take back a little of the last change, replacing the last half of the second side of the equilateral triangle (length 1/3 0.58) with a little part of the original semicircle (length /6 0.52). What you're left with is the shortest path which can be rotated any way and still touch at least one edge.
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mikedagr8
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Re: Reach the shore
« Reply #7 on: Jul 27th, 2007, 2:12am » |
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i would have thought 1 mile any direction, the a 90 degree turn and walk another <2 miles would be the worst?
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pex
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on Jul 27th, 2007, 2:12am, mikedagr8 wrote:| i would have thought 1 mile any direction, the a 90 degree turn and walk another <2 miles would be the worst? |
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Something like this, you mean? 
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mikedagr8
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Re: Reach the shore
« Reply #9 on: Jul 27th, 2007, 2:34am » |
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Yes something like that. Well it doesn't look like i always make the bed, so instead of <2 miles, make it the remaining distance to shore. Also I am assuming that you can see the banks, otherwise, it is silly, like if you miss the bank by 50 feet, why not just make the extra 50 feet?
P.S. Nice diagrams, very handy pex
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| « Last Edit: Jul 27th, 2007, 2:40am by mikedagr8 » |
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pex
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Re: Reach the shore
« Reply #10 on: Jul 27th, 2007, 2:47am » |
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I drew it so you could see it wouldn't work. In principle, the distance you cover can be arbitrarily long using your method: suppose that, after the first mile, you're very close to the shore; just too far to see/feel/note it for whatever reason. Then, after a 90o turn, you move almost parallel to the shore - so it will take very long to reach the other shore.
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mikedagr8
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Re: Reach the shore
« Reply #11 on: Jul 27th, 2007, 2:49am » |
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fine, 1.1 miles 
Fair enough, i guess not
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towr
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Re: Reach the shore
« Reply #12 on: Jul 27th, 2007, 2:55am » |
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on Jul 27th, 2007, 2:34am, mikedagr8 wrote:| Also I am assuming that you can see the banks, otherwise, it is silly, like if you miss the bank by 50 feet, why not just make the extra 50 feet? |
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Maybe there's a very thick fog over the river, and you can't see more than two feet in any direction.
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mikedagr8
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Re: Reach the shore
« Reply #13 on: Jul 27th, 2007, 2:57am » |
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In that case, stay where you are, and wait for the fog to clear
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Grimbal
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Re: Reach the shore
« Reply #14 on: Jul 27th, 2007, 9:01am » |
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Maybe it is night. With no moon. And the borders are steep. In fact, vertical.
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mikedagr8
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Re: Reach the shore
« Reply #15 on: Jul 27th, 2007, 5:30pm » |
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So wait until day time, and assuming you can touch the floor of the "river", throw something aas far as you can infront of you so that you have an idea what is a head of you.
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username101
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Re: Reach the shore
« Reply #16 on: Aug 14th, 2007, 12:16pm » |
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well if it was a superhuman it would fly up in the air till it see's each side of the river then simple fly towards land 
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Aurora
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Re: Reach the shore
« Reply #17 on: Aug 15th, 2007, 8:19am » |
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*Wondering what on earth someone would be doing in the middle of a 2 mile wide river* 
Seeing as it's such a strange situaton to be in, I'd suggest that you go back to the shore by whatever means you got there. 
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Grimbal
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Re: Reach the shore
« Reply #18 on: Aug 15th, 2007, 9:09am » |
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Maybe you have been a bad person and as a punishment, you have been dumped in the middle of the river by helicopter. You just managed to undo the ropes, get out of the bag and to break off the bucket of concrete that has been poured around your feet. Now you have to go back to a shore. Fast, because of the crocodiles.
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| « Last Edit: Aug 16th, 2007, 12:58am by Grimbal » |
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Obob
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Re: Reach the shore
« Reply #19 on: Aug 15th, 2007, 12:09pm » |
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Plus, your captors took your eyes.
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Grimbal
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Re: Reach the shore
« Reply #20 on: Aug 15th, 2007, 1:47pm » |
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How cruel you are! No, it is just night and foggy.
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mikedagr8
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Re: Reach the shore
« Reply #21 on: Aug 15th, 2007, 4:56pm » |
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If they dropped you here by helicopter, and literally dropped you, you'd be injured, and depending on the height you were dropped and where you landed depends on the injury. That being said, you wouldn't travel as fast. Also instead of crocodiles/alligators who only attack blood essentially or when they get annoyed or feel threatened why not SCUBA equipped sheep who only eat humans?
P.S. Black Sheep comes out today in Australia WOOOOOOOOOOOOOOOOOOOOOOOH!!!!!!!!!!!!!
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Zatanna
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Re: Reach the shore
« Reply #22 on: Aug 19th, 2007, 12:57pm » |
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The "river" part ruins the riddle because you have a refrence point. I would row slightly against the current (no matter how slow) to reach the other side in the shortest time.
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Grimbal
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Re: Reach the shore
« Reply #23 on: Aug 19th, 2007, 2:20pm » |
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If you are in the river and you cannot see the shore, you cannot tell which way the current goes.
In fact, if you consider the water as your fixed reference point, the only thing that moves is that the shore. And that shouldn't matter.
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JiNbOtAk
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Re: Reach the shore
« Reply #24 on: Aug 19th, 2007, 6:07pm » |
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on Aug 19th, 2007, 2:20pm, Grimbal wrote:| In fact, if you consider the water as your fixed reference point, the only thing that moves is that the shore. |
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Better yet, consider the water as your fixed reference, and assume you are the one that's moving. How long for you to reach the banks then ?
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