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        riddles >> medium >> Reach the shore
        
(Message started by: tseuG on Jul 25^th, 2007, 11:21am)

Title: Reach the shore
Post by tseuG on Jul 25^th, 2007, 11:21am

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?

Title: Re: Reach the shore
Post by SMQ on Jul 25^th, 2007, 12:01pm

We've seen several similar problems before (for example, this one (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1144965993) and this one (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_easy;action=display;num=1143054255) [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:

[hide]1. walk 2/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 miles in any direction
2. turn 120^o and walk 1/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 miles
3. continue in an arc of radius 1 mi around your starting point for 30^o
4. walk straight 1 mile. (at an angle of 150^o from the direction you started walking)[/hide]

Wihch has a worst-case of walking [hide]http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/6 + 1 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/approx.gif 3.26[/hide] miles.

--SMQ

Title: Re: Reach the shore
Post by towr on Jul 25^th, 2007, 12:27pm

Must be a shallow river if you're walking ;)

Title: Re: Reach the shore
Post by Sameer on Jul 25^th, 2007, 12:29pm

So would an infinitely long river have still or very fast moving water?

Title: Re: Reach the shore
Post by mikedagr8 on Jul 26^th, 2007, 4:04am

very fast, otherwise it would be an infinite body of water, so a lake?

Title: Re: Reach the shore
Post by tseuG on Jul 26^th, 2007, 10:25am

Hi SMQ (or anyone else),

Could you explain semi-intuitively why that works?

Title: Re: Reach the shore
Post by SMQ on Jul 26^th, 2007, 11:49am

Sure. Start off with an "obvious" answer: walk/swim/row/fly a mile in any direction, then turn 90^o 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 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif + 1 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/approx.gif 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 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/2 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/approx.gif 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 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/2 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/approx.gif 2.57) with two sides of an equilateral triangle (length 4/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/approx.gif 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/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/approx.gif 0.58) with a little part of the original semicircle (length http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/6 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/approx.gif 0.52). What you're left with is the shortest path which can be rotated any way and still touch at least one edge.

--SMQ

Title: Re: Reach the shore
Post by mikedagr8 on Jul 27^th, 2007, 2:12am

i would have thought 1 mile any direction, the a 90 degree turn and walk another <2 miles would be the worst?

Title: Re: Reach the shore
Post by pex on Jul 27^th, 2007, 2:29am

on 07/27/07 at 02:12:59, mikedagr8 wrote:

i would have thought 1 mile any direction, the a 90 degree turn and walk another <2 miles would be the worst?

Something like this, you mean? ;)

Title: Re: Reach the shore
Post by mikedagr8 on Jul 27^th, 2007, 2:34am

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

Title: Re: Reach the shore
Post by pex on Jul 27^th, 2007, 2:47am

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 90^o turn, you move almost parallel to the shore - so it will take very long to reach the other shore.

Title: Re: Reach the shore
Post by mikedagr8 on Jul 27^th, 2007, 2:49am

fine, 1.1 miles :P

Fair enough, i guess not

Title: Re: Reach the shore
Post by towr on Jul 27^th, 2007, 2:55am

on 07/27/07 at 02:34:31, 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?

Maybe there's a very thick fog over the river, and you can't see more than two feet in any direction.

Title: Re: Reach the shore
Post by mikedagr8 on Jul 27^th, 2007, 2:57am

In that case, stay where you are, and wait for the fog to clear ;)

Title: Re: Reach the shore
Post by Grimbal on Jul 27^th, 2007, 9:01am

Maybe it is night. With no moon. And the borders are steep. In fact, vertical.

Title: Re: Reach the shore
Post by mikedagr8 on Jul 27^th, 2007, 5:30pm

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.

Title: Re: Reach the shore
Post by username101 on Aug 14^th, 2007, 12:16pm

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 ;D

Title: Re: Reach the shore
Post by Aurora on Aug 15^th, 2007, 8:19am

*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. ::)

Title: Re: Reach the shore
Post by Grimbal on Aug 15^th, 2007, 9:09am

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.

Title: Re: Reach the shore
Post by Obob on Aug 15^th, 2007, 12:09pm

Plus, your captors took your eyes.

Title: Re: Reach the shore
Post by Grimbal on Aug 15^th, 2007, 1:47pm

How cruel you are! No, it is just night and foggy.

Title: Re: Reach the shore
Post by mikedagr8 on Aug 15^th, 2007, 4:56pm

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!!!!!!!!!!!!!

Title: Re: Reach the shore
Post by Zatanna on Aug 19^th, 2007, 12:57pm

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.

Title: Re: Reach the shore
Post by Grimbal on Aug 19^th, 2007, 2:20pm

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.

Title: Re: Reach the shore
Post by JiNbOtAk on Aug 19^th, 2007, 6:07pm

on 08/19/07 at 14:20:03, Grimbal wrote:

In fact, if you consider the water as your fixed reference point, the only thing that moves is that the shore.

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 ? ;D

Title: Re: Reach the shore
Post by Zatanna on Aug 26^th, 2007, 12:42pm

on 08/19/07 at 14:20:03, Grimbal wrote:

No, you're wrong. The boat will not move EXACTLY as fast as the waves. Throw a stick into a stream and you'll see what I mean.

Title: Re: Reach the shore
Post by towr on Aug 26^th, 2007, 1:13pm

on 08/26/07 at 12:42:50, Zatanna wrote:

No, you're wrong. The boat will not move EXACTLY as fast as the waves. Throw a stick into a stream and you'll see what I mean.

What boat and what waves?
Heck, if there are waves they may even be moving upstream; it has practically nothing to do with the flow of the river. Of course, the flow of the river is not uniform either, it will move faster near the middle, probably slower near the bottom etc.

Title: Re: Reach the shore
Post by Grimbal on Aug 26^th, 2007, 3:14pm

In fact the water can carry you at any speed upstream and downstream, non-uniformly, or even in a time-dependent way because of some tides, as long as the movements are parallel to the sides, (and the sides are straight, parallel and infinite in length) it won't change anything to you. Provided you always know which direction is north.